83:
366:
sampling mining galleries crossing either the host rock and the mineralized veins, the distribution of geochemical variables would be bimodal. Bimodal distributions are also seen in traffic analysis, where traffic peaks in during the AM rush hour and then again in the PM rush hour. This phenomenon is also seen in daily water distribution, as water demand, in the form of showers, cooking, and toilet use, generally peak in the morning and evening periods.
9157:
61:
72:
9167:
5373:
4970:
38:
2601:
Although several have been suggested, there is no presently generally agreed summary statistic (or set of statistics) to quantify the parameters of a general bimodal distribution. For a mixture of two normal distributions the means and standard deviations along with the mixing parameter (the weight
641:
Bimodal distributions have the peculiar property that – unlike the unimodal distributions – the mean may be a more robust sample estimator than the median. This is clearly the case when the distribution is U-shaped like the arcsine distribution. It may not be true when the distribution has one or
365:
animals that are active both in morning and evening twilight. In fishery science multimodal length distributions reflect the different year classes and can thus be used for age distribution- and growth estimates of the fish population. Sediments are usually distributed in a bimodal fashion. When
633:
A mixture of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their
4555:
In the study of sediments, particle size is frequently bimodal. Empirically, it has been found useful to plot the frequency against the log( size ) of the particles. This usually gives a clear separation of the particles into a bimodal distribution. In geological applications the
5346:
is available for testing for bimodality. This package assumes that the data are distributed as a sum of two normal distributions. If this assumption is not correct the results may not be reliable. It also includes functions for fitting a sum of two normal distributions to the data.
539:
Mixtures with two distinct components need not be bimodal and two component mixtures of unimodal component densities can have more than two modes. There is no immediate connection between the number of components in a mixture and the number of modes of the resulting density.
5278:
Silverman introduced a bootstrap method for the number of modes. The test uses a fixed bandwidth which reduces the power of the test and its interpretability. Under smoothed densities may have an excessive number of modes whose count during bootstrapping is unstable.
2597:
can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value. The standard deviation is also larger than deviation of each normal distribution.
548:
Bimodal distributions, despite their frequent occurrence in data sets, have only rarely been studied. This may be because of the difficulties in estimating their parameters either with frequentist or
Bayesian methods. Among those that have been studied are
1921:
4770:
2237:
5350:
Assuming that the distribution is a mixture of two normal distributions then the expectation-maximization algorithm may be used to determine the parameters. Several programmes are available for this including
Cluster, and the R package nor1mix.
2715:
4778:
5072:
2575:
5270:. The p-values for the dip statistic values range between 0 and 1. P-values less than 0.05 indicate significant multimodality and p-values greater than 0.05 but less than 0.10 suggest multimodality with marginal significance.
4210:
4499:
56:(p.d.f.), which is an equally-weighted average of the bell-shaped p.d.f.s of the two normal distributions. If the weights were not equal, the resulting distribution could still be bimodal but with peaks of different heights.
1996:
5392:
The CumFreqA program for the fitting of composite probability distributions to a data set (X) can divide the set into two parts with a different distribution. The figure shows an example of a double generalized mirrored
4658:
3100:
3309:
2426:
3662:
To use this index, the log of the values are taken. The data is then divided into interval of width Φ whose value is log 2. The width of the peaks are taken to be four times 1/4Φ centered on their maximum values.
4567:
An alternative method is to plot the log of the particle size against the cumulative frequency. This graph will usually consist two reasonably straight lines with a connecting line corresponding to the antimode.
3349:. Values greater than 5/9 may indicate a bimodal or multimodal distribution, though corresponding values can also result for heavily skewed unimodal distributions. The maximum value (1.0) is reached only by a
2333:
3188:
lies between 0 and 1. The logic behind this coefficient is that a bimodal distribution with light tails will have very low kurtosis, an asymmetric character, or both – all of which increase this coefficient.
5200:
Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences. Larkin later introduced a test based on the F test; Benett created one based on
4398:
The authors recommended a cut off value of 1.5 with B being greater than 1.5 for a bimodal distribution and less than 1.5 for a unimodal distribution. No statistical justification for this value was given.
3746:
4355:
2085:
3623:
1712:
It is not uncommon to encounter situations where an investigator believes that the data comes from a mixture of two normal distributions. Because of this, this mixture has been studied in some detail.
5326:
Assuming that the distribution is known to be bimodal or has been shown to be bimodal by one or more of the tests above, it is frequently desirable to fit a curve to the data. This may be difficult.
3163:
2957:
5359:
The mixtools package available for R can test for and estimate the parameters of a number of different distributions. A package for a mixture of two right-tailed gamma distributions is available.
2844:
1737:
A mixture of two approximately equal mass normal distributions has a negative kurtosis since the two modes on either side of the center of mass effectively reduces the tails of the distribution.
3445:
1784:
1669:
5303:
A study of a mixture density of two normal distributions data found that separation into the two normal distributions was difficult unless the means were separated by 4–6 standard deviations.
2094:
743:
2878:
With distributions other than these the data must be divided into 'layers'. Within a layer the responses are either equal or zero. The categories do not have to be contiguous. A value for
1734:
of the combined distribution were derived by
Eisenberger. Necessary and sufficient conditions for a mixture of normal distributions to be bimodal have been identified by Ray and Lindsay.
837:
122:, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal.
3821:
1592:
5116:
Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions. This method required the solution of a ninth order
3529:
4004:
1740:
A mixture of two normal distributions with highly unequal mass has a positive kurtosis since the smaller distribution lengthens the tail of the more dominant normal distribution.
274:
5208:
A method based on the score and Wald tests has been proposed. This method can distinguish between unimodal and bimodal distributions when the underlying distributions are known.
1723:
is bimodal only if their means differ by at least twice the common standard deviation. Estimates of the parameters is simplified if the variances can be assumed to be equal (the
1543:
2456:
314:
distributed random variable is bimodal when the degrees of freedom are more than one. Similarly the reciprocal of a normally distributed variable is also bimodally distributed.
5167:
4090:
534:
5194:. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1.
4666:
1489:
1183:
969:
7815:
3360:
The distribution of this statistic is unknown. It is related to a statistic proposed earlier by
Pearson – the difference between the kurtosis and the square of the skewness (
2616:
494:
4965:{\displaystyle {\mathit {Skew}}={\frac {\phi _{84}+\phi _{16}-2\phi _{50}}{2(\phi _{84}-\phi _{16})}}+{\frac {\phi _{95}+\phi _{5}-2\phi _{50}}{2(\phi _{95}-\phi _{5})}}}
4413:
2451:
1779:
439:
130:
When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the
1930:
4978:
459:
419:
7944:
7581:
2340:
7555:
8427:
7612:
5310:
the Kernel Mean
Matching algorithm is used to decide if a data set belongs to a single normal distribution or to a mixture of two normal distributions.
4105:
2246:
7659:
Young, Derek; Benaglia, Tatiana; Chauveau, Didier; Hunter, David; Elmore, Ryan; Hettmansperger, Thomas; Thomas, Hoben; Xuan, Fengjuan (10 March 2017).
8335:
9122:
4582:
4280:
2030:
1715:
A mixture of two normal distributions has five parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two
8988:
8200:
7959:
7808:
4395:-size is defined as minus one times the log of the data size taken to the base 2. This transformation is commonly used in the study of sediments.
8883:
8647:
3019:
8321:
3198:
8642:
8586:
8246:
7884:
6575:
Scargle, JD (1982). "Studies in astronomical time series analysis. II – Statistical aspects of spectral analysis of unevenly spaced data".
4539:) have applied this quantity more broadly as an index for detecting bimodality, with a small value indicating a more bimodal distribution.
5688:
Proceedings of the 2013 International
Conference on Information, Operations Management and Statistics (ICIOMS2013), Kuala Lumpur, Malaysia
8392:
8928:
8662:
8515:
8190:
7934:
563:
9170:
8387:
3680:
assumes that the distribution is a sum of two normal distributions with equal variances but differing means. It is defined as follows:
9160:
8832:
8808:
7801:
4373:
are the proportion contained in the primary (that with the greater amplitude) and secondary (that with the lesser amplitude) mode and
9029:
8657:
9191:
8906:
8867:
8839:
8813:
8731:
8080:
7828:
7387:
7126:
5945:
5452:
3686:
3342:
2977:
One theoretical problem with this index is that it assumes that the intervals are equally spaced. This may limit its applicability.
401:
distributions (i.e. distributions having only one mode). In other words, the bimodally distributed random variable X is defined as
300:
5741:"Mutational fitness effects in RNA and single-stranded DNA viruses: common patterns revealed by site-directed mutagenesis studies"
9017:
8983:
8849:
8844:
8689:
8497:
8195:
7949:
3566:
8767:
8680:
8652:
8561:
8510:
8484:
8382:
8165:
8130:
2963:
610:
8781:
8698:
8535:
8282:
8160:
8135:
7999:
7994:
7989:
6475:"The bimodality index: a criterion for discovering and ranking bimodal signatures from cancer gene expression profiling data"
6084:"A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU"
3120:
338:
8459:
7969:
7964:
2902:
7507:"Testing for bimodality in frequency distributions of data suggesting polymorphisms of drug metabolism--hypothesis testing"
5680:
2874:
the responses are evenly distributed among two or more contiguous categories, with the other categories with zero responses
9097:
8963:
8671:
8520:
8452:
8437:
8330:
8304:
8236:
8075:
7906:
7891:
4560:
is normally taken to the base 2. The log transformed values are referred to as phi (Φ) units. This system is known as the
2781:
8614:
6710:
Chaudhuri, D; Agrawal, A (2010). "Split-and-merge procedure for image segmentation using bimodality detection approach".
5120:. In a subsequent paper Pearson reported that for any distribution skewness + 1 < kurtosis. Later Pearson showed that
3378:
8993:
8933:
8923:
8540:
8241:
8100:
1600:
119:
53:
8342:
8085:
8014:
3651:
is the logarithm taken to the base 2 of the proportion of the distribution in the i interval. The maximal value of the
8978:
8973:
8918:
8854:
8619:
8397:
8294:
7879:
5362:
Several other packages for R are available to fit mixture models; these include flexmix, mcclust, agrmt, and mixdist.
8798:
8606:
7470:
Bajgier SM; Aggarwal LK (1991). "Powers of goodness-of-fit tests in detecting balanced mixed normal distributions".
5205:. Tokeshi has proposed a fourth test. A test based on a likelihood ratio has been proposed by Holzmann and Vollmer.
656:
9112:
8888:
8707:
8489:
8442:
8311:
8287:
8267:
8110:
7984:
7864:
7588:
1730:
If the means of the two normal distributions are equal, then the combined distribution is unimodal. Conditions for
9117:
8060:
778:
8901:
8862:
8736:
8573:
8417:
8362:
8260:
8224:
8095:
7564:
5901:
5236:
6295:
Ellison, AM (1987). "Effect of seed dimorphism on the density-dependent dynamics of experimental populations of
4547:
A number of tests are available to determine if a data set is distributed in a bimodal (or multimodal) fashion.
3775:
8803:
8591:
8357:
8316:
8231:
8185:
8125:
8090:
7979:
7874:
7824:
7554:
Famoye, Felix; Lee, Carl; Eugene, Nicholas. "Beta-normal distribution: Bimodality properties and application".
5343:
5267:
4536:
3346:
1551:
218:
111:
7916:
5318:
This distribution is bimodal for certain values of is parameters. A test for these values has been described.
3487:
6339:"Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation"
1758:
9102:
9044:
8715:
8502:
8412:
8367:
8352:
8272:
8170:
8120:
8115:
7896:
1916:{\displaystyle \left\vert \log(1-p)-\log(p)\right\vert \geq 2\log(d-{\sqrt {d^{2}-1}})+2d{\sqrt {d^{2}-1}},}
606:
workers arises due to existence of two distinct classes of workers, namely major workers and minor workers.
6808:"Contributions to the mathematical theory of evolution: On the dissection of asymmetrical frequency-curves"
6522:
Sturrock, P (2008). "Analysis of bimodality in histograms formed from GALLEX and GNO solar neutrino data".
5967:"On more robust estimation of skewness and kurtosis: Simulation and application to the S & P 500 index"
3851:
118:(i.e., more than one local peak of the distribution). These appear as distinct peaks (local maxima) in the
8968:
8956:
8945:
8827:
8723:
8530:
7974:
7954:
7859:
7083:
6459:
5187:
4028:. It suffers from the usual problems of estimation and spectral leakage common to this form of statistic.
3350:
230:
7702:
Gruen, Bettina; Leisch, Friedrich; Sarkar, Deepayan; Mortier, Frederic; Picard, Nicolas (28 April 2017).
5248:
5227:
is commonly employed in computer graphics to determine the optimal separation between two distributions.
4765:{\displaystyle {\mathit {StdDev}}={\frac {\phi _{84}-\phi _{16}}{4}}+{\frac {\phi _{95}-\phi _{5}}{6.6}}}
1500:
9092:
9049:
8893:
8568:
8422:
8402:
8299:
7869:
7719:"mclust: Gaussian Mixture Modelling for Model-Based Clustering, Classification, and Density Estimation"
5126:
4046:
2232:{\displaystyle S={\frac {\sqrt {-2+3r+3r^{2}-2r^{3}+2(1-r+r^{2})^{1.5}}}{{\sqrt {r}}(1+{\sqrt {r}})}}.}
507:
346:
82:
7717:
Fraley, Chris; Raftery, Adrian E.; Scrucca, Luca; Murphy, Thomas
Brendan; Fop, Michael (21 May 2017).
7674:
5244:
2710:{\displaystyle D={\frac {\left|\mu _{1}-\mu _{2}\right|}{\sqrt {2(\sigma _{1}^{2}+\sigma _{2}^{2})}}}}
1191:
977:
845:
9142:
9137:
9132:
9127:
9064:
9034:
8913:
8556:
8447:
8347:
8050:
8009:
8004:
7901:
7154:
6819:
6753:
6682:
6623:
6612:"Tree cover bimodality in savannas and forests emerging from the switching between two fire dynamics"
6584:
6541:
6350:
6261:
6178:
5844:
5549:
5545:
5424:
5398:
5191:
3354:
387:
307:
202:
45:
7403:
Ringach, Martin
Maechler (originally from Fortran and S.-plus by Dario; NYU.edu) (5 December 2016).
7121:. Studies in Classification, Data Analysis, and Knowledge Organization. Springer. pp. 169–181.
7088:
9076:
8601:
8581:
8551:
8525:
8479:
8407:
8219:
8155:
5394:
1716:
322:
49:
5788:
Eyre-Walker, A; Keightley, PD (Aug 2007). "The distribution of fitness effects of new mutations".
5235:
To test if a distribution is other than unimodal, several additional tests have been devised: the
3184:. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of
464:
9107:
8596:
8377:
8372:
8277:
8214:
8209:
8065:
8055:
7939:
7487:
7452:
7341:
7273:
7238:
7170:
7015:
6557:
6531:
6368:
6316:
6277:
6229:
6194:
6168:
6141:
6103:
6061:
6043:
5918:
5813:
5661:
5574:
5557:
5516:
5376:
Number of joggers in a park by time of the day (X in hours) in a bimodal probability distribution
5260:
2594:
1720:
635:
571:
6792:
6765:
7762:
5256:
5252:
333:
Examples of variables with bimodal distributions include the time between eruptions of certain
143:
139:
9005:
8432:
8175:
8105:
8070:
8019:
7660:
7606:
7536:
7444:
7383:
7122:
7114:
6916:
6651:
6504:
6428:
5941:
5872:
5805:
5770:
5448:
5240:
2985:
This index assumes that the distribution is a mixture of two normal distributions with means (
2433:
582:
In biology five factors are known to contribute to bimodal distributions of population sizes:
206:
131:
115:
7748:
7703:
7645:
7404:
424:
8180:
7854:
7718:
7526:
7518:
7479:
7434:
7333:
7304:
7265:
7230:
7201:
7162:
7093:
7050:
7042:
7007:
6980:
6947:
6908:
6879:
6827:
6788:
6761:
6719:
6690:
6641:
6631:
6592:
6549:
6494:
6486:
6455:
6418:
6408:
6358:
6308:
6269:
6221:
6186:
6133:
6095:
6053:
6016:
5989:
5910:
5862:
5852:
5797:
5760:
5752:
5721:
5653:
5624:
5566:
5508:
5477:
5224:
5067:{\displaystyle {\mathit {Kurt}}={\frac {\phi _{95}-\phi _{5}}{2.44(\phi _{75}-\phi _{25})}}}
4407:
358:
7733:
7626:
6998:
Tokeshi, M (1992). "Dynamics and distribution in animal communities; theory and analysis".
4410:
for finding a threshold for separation between two modes relies on minimizing the quantity
2570:{\displaystyle |\mu _{1}-\mu _{2}|\leq 2\sigma {\sqrt {1+{\frac {|\log p-\ln(1-p)|}{2}}}}.}
7033:
Barreto, S; Borges, PAV; Guo, Q (2003). "A typing error in
Tokeshi's test of bimodality".
3331:
3323:
2763:
This measure is a weighted average of the degree of agreement the frequency distribution.
5644:
Hassan, M. Y.; El-Bassiouni, M. Y. (2016). "Bimodal skew-symmetric normal distribution".
7158:
7145:
Silverman, B. W. (1981). "Using kernel density estimates to investigate multimodality".
6823:
6757:
6686:
6627:
6588:
6545:
6354:
6265:
6182:
5848:
5533:
2889:) is calculated and a weighted average for the distribution is determined. The weights (
2864:
The value of U is 1 if the distribution has any of the three following characteristics:
2585:
Bimodal distributions are a commonly used example of how summary statistics such as the
71:
8253:
7786:
CumFreq, free program for fitting of probability distributions to a data set. On line:
7531:
7522:
7506:
7166:
6912:
6646:
6611:
6499:
6474:
6423:
6396:
6083:
5896:
5867:
5832:
5765:
5740:
5702:
5413:
1753:
444:
404:
7221:
Mueller, DW; Sawitzki, G (1991). "Excess mass estimates and tests for multimodality".
9185:
8876:
8624:
7911:
7491:
7345:
7277:
7046:
6561:
6233:
6159:
Ashman KM; Bird CM; Zepf SE (1994). "Detecting bimodality in astronomical datasets".
5962:
5665:
5512:
5418:
5287:
Bajgier and
Aggarwal have proposed a test based on the kurtosis of the distribution.
1724:
7456:
7419:
7019:
6936:"An algorithm for assessing bimodality vs. unimodality in a univariate distribution"
6281:
6198:
6107:
6065:
5922:
5817:
5629:
5612:
5578:
5468:
Fieller E (1932). "The distribution of the index in a normal bivariate population".
4036:
Another bimodality index has been proposed by de Michele and Accatino. Their index (
295:
are distributed as normal variables with a mean of 0 and a standard deviation of 1.
17:
7793:
7234:
6984:
6020:
5993:
5980:
Robertson, CA; Fryer, JG (1969). "Some descriptive properties of normal mixtures".
5550:"A remark on bimodality and weak instrumentation in structural equation estimation"
5496:
5381:
5366:
375:
7557:
Joint Statistical Meetings - Section on Physical & Engineering Sciences (SPES)
5520:
4205:{\displaystyle \mu _{M}={\frac {\sum _{i=1}^{L}m_{i}x_{i}}{\sum _{i=1}^{L}m_{i}}}}
60:
6636:
5657:
5197:
Baker proposed a transformation to convert a bimodal to a unimodal distribution.
4576:
Approximate values for several statistics can be derived from the graphic plots.
4494:{\displaystyle {\frac {n_{1}\sigma _{1}^{2}+n_{2}\sigma _{2}^{2}}{m\sigma ^{2}}}}
4260:< 0.1) distribution. No statistical justification was offered for this value.
7378:
Hartigan, J. A. (1988). "The Span Test of Multimodality". In Bock, H. H. (ed.).
6741:
5745:
Philosophical Transactions of the Royal Society of London B: Biological Sciences
5481:
4025:
1752:
When the components of the mixture have equal variances the mixture is unimodal
1731:
362:
342:
6124:
Behboodian, J (1970). "On the modes of a mixture of two normal distributions".
6057:
5837:
Proceedings of the National Academy of Sciences of the United States of America
1991:{\displaystyle d={\frac {\left\vert \mu _{1}-\mu _{2}\right\vert }{2\sigma }},}
1743:
Mixtures of other distributions require additional parameters to be estimated.
625:
being either neutral or lethal with relatively few having intermediate effect.
185:
Under this classification bimodal distributions are classified as type S or U.
7439:
7098:
7071:
6884:
6867:
6553:
6252:
Zhang, C; Mapes, BE; Soden, BJ (2003). "Bimodality in tropical water vapour".
6225:
6099:
6034:
Ray, S; Lindsay, BG (2005). "The topography of multivariate normal mixtures".
5570:
5117:
4653:{\displaystyle {\mathit {Mean}}={\frac {\phi _{16}+\phi _{50}+\phi _{84}}{3}}}
4561:
603:
100:
31:
7483:
7359:
Andrushkiw RI; Klyushin DD; Petunin YI (2008). "A new test for unimodality".
7309:
7292:
7206:
7189:
6413:
193:
Bimodal distributions occur both in mathematics and in the natural sciences.
5966:
5857:
5369:
can also fit a variety of mixed distributions with the PROC FREQ procedure.
5307:
4557:
622:
350:
135:
7448:
6920:
6832:
6807:
6655:
6508:
6432:
6363:
6338:
5876:
5809:
5774:
5756:
5592:
Hassan, MY; Hijazi, RH (2010). "A bimodal exponential power distribution".
95:
distribution, that would become multimodal if conditioned on either x or y.
7540:
5725:
7420:"Assessing bimodality to detect the presence of a dual cognitive process"
6173:
3181:
3173:
3095:{\displaystyle S={\frac {\mu _{1}-\mu _{2}}{2(\sigma _{1}+\sigma _{2})}}}
2772:
1697:
1693:
398:
224:
The ratio of two normal distributions is also bimodally distributed. Let
91:
7787:
7055:
6899:
Haldane, JBS (1951). "Simple tests for bimodality and bitangentiality".
6742:"Brazos River bar: a study in the significance of grain size parameters"
6723:
6385:
SAS Institute Inc. (2012). SAS/STAT 12.1 user’s guide. Cary, NC: Author.
6212:
Van der Eijk, C (2001). "Measuring agreement in ordered rating scales".
5914:
7337:
7269:
7242:
7174:
7011:
6952:
6935:
6320:
6145:
4535:
is the sample variance. Some researchers (particularly in the field of
3304:{\displaystyle b={\frac {g^{2}+1}{k+{\frac {3(n-1)^{2}}{(n-2)(n-3)}}}}}
2896:) for each layer are the number of responses in that layer. In symbols
2768:
2421:{\displaystyle |\mu _{1}-\mu _{2}|\leq 2\min(\sigma _{1},\sigma _{2}).}
7747:
Macdonald, Peter; Du, with contributions from Juan (29 October 2012).
6695:
6670:
6490:
6446:
Wilcock, PR (1993). "The critical shear stress of natural sediments".
6273:
6671:"Measuring and defining bimodal sediments: Problems and implications"
6372:
6048:
5202:
5186:
is the square of the skewness. Equality holds only for the two point
2590:
638:
to produce bimodality when the two distribution curves are combined.
614:
397:
A bimodal distribution commonly arises as a mixture of two different
334:
154:
Galtung introduced a classification system (AJUS) for distributions:
7405:"diptest: Hartigan's Dip Test Statistic for Unimodality - Corrected"
7324:
Hartigan, JA; Mohanty, S (1992). "The RUNT test for multimodality".
6312:
6137:
5801:
2602:
for the combination) are usually used – a total of five parameters.
6596:
6395:
Pfister, R; Schwarz, KA; Janczyk, M.; Dale, R; Freeman, JB (2013).
6190:
5372:
134:. The difference between the major and minor modes is known as the
37:
6536:
5404:
X < 8.10 : CDF = 1 - exp X > 8.10 : CDF = 1 - exp
5371:
2328:{\displaystyle |\mu _{1}-\mu _{2}|<S|\sigma _{1}+\sigma _{2}|.}
592:
the size and time dependence of the growth rate of each individual
81:
70:
59:
52:
with the same variance but different means. The figure shows the
36:
7256:
Rozál, GPM Hartigan JA (1994). "The MAP test for multimodality".
6397:"Good things peak in pairs: A note on the bimodality coefficient"
7563:. American Statistical Society. pp. 951–956. Archived from
3642:
are the amplitudes of the left and right peaks respectively and
2755:> 2 is required for a clean separation of the distributions.
2586:
618:
354:
7797:
7646:"nor1mix: Normal (1-d) Mixture Models (S3 Classes and Methods)"
2430:
If the two normal distributions have equal standard deviations
7072:"One sample tests for the location of modes of nonnormal data"
6473:
Wang, J; Wen, S; Symmans, WF; Pusztai, L; Coombes, KR (2009).
5295:
Additional tests are available for a number of special cases:
3741:{\displaystyle \delta ={\frac {|\mu _{1}-\mu _{2}|}{\sigma }}}
3481:
This is the ratio of the left and right peaks. Mathematically
3473:
is always < 1. Larger values indicate more distinct peaks.
4350:{\displaystyle B=|\phi _{2}-\phi _{1}|{\frac {p_{2}}{p_{1}}}}
3548:
are the amplitudes of the left and right peaks respectively.
2871:
the responses are evenly distributed among all the categories
2080:{\displaystyle r={\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}.}
6779:
Dyer, KR (1970). "Grain-size parameters for sandy gravels".
4993:
4990:
4987:
4984:
4793:
4790:
4787:
4784:
4687:
4684:
4681:
4678:
4675:
4672:
4597:
4594:
4591:
4588:
3838:
A different bimodality index has been proposed by Sturrock.
6007:
Eisenberger, I (1964). "Genesis of bimodal distributions".
5938:
Data Analysis and Regression: A Second Course in Statistics
2857:
the number of categories that have nonzero frequencies and
595:
mortality rates that may affect each size class differently
5497:"Bimodal t-ratios: the impact of thick tails on inference"
3618:{\displaystyle B={\sqrt {\frac {A_{r}}{A_{l}}}}\sum P_{i}}
3353:
with only two distinct values or the sum of two different
6971:
and other pterosaurs, with comments on cranial crests".
6669:
Sambrook Smith, GH; Nicholas, AP; Ferguson, RI (1997).
5899:; Watkins, William (2002). "Is Human Height Bimodal?".
5401:
with cumulative distribution function (CDF) equations:
5266:
An implementation of the dip test is available for the
3158:{\displaystyle \beta ={\frac {\gamma ^{2}+1}{\kappa }}}
30:"Bimodal" redirects here. For the musical concept, see
2952:{\displaystyle A_{\text{overall}}=\sum _{i}w_{i}A_{i}}
589:
the distribution of growth rates among the individuals
171:
This classification has since been modified slightly:
7661:"mixtools: Tools for Analyzing Finite Mixture Models"
6254:
Quarterly Journal of the Royal Meteorological Society
5679:
Bosea, S.; Shmuelib, G.; Sura, P.; Dubey, P. (2013).
5495:
Fiorio, CV; HajivassILiou, VA; Phillips, PCB (2010).
5129:
4981:
4781:
4669:
4585:
4416:
4283:
4108:
4049:
3854:
3778:
3689:
3569:
3490:
3381:
3201:
3123:
3022:
2905:
2784:
2619:
2459:
2436:
2343:
2249:
2097:
2033:
1933:
1787:
1761:
1603:
1554:
1503:
1194:
980:
848:
781:
659:
510:
467:
447:
427:
407:
378:
models, the parameters may be bimodally distributed.
233:
7076:
Journal of Applied Mathematics and Decision Sciences
5681:"Fitting Com-Poisson mixtures to bimodal count data"
2839:{\displaystyle A=U\left(1-{\frac {S-1}{K-1}}\right)}
2243:= 1. The mixture density is unimodal if and only if
9085:
9043:
8944:
8780:
8758:
8749:
8633:
8468:
8144:
8041:
8032:
7925:
7845:
7836:
7380:
Classification and Related Methods of Data Analysis
5831:Hietpas, RT; Jensen, JD; Bolon, DN (May 10, 2011).
5329:Bayesian methods may be useful in difficult cases.
3440:{\displaystyle A_{B}={\frac {A_{1}-A_{an}}{A_{1}}}}
2970:= 0: when all the responses fall into one category
7147:Journal of the Royal Statistical Society, Series B
5833:"Experimental illumination of a fitness landscape"
5161:
5066:
4964:
4764:
4652:
4493:
4349:
4204:
4084:
3998:
3815:
3740:
3617:
3523:
3439:
3303:
3157:
3094:
2951:
2838:
2709:
2569:
2445:
2420:
2327:
2231:
2079:
2012:are the means of the two normal distributions and
1990:
1915:
1773:
1664:{\displaystyle \nu _{r}=\int (x-\mu )^{r}f(x)\,dx}
1663:
1586:
1537:
1483:
1177:
963:
831:
737:
528:
488:
453:
433:
413:
268:
7293:"Nonparametric testing of the existence of modes"
6812:Philosophical Transactions of the Royal Society A
6343:Philosophical Transactions of the Royal Society A
6119:
6117:
5646:Communications in Statistics - Theory and Methods
4248:The authors suggested a cut off value of 0.1 for
621:is also frequently found to be bimodal with most
7140:
7138:
5890:
5888:
5886:
2383:
27:Probability distribution with more than one mode
7223:Journal of the American Statistical Association
6940:Behavior Research Methods & Instrumentation
6077:
6075:
321:statistic generated from data set drawn from a
299:has a known density that can be expressed as a
7382:. Amsterdam: North-Holland. pp. 229–236.
6332:
6330:
5714:Annals of the Entomological Society of America
5534:Introduction to tropical fish stock assessment
5216:Statistical tests for the antimode are known.
4391:-sizes of the primary and secondary mode. The
2610:A statistic that may be useful is Ashman's D:
738:{\displaystyle f(x)=pg_{1}(x)+(1-p)g_{2}(x)\,}
598:the DNA methylation in human and mouse genome.
138:. In time series the major mode is called the
44:A simple bimodal distribution, in this case a
7809:
7749:"mixdist: Finite Mixture Distribution Models"
158:A: unimodal distribution – peak in the middle
8:
6247:
6245:
6243:
832:{\displaystyle \mu =p\mu _{1}+(1-p)\mu _{2}}
586:the initial distribution of individual sizes
201:Important bimodal distributions include the
7505:Jackson, PR; Tucker, GT; Woods, HF (1989).
6793:10.1306/74D71FE6-2B21-11D7-8648000102C1865D
6766:10.1306/74d70646-2b21-11d7-8648000102c1865d
6735:
6733:
6082:Holzmann, Hajo; Vollmer, Sebastian (2008).
559:Bimodal skew-symmetric normal distribution.
8755:
8038:
7842:
7816:
7802:
7794:
6967:Bennett, SC (1992). "Sexual dimorphism of
6868:"Transformations of bimodal distributions"
3816:{\displaystyle BI=\delta {\sqrt {p(1-p)}}}
2751:For a mixture of two normal distributions
7530:
7472:Educational and Psychological Measurement
7438:
7308:
7205:
7097:
7087:
7054:
6951:
6883:
6831:
6694:
6645:
6635:
6535:
6498:
6422:
6412:
6362:
6172:
6047:
5866:
5856:
5764:
5707:worker and an anomaly (Hym.: Formicidae)"
5628:
5147:
5134:
5128:
5052:
5039:
5021:
5008:
5001:
4983:
4982:
4980:
4950:
4937:
4919:
4903:
4890:
4883:
4868:
4855:
4837:
4821:
4808:
4801:
4783:
4782:
4780:
4750:
4737:
4730:
4715:
4702:
4695:
4671:
4670:
4668:
4638:
4625:
4612:
4605:
4587:
4586:
4584:
4482:
4467:
4462:
4452:
4439:
4434:
4424:
4417:
4415:
4339:
4329:
4323:
4318:
4312:
4299:
4290:
4282:
4193:
4183:
4172:
4160:
4150:
4140:
4129:
4122:
4113:
4107:
4099:is the arithmetic mean of the sample and
4077:
4071:
4056:
4048:
3985:
3950:
3945:
3926:
3891:
3886:
3861:
3853:
3791:
3777:
3727:
3721:
3708:
3699:
3696:
3688:
3609:
3593:
3583:
3576:
3568:
3513:
3503:
3497:
3489:
3457:is the amplitude of the smaller peak and
3429:
3415:
3402:
3395:
3386:
3380:
3257:
3235:
3215:
3208:
3200:
3137:
3130:
3122:
3080:
3067:
3049:
3036:
3029:
3021:
2943:
2933:
2923:
2910:
2904:
2805:
2783:
2695:
2690:
2677:
2672:
2650:
2637:
2626:
2618:
2551:
2513:
2510:
2502:
2488:
2482:
2469:
2460:
2458:
2453:a sufficient condition for unimodality is
2435:
2406:
2393:
2372:
2366:
2353:
2344:
2342:
2337:A sufficient condition for unimodality is
2317:
2311:
2298:
2289:
2278:
2272:
2259:
2250:
2248:
2213:
2197:
2189:
2179:
2148:
2132:
2104:
2096:
2066:
2061:
2051:
2046:
2040:
2032:
1964:
1951:
1940:
1932:
1896:
1890:
1864:
1858:
1786:
1760:
1654:
1636:
1608:
1602:
1587:{\displaystyle \delta _{i}=\mu _{i}-\mu }
1572:
1559:
1553:
1528:
1502:
1472:
1467:
1454:
1449:
1439:
1434:
1418:
1413:
1403:
1393:
1377:
1372:
1362:
1328:
1323:
1310:
1305:
1295:
1290:
1274:
1269:
1259:
1249:
1233:
1228:
1218:
1199:
1193:
1166:
1161:
1148:
1143:
1133:
1117:
1112:
1102:
1068:
1063:
1050:
1045:
1035:
1019:
1014:
1004:
985:
979:
952:
947:
934:
929:
895:
890:
877:
872:
853:
847:
823:
795:
780:
734:
719:
682:
658:
509:
466:
446:
426:
406:
240:
232:
167:S: bimodal or multimodal – multiple peaks
7511:British Journal of Clinical Pharmacology
7117:. In Gaul W; Opitz O; Schader M (eds.).
3524:{\displaystyle R={\frac {A_{r}}{A_{l}}}}
2853:is the unimodality of the distribution,
570:Bimodality also naturally arises in the
353:in US adults, the absolute magnitude of
349:, the speed of inactivation of the drug
6460:10.1061/(asce)0733-9429(1993)119:4(491)
5435:
5322:Parameter estimation and fitting curves
3345:is 5/9. This is also its value for the
7611:: CS1 maint: archived copy as title (
7604:
4272:) has been proposed by Sambrook Smith
3676:The bimodality index proposed by Wang
3318:is the number of items in the sample,
2868:all responses are in a single category
566:has been fitted to bimodal count data.
6847:Pearson, K (1929). "Editorial note".
6088:AStA Advances in Statistical Analysis
5445:Theory and methods of social research
5384:contains a tool for mixture modeling
5365:The statistical programming language
3999:{\displaystyle B={\frac {1}{N}}\left}
602:The bimodal distribution of sizes of
217:are less than 1). Others include the
7:
9166:
7704:"flexmix: Flexible Mixture Modeling"
5936:Mosteller, F.; Tukey, J. W. (1977).
4531:is the total size of the sample and
4510:is the number of data points in the
564:Conway-Maxwell-Poisson distributions
269:{\displaystyle R={\frac {a+x}{b+y}}}
78:A bivariate, multimodal distribution
7188:Hartigan, JA; Hartigan, PM (1985).
6610:De Michele, C; Accatino, F (2014).
5617:Proyecciones Journal of Mathematics
5299:Mixture of two normal distributions
3192:The formula for a finite sample is
1708:Mixture of two normal distributions
1538:{\displaystyle \mu =\int xf(x)\,dx}
7644:Mächler, Martin (25 August 2016).
7523:10.1111/j.1365-2125.1989.tb03558.x
7167:10.1111/j.2517-6161.1981.tb01155.x
6973:Journal of Vertebrate Paleontology
6913:10.1111/j.1469-1809.1951.tb02488.x
4252:to distinguish between a bimodal (
3769:is the common standard deviation.
3464:is the amplitude of the antimode.
757:is a probability distribution and
504:are unimodal random variables and
25:
7070:Carolan, AM; Rayner, JCW (2001).
6872:Annals of Mathematical Statistics
5940:. Reading, Mass: Addison-Wesley.
5162:{\displaystyle b_{2}-b_{1}\geq 1}
5112:Unimodal vs. bimodal distribution
4085:{\displaystyle B=|\mu -\mu _{M}|}
2023:= 1/2 was described by Schilling
553:Bimodal exponential distribution.
529:{\displaystyle 0<\alpha <1}
301:confluent hypergeometric function
9165:
9156:
9155:
7047:10.1046/j.1466-822x.2003.00018.x
7000:Researches on Population Ecology
6448:Journal of Hydraulic Engineering
5613:"Alpha-skew-normal distribution"
5513:10.1111/j.1368-423X.2010.00315.x
5108:percentage of the distribution.
4224:is number of data points in the
2861:the total number of categories.
2239:If the variances are equal then
2019:The following test for the case
1484:{\displaystyle \nu _{4}=p+(1-p)}
1178:{\displaystyle \nu _{3}=p+(1-p)}
964:{\displaystyle \nu _{2}=p+(1-p)}
161:J: unimodal – peak at either end
7732:Ruedin, Didier (2 April 2016).
7035:Global Ecology and Biogeography
6781:Journal of Sedimentary Research
6746:Journal of Sedimentary Research
5630:10.4067/s0716-09172010000300006
5421:- Gaussian Mixture Models (GMM)
4032:de Michele and Accatino's index
3110:Sarle's bimodality coefficient
611:distribution of fitness effects
556:Alpha-skew-normal distribution.
164:U: bimodal – peaks at both ends
7361:Theory of Stochastic Processes
7235:10.1080/01621459.1991.10475103
6985:10.1080/02724634.1992.10011472
6021:10.1080/00401706.1964.10490199
5994:10.1080/03461238.1969.10404590
5982:Skandinavisk Aktuarietidskrift
5594:Pakistan Journal of Statistics
5447:. Oslo: Universitetsforlaget.
5058:
5032:
4956:
4930:
4874:
4848:
4319:
4291:
4078:
4057:
4021:is exponentially distributed.
3977:
3962:
3918:
3903:
3808:
3796:
3728:
3700:
3292:
3280:
3277:
3265:
3254:
3241:
3086:
3060:
2701:
2665:
2552:
2548:
2536:
2514:
2489:
2461:
2412:
2386:
2373:
2345:
2318:
2290:
2279:
2251:
2220:
2204:
2186:
2160:
1878:
1849:
1829:
1823:
1811:
1799:
1651:
1645:
1633:
1620:
1525:
1519:
1478:
1355:
1352:
1340:
1334:
1211:
1172:
1095:
1092:
1080:
1074:
997:
958:
922:
919:
907:
901:
865:
816:
804:
731:
725:
712:
700:
694:
688:
669:
663:
480:
468:
149:
1:
7190:"The dip test of unimodality"
2748:are the standard deviations.
572:cusp catastrophe distribution
175:J: (modified) – peak on right
6637:10.1371/journal.pone.0091195
5658:10.1080/03610926.2014.882950
5388:Example software application
5190:or the sum of two different
5100:is the value of the variate
4024:This statistic is a form of
2016:is their standard deviation.
1927:is the mixing parameter and
613:of mutations for both whole
489:{\displaystyle (1-\alpha ),}
120:probability density function
54:probability density function
6740:Folk, RL; Ward, WC (1957).
5739:Sanjuán, R (Jun 27, 2010).
5703:"Dimorphism in the African
5085:is the standard deviation,
3357:(a bi-delta distribution).
2999:) and standard deviations (
359:circadian activity patterns
9208:
8989:Wrapped asymmetric Laplace
7960:Extended negative binomial
6301:American Journal of Botany
6058:10.1214/009053605000000417
4017:is uniformly distributed,
3659:may be greater than this.
536:is a mixture coefficient.
385:
345:, the age of incidence of
178:L: unimodal – peak on left
29:
9151:
8648:Generalized extreme value
8428:Relativistic Breit–Wigner
7825:Probability distributions
7763:"Gaussian mixture models"
7440:10.3758/s13428-012-0225-x
7427:Behavior Research Methods
7326:Journal of Classification
7258:Journal of Classification
7099:10.1155/s1173912601000013
6577:The Astrophysical Journal
6554:10.1007/s11207-008-9170-3
6100:10.1007/s10182-008-0057-2
5902:The American Statistician
5571:10.1017/S0266466606060439
5482:10.1093/biomet/24.3-4.428
3830:is the mixing parameter.
761:is the mixing parameter.
197:Probability distributions
9192:Continuous distributions
7484:10.1177/0013164491512001
7113:Hartigan, J. A. (2000).
6675:Water Resources Research
6414:10.3389/fpsyg.2013.00700
6161:The Astronomical Journal
5611:Elal-Olivero, D (2010).
5501:The Econometrics Journal
5338:Two normal distributions
5314:Beta-normal distribution
4537:digital image processing
3347:exponential distribution
2767:ranges from -1 (perfect
2446:{\displaystyle \sigma ,}
544:Particular distributions
306:The distribution of the
219:U-quadratic distribution
150:Galtung's classification
112:probability distribution
8643:Generalized chi-squared
8587:Normal-inverse Gaussian
7751:– via R-Packages.
7721:– via R-Packages.
7706:– via R-Packages.
7663:– via R-Packages.
7648:– via R-Packages.
7407:– via R-Packages.
7115:"Testing for Antimodes"
6885:10.1214/aoms/1177733063
6712:Defence Science Journal
6401:Frontiers in Psychology
6226:10.1023/a:1010374114305
5858:10.1073/pnas.1016024108
5790:Nature Reviews Genetics
5380:In Python, the package
4523:is the variance of the
4256:> 0.1)and unimodal (
4245:is the number of bins.
2087:The separation factor (
1774:{\displaystyle d\leq 1}
434:{\displaystyle \alpha }
67:A bimodal distribution.
8955:Univariate (circular)
8516:Generalized hyperbolic
7945:Conway–Maxwell–Poisson
7935:Beta negative binomial
7631:engineering.purdue.edu
7418:Freeman; Dale (2012).
7310:10.1214/aos/1031594735
7207:10.1214/aos/1176346577
6833:10.1098/rsta.1894.0003
6364:10.1098/rsta.1916.0009
6214:Quality & Quantity
5757:10.1098/rstb.2010.0063
5377:
5268:R programming language
5188:Bernoulli distribution
5163:
5068:
4966:
4766:
4654:
4495:
4351:
4264:Sambrook Smith's index
4206:
4188:
4145:
4086:
4000:
3955:
3896:
3817:
3742:
3655:is 1 but the value of
3619:
3525:
3441:
3351:Bernoulli distribution
3305:
3159:
3106:Bimodality coefficient
3096:
2953:
2840:
2711:
2571:
2447:
2422:
2329:
2233:
2081:
1992:
1917:
1775:
1665:
1588:
1539:
1485:
1179:
965:
833:
739:
530:
490:
455:
435:
415:
270:
96:
79:
68:
57:
9000:Bivariate (spherical)
8498:Kaniadakis κ-Gaussian
7736:. cran.r-project.org.
7291:Minnotte, MC (1997).
6297:Atriplex triangularis
5375:
5283:Bajgier-Aggarwal test
5192:Dirac delta functions
5164:
5069:
4967:
4767:
4655:
4496:
4352:
4237:is the center of the
4207:
4168:
4125:
4087:
4001:
3941:
3882:
3818:
3743:
3620:
3560:) is due to Wilcock.
3526:
3442:
3355:Dirac delta functions
3306:
3160:
3097:
2954:
2841:
2712:
2572:
2448:
2423:
2330:
2234:
2082:
1993:
1918:
1776:
1747:Tests for unimodality
1666:
1589:
1540:
1486:
1180:
966:
834:
740:
531:
491:
456:
436:
416:
341:, the size of worker
329:Occurrences in nature
271:
209:(iff both parameters
142:and the antimode the
85:
74:
63:
40:
9065:Dirac delta function
9012:Bivariate (toroidal)
8969:Univariate von Mises
8840:Multivariate Laplace
8732:Shifted log-logistic
8081:Continuous Bernoulli
7297:Annals of Statistics
7194:Annals of Statistics
6349:(538–548): 429–457.
6036:Annals of Statistics
5895:Schilling, Mark F.;
5443:Galtung, J. (1969).
5425:Mixture distribution
5399:distribution fitting
5273:
5247:, the MAP test, the
5179:is the kurtosis and
5127:
5093:is the kurtosis and
4979:
4779:
4667:
4583:
4414:
4281:
4106:
4047:
3852:
3776:
3687:
3567:
3552:Bimodality parameter
3488:
3379:
3368:Bimodality amplitude
3343:uniform distribution
3199:
3121:
3020:
2964:uniform distribution
2903:
2782:
2775:). It is defined as
2617:
2457:
2434:
2341:
2247:
2095:
2031:
1931:
1785:
1759:
1717:normal distributions
1601:
1552:
1501:
1192:
978:
846:
779:
657:
508:
465:
445:
425:
405:
388:Mixture distribution
231:
203:arcsine distribution
50:normal distributions
18:Bimodal distribution
9113:Natural exponential
9018:Bivariate von Mises
8984:Wrapped exponential
8850:Multivariate stable
8845:Multivariate normal
8166:Benktander 2nd kind
8161:Benktander 1st kind
7950:Discrete phase-type
7627:"Cluster home page"
7159:1981JRSSB..43...97S
6934:Larkin, RP (1979).
6824:1894RSPTA.185...71P
6806:Pearson, K (1894).
6758:1957JSedR..27....3F
6724:10.14429/dsj.60.356
6687:1997WRR....33.1179S
6628:2014PLoSO...991195D
6589:1982ApJ...263..835S
6546:2008SoPh..249....1S
6355:1916RSPTA.216..429P
6337:Pearson, K (1916).
6299:(Chenopodiaceae)".
6266:2003QJRMS.129.2847Z
6183:1994AJ....108.2348A
5915:10.1198/00031300265
5849:2011PNAS..108.7896H
5726:10.1093/aesa/39.1.7
5395:Gumbel distribution
5355:Other distributions
5249:mode existence test
4472:
4444:
3372:This is defined as
2700:
2682:
2071:
2056:
1721:standard deviations
1477:
1459:
1444:
1423:
1382:
1333:
1315:
1300:
1279:
1238:
1171:
1153:
1122:
1073:
1055:
1024:
957:
939:
900:
882:
646:Moments of mixtures
636:standard deviations
323:Cauchy distribution
114:with more than one
8768:Rectified Gaussian
8653:Generalized Pareto
8511:Generalized normal
8383:Matrix-exponential
7682:cran.r-project.org
7338:10.1007/bf02618468
7270:10.1007/BF01201021
7012:10.1007/bf02514796
6953:10.3758/BF03205709
6901:Annals of Eugenics
6866:Baker, GA (1930).
6479:Cancer Informatics
6260:(594): 2847–2866.
5701:Weber, NA (1946).
5558:Econometric Theory
5546:Phillips, P. C. B.
5378:
5159:
5064:
4962:
4762:
4650:
4491:
4458:
4430:
4347:
4202:
4082:
3996:
3813:
3765:are the means and
3738:
3667:Bimodality indices
3615:
3521:
3437:
3301:
3155:
3092:
2981:Bimodal separation
2949:
2928:
2836:
2734:are the means and
2707:
2686:
2668:
2595:standard deviation
2581:Summary statistics
2567:
2443:
2418:
2325:
2229:
2077:
2057:
2042:
1988:
1913:
1771:
1661:
1584:
1535:
1481:
1463:
1445:
1430:
1409:
1368:
1319:
1301:
1286:
1265:
1224:
1175:
1157:
1139:
1108:
1059:
1041:
1010:
961:
943:
925:
886:
868:
829:
735:
629:General properties
526:
486:
451:
431:
411:
347:Hodgkin's lymphoma
266:
97:
80:
69:
58:
9179:
9178:
8776:
8775:
8745:
8744:
8636:whose type varies
8582:Normal (Gaussian)
8536:Hyperbolic secant
8485:Exponential power
8388:Maxwell–Boltzmann
8136:Wigner semicircle
8028:
8027:
8000:Parabolic fractal
7990:Negative binomial
6696:10.1029/97wr00365
6491:10.4137/CIN.S2846
6274:10.1256/qj.02.166
5751:(1548): 1975–82.
5089:is the skewness,
5062:
4960:
4878:
4760:
4725:
4648:
4551:Graphical methods
4543:Statistical tests
4489:
4345:
4268:A further index (
4200:
3869:
3811:
3736:
3600:
3599:
3519:
3435:
3299:
3296:
3153:
3090:
2919:
2913:
2829:
2771:) to +1 (perfect
2758:
2705:
2704:
2605:
2562:
2560:
2224:
2218:
2202:
2195:
2072:
1983:
1908:
1876:
642:more long tails.
461:with probability
454:{\displaystyle Z}
421:with probability
414:{\displaystyle Y}
339:color of galaxies
287:are constant and
264:
207:beta distribution
181:F: no peak (flat)
89:A non-example: a
16:(Redirected from
9199:
9169:
9168:
9159:
9158:
9098:Compound Poisson
9073:
9061:
9030:von Mises–Fisher
9026:
9014:
9002:
8964:Circular uniform
8960:
8880:
8824:
8795:
8756:
8658:Marchenko–Pastur
8521:Geometric stable
8438:Truncated normal
8331:Inverse Gaussian
8237:Hyperexponential
8076:Beta rectangular
8044:bounded interval
8039:
7907:Discrete uniform
7892:Poisson binomial
7843:
7818:
7811:
7804:
7795:
7789:
7784:
7778:
7777:
7775:
7773:
7767:scikit-learn.org
7759:
7753:
7752:
7744:
7738:
7737:
7729:
7723:
7722:
7714:
7708:
7707:
7699:
7693:
7692:
7690:
7688:
7679:
7671:
7665:
7664:
7656:
7650:
7649:
7641:
7635:
7634:
7623:
7617:
7616:
7610:
7602:
7600:
7599:
7593:
7587:. Archived from
7586:
7578:
7572:
7571:
7569:
7562:
7551:
7545:
7544:
7534:
7502:
7496:
7495:
7467:
7461:
7460:
7442:
7424:
7415:
7409:
7408:
7400:
7394:
7393:
7375:
7369:
7368:
7356:
7350:
7349:
7321:
7315:
7314:
7312:
7303:(4): 1646–1660.
7288:
7282:
7281:
7253:
7247:
7246:
7229:(415): 738–746.
7218:
7212:
7211:
7209:
7185:
7179:
7178:
7142:
7133:
7132:
7110:
7104:
7103:
7101:
7091:
7067:
7061:
7060:
7058:
7030:
7024:
7023:
6995:
6989:
6988:
6964:
6958:
6957:
6955:
6931:
6925:
6924:
6896:
6890:
6889:
6887:
6863:
6857:
6856:
6844:
6838:
6837:
6835:
6803:
6797:
6796:
6776:
6770:
6769:
6737:
6728:
6727:
6707:
6701:
6700:
6698:
6681:(5): 1179–1185.
6666:
6660:
6659:
6649:
6639:
6607:
6601:
6600:
6572:
6566:
6565:
6539:
6519:
6513:
6512:
6502:
6470:
6464:
6463:
6443:
6437:
6436:
6426:
6416:
6392:
6386:
6383:
6377:
6376:
6366:
6334:
6325:
6324:
6307:(8): 1280–1288.
6292:
6286:
6285:
6249:
6238:
6237:
6209:
6203:
6202:
6176:
6174:astro-ph/9408030
6156:
6150:
6149:
6121:
6112:
6111:
6079:
6070:
6069:
6051:
6042:(5): 2042–2065.
6031:
6025:
6024:
6004:
5998:
5997:
5988:(3–4): 137–146.
5977:
5971:
5970:
5958:
5952:
5951:
5933:
5927:
5926:
5892:
5881:
5880:
5870:
5860:
5843:(19): 7896–901.
5828:
5822:
5821:
5785:
5779:
5778:
5768:
5736:
5730:
5729:
5711:
5698:
5692:
5691:
5685:
5676:
5670:
5669:
5652:(5): 1527–1541.
5641:
5635:
5634:
5632:
5608:
5602:
5601:
5589:
5583:
5582:
5554:
5542:
5536:
5531:
5525:
5524:
5492:
5486:
5485:
5476:(3–4): 428–440.
5465:
5459:
5458:
5440:
5274:Silverman's test
5245:excess mass test
5168:
5166:
5165:
5160:
5152:
5151:
5139:
5138:
5073:
5071:
5070:
5065:
5063:
5061:
5057:
5056:
5044:
5043:
5027:
5026:
5025:
5013:
5012:
5002:
4997:
4996:
4971:
4969:
4968:
4963:
4961:
4959:
4955:
4954:
4942:
4941:
4925:
4924:
4923:
4908:
4907:
4895:
4894:
4884:
4879:
4877:
4873:
4872:
4860:
4859:
4843:
4842:
4841:
4826:
4825:
4813:
4812:
4802:
4797:
4796:
4771:
4769:
4768:
4763:
4761:
4756:
4755:
4754:
4742:
4741:
4731:
4726:
4721:
4720:
4719:
4707:
4706:
4696:
4691:
4690:
4659:
4657:
4656:
4651:
4649:
4644:
4643:
4642:
4630:
4629:
4617:
4616:
4606:
4601:
4600:
4564:(or phi) scale.
4500:
4498:
4497:
4492:
4490:
4488:
4487:
4486:
4473:
4471:
4466:
4457:
4456:
4443:
4438:
4429:
4428:
4418:
4356:
4354:
4353:
4348:
4346:
4344:
4343:
4334:
4333:
4324:
4322:
4317:
4316:
4304:
4303:
4294:
4211:
4209:
4208:
4203:
4201:
4199:
4198:
4197:
4187:
4182:
4166:
4165:
4164:
4155:
4154:
4144:
4139:
4123:
4118:
4117:
4091:
4089:
4088:
4083:
4081:
4076:
4075:
4060:
4005:
4003:
4002:
3997:
3995:
3991:
3990:
3989:
3984:
3980:
3954:
3949:
3931:
3930:
3925:
3921:
3895:
3890:
3870:
3862:
3845:) is defined as
3834:Sturrock's index
3822:
3820:
3819:
3814:
3812:
3792:
3747:
3745:
3744:
3739:
3737:
3732:
3731:
3726:
3725:
3713:
3712:
3703:
3697:
3624:
3622:
3621:
3616:
3614:
3613:
3601:
3598:
3597:
3588:
3587:
3578:
3577:
3556:This parameter (
3530:
3528:
3527:
3522:
3520:
3518:
3517:
3508:
3507:
3498:
3446:
3444:
3443:
3438:
3436:
3434:
3433:
3424:
3423:
3422:
3407:
3406:
3396:
3391:
3390:
3310:
3308:
3307:
3302:
3300:
3298:
3297:
3295:
3263:
3262:
3261:
3236:
3227:
3220:
3219:
3209:
3164:
3162:
3161:
3156:
3154:
3149:
3142:
3141:
3131:
3101:
3099:
3098:
3093:
3091:
3089:
3085:
3084:
3072:
3071:
3055:
3054:
3053:
3041:
3040:
3030:
2958:
2956:
2955:
2950:
2948:
2947:
2938:
2937:
2927:
2915:
2914:
2911:
2882:for each layer (
2845:
2843:
2842:
2837:
2835:
2831:
2830:
2828:
2817:
2806:
2759:van der Eijk's A
2716:
2714:
2713:
2708:
2706:
2699:
2694:
2681:
2676:
2661:
2660:
2656:
2655:
2654:
2642:
2641:
2627:
2576:
2574:
2573:
2568:
2563:
2561:
2556:
2555:
2517:
2511:
2503:
2492:
2487:
2486:
2474:
2473:
2464:
2452:
2450:
2449:
2444:
2427:
2425:
2424:
2419:
2411:
2410:
2398:
2397:
2376:
2371:
2370:
2358:
2357:
2348:
2334:
2332:
2331:
2326:
2321:
2316:
2315:
2303:
2302:
2293:
2282:
2277:
2276:
2264:
2263:
2254:
2238:
2236:
2235:
2230:
2225:
2223:
2219:
2214:
2203:
2198:
2194:
2193:
2184:
2183:
2153:
2152:
2137:
2136:
2106:
2105:
2086:
2084:
2083:
2078:
2073:
2070:
2065:
2055:
2050:
2041:
1997:
1995:
1994:
1989:
1984:
1982:
1974:
1970:
1969:
1968:
1956:
1955:
1941:
1922:
1920:
1919:
1914:
1909:
1901:
1900:
1891:
1877:
1869:
1868:
1859:
1836:
1832:
1780:
1778:
1777:
1772:
1670:
1668:
1667:
1662:
1641:
1640:
1613:
1612:
1593:
1591:
1590:
1585:
1577:
1576:
1564:
1563:
1544:
1542:
1541:
1536:
1490:
1488:
1487:
1482:
1476:
1471:
1458:
1453:
1443:
1438:
1422:
1417:
1408:
1407:
1398:
1397:
1381:
1376:
1367:
1366:
1332:
1327:
1314:
1309:
1299:
1294:
1278:
1273:
1264:
1263:
1254:
1253:
1237:
1232:
1223:
1222:
1204:
1203:
1184:
1182:
1181:
1176:
1170:
1165:
1152:
1147:
1138:
1137:
1121:
1116:
1107:
1106:
1072:
1067:
1054:
1049:
1040:
1039:
1023:
1018:
1009:
1008:
990:
989:
970:
968:
967:
962:
956:
951:
938:
933:
899:
894:
881:
876:
858:
857:
838:
836:
835:
830:
828:
827:
800:
799:
744:
742:
741:
736:
724:
723:
687:
686:
535:
533:
532:
527:
495:
493:
492:
487:
460:
458:
457:
452:
440:
438:
437:
432:
420:
418:
417:
412:
275:
273:
272:
267:
265:
263:
252:
241:
21:
9207:
9206:
9202:
9201:
9200:
9198:
9197:
9196:
9182:
9181:
9180:
9175:
9147:
9123:Maximum entropy
9081:
9069:
9057:
9047:
9039:
9022:
9010:
8998:
8953:
8940:
8877:Matrix-valued:
8874:
8820:
8791:
8783:
8772:
8760:
8751:
8741:
8635:
8629:
8546:
8472:
8470:
8464:
8393:Maxwell–Jüttner
8242:Hypoexponential
8148:
8146:
8145:supported on a
8140:
8101:Noncentral beta
8061:Balding–Nichols
8043:
8042:supported on a
8034:
8024:
7927:
7921:
7917:Zipf–Mandelbrot
7847:
7838:
7832:
7822:
7792:
7785:
7781:
7771:
7769:
7761:
7760:
7756:
7746:
7745:
7741:
7731:
7730:
7726:
7716:
7715:
7711:
7701:
7700:
7696:
7686:
7684:
7677:
7673:
7672:
7668:
7658:
7657:
7653:
7643:
7642:
7638:
7625:
7624:
7620:
7603:
7597:
7595:
7591:
7584:
7582:"Archived copy"
7580:
7579:
7575:
7567:
7560:
7553:
7552:
7548:
7504:
7503:
7499:
7469:
7468:
7464:
7422:
7417:
7416:
7412:
7402:
7401:
7397:
7390:
7377:
7376:
7372:
7358:
7357:
7353:
7323:
7322:
7318:
7290:
7289:
7285:
7255:
7254:
7250:
7220:
7219:
7215:
7187:
7186:
7182:
7144:
7143:
7136:
7129:
7112:
7111:
7107:
7089:10.1.1.504.4999
7069:
7068:
7064:
7032:
7031:
7027:
6997:
6996:
6992:
6966:
6965:
6961:
6933:
6932:
6928:
6898:
6897:
6893:
6865:
6864:
6860:
6846:
6845:
6841:
6805:
6804:
6800:
6778:
6777:
6773:
6739:
6738:
6731:
6709:
6708:
6704:
6668:
6667:
6663:
6609:
6608:
6604:
6574:
6573:
6569:
6521:
6520:
6516:
6472:
6471:
6467:
6445:
6444:
6440:
6394:
6393:
6389:
6384:
6380:
6336:
6335:
6328:
6313:10.2307/2444163
6294:
6293:
6289:
6251:
6250:
6241:
6211:
6210:
6206:
6158:
6157:
6153:
6138:10.2307/1267357
6123:
6122:
6115:
6081:
6080:
6073:
6033:
6032:
6028:
6006:
6005:
6001:
5979:
5978:
5974:
5960:
5959:
5955:
5948:
5935:
5934:
5930:
5897:Watkins, Ann E.
5894:
5893:
5884:
5830:
5829:
5825:
5802:10.1038/nrg2146
5787:
5786:
5782:
5738:
5737:
5733:
5709:
5700:
5699:
5695:
5690:. pp. 1–8.
5683:
5678:
5677:
5673:
5643:
5642:
5638:
5610:
5609:
5605:
5591:
5590:
5586:
5552:
5544:
5543:
5539:
5532:
5528:
5494:
5493:
5489:
5467:
5466:
5462:
5455:
5442:
5441:
5437:
5433:
5410:
5405:
5390:
5335:
5324:
5293:
5285:
5276:
5233:
5214:
5203:Fisher's G test
5185:
5178:
5143:
5130:
5125:
5124:
5114:
5099:
5048:
5035:
5028:
5017:
5004:
5003:
4977:
4976:
4946:
4933:
4926:
4915:
4899:
4886:
4885:
4864:
4851:
4844:
4833:
4817:
4804:
4803:
4777:
4776:
4746:
4733:
4732:
4711:
4698:
4697:
4665:
4664:
4634:
4621:
4608:
4607:
4581:
4580:
4553:
4545:
4527:subpopulation,
4522:
4514:subpopulation,
4509:
4478:
4474:
4448:
4420:
4419:
4412:
4411:
4386:
4379:
4372:
4365:
4335:
4325:
4308:
4295:
4279:
4278:
4236:
4223:
4189:
4167:
4156:
4146:
4124:
4109:
4104:
4103:
4067:
4045:
4044:
3940:
3936:
3935:
3881:
3877:
3876:
3875:
3871:
3850:
3849:
3774:
3773:
3764:
3757:
3717:
3704:
3698:
3685:
3684:
3669:
3650:
3641:
3634:
3605:
3589:
3579:
3565:
3564:
3554:
3547:
3540:
3509:
3499:
3486:
3485:
3479:
3472:
3463:
3456:
3425:
3411:
3398:
3397:
3382:
3377:
3376:
3370:
3332:excess kurtosis
3324:sample skewness
3264:
3253:
3237:
3228:
3211:
3210:
3197:
3196:
3133:
3132:
3119:
3118:
3108:
3076:
3063:
3056:
3045:
3032:
3031:
3018:
3017:
3012:
3005:
2998:
2991:
2983:
2939:
2929:
2906:
2901:
2900:
2895:
2888:
2818:
2807:
2798:
2794:
2780:
2779:
2761:
2747:
2740:
2733:
2726:
2646:
2633:
2632:
2628:
2615:
2614:
2608:
2583:
2512:
2478:
2465:
2455:
2454:
2432:
2431:
2402:
2389:
2362:
2349:
2339:
2338:
2307:
2294:
2268:
2255:
2245:
2244:
2196:
2185:
2175:
2144:
2128:
2093:
2092:
2029:
2028:
2011:
2004:
1975:
1960:
1947:
1946:
1942:
1929:
1928:
1892:
1860:
1792:
1788:
1783:
1782:
1757:
1756:
1749:
1710:
1691:
1682:
1632:
1604:
1599:
1598:
1568:
1555:
1550:
1549:
1499:
1498:
1399:
1389:
1358:
1255:
1245:
1214:
1195:
1190:
1189:
1129:
1098:
1031:
1000:
981:
976:
975:
849:
844:
843:
819:
791:
777:
776:
764:The moments of
756:
715:
678:
655:
654:
648:
631:
617:and individual
580:
546:
506:
505:
463:
462:
443:
442:
423:
422:
403:
402:
395:
390:
384:
372:
331:
253:
242:
229:
228:
199:
191:
152:
128:
35:
28:
23:
22:
15:
12:
11:
5:
9205:
9203:
9195:
9194:
9184:
9183:
9177:
9176:
9174:
9173:
9163:
9152:
9149:
9148:
9146:
9145:
9140:
9135:
9130:
9125:
9120:
9118:Location–scale
9115:
9110:
9105:
9100:
9095:
9089:
9087:
9083:
9082:
9080:
9079:
9074:
9067:
9062:
9054:
9052:
9041:
9040:
9038:
9037:
9032:
9027:
9020:
9015:
9008:
9003:
8996:
8991:
8986:
8981:
8979:Wrapped Cauchy
8976:
8974:Wrapped normal
8971:
8966:
8961:
8950:
8948:
8942:
8941:
8939:
8938:
8937:
8936:
8931:
8929:Normal-inverse
8926:
8921:
8911:
8910:
8909:
8899:
8891:
8886:
8881:
8872:
8871:
8870:
8860:
8852:
8847:
8842:
8837:
8836:
8835:
8825:
8818:
8817:
8816:
8811:
8801:
8796:
8788:
8786:
8778:
8777:
8774:
8773:
8771:
8770:
8764:
8762:
8753:
8747:
8746:
8743:
8742:
8740:
8739:
8734:
8729:
8721:
8713:
8705:
8696:
8687:
8678:
8669:
8660:
8655:
8650:
8645:
8639:
8637:
8631:
8630:
8628:
8627:
8622:
8620:Variance-gamma
8617:
8612:
8604:
8599:
8594:
8589:
8584:
8579:
8571:
8566:
8565:
8564:
8554:
8549:
8544:
8538:
8533:
8528:
8523:
8518:
8513:
8508:
8500:
8495:
8487:
8482:
8476:
8474:
8466:
8465:
8463:
8462:
8460:Wilks's lambda
8457:
8456:
8455:
8445:
8440:
8435:
8430:
8425:
8420:
8415:
8410:
8405:
8400:
8398:Mittag-Leffler
8395:
8390:
8385:
8380:
8375:
8370:
8365:
8360:
8355:
8350:
8345:
8340:
8339:
8338:
8328:
8319:
8314:
8309:
8308:
8307:
8297:
8295:gamma/Gompertz
8292:
8291:
8290:
8285:
8275:
8270:
8265:
8264:
8263:
8251:
8250:
8249:
8244:
8239:
8229:
8228:
8227:
8217:
8212:
8207:
8206:
8205:
8204:
8203:
8193:
8183:
8178:
8173:
8168:
8163:
8158:
8152:
8150:
8147:semi-infinite
8142:
8141:
8139:
8138:
8133:
8128:
8123:
8118:
8113:
8108:
8103:
8098:
8093:
8088:
8083:
8078:
8073:
8068:
8063:
8058:
8053:
8047:
8045:
8036:
8030:
8029:
8026:
8025:
8023:
8022:
8017:
8012:
8007:
8002:
7997:
7992:
7987:
7982:
7977:
7972:
7967:
7962:
7957:
7952:
7947:
7942:
7937:
7931:
7929:
7926:with infinite
7923:
7922:
7920:
7919:
7914:
7909:
7904:
7899:
7894:
7889:
7888:
7887:
7880:Hypergeometric
7877:
7872:
7867:
7862:
7857:
7851:
7849:
7840:
7834:
7833:
7823:
7821:
7820:
7813:
7806:
7798:
7791:
7790:
7779:
7754:
7739:
7724:
7709:
7694:
7666:
7651:
7636:
7618:
7573:
7570:on 2016-03-04.
7546:
7517:(6): 655–662.
7497:
7478:(2): 253–269.
7462:
7410:
7395:
7388:
7370:
7351:
7316:
7283:
7248:
7213:
7180:
7134:
7127:
7105:
7062:
7041:(2): 173–174.
7025:
7006:(2): 249–273.
6990:
6979:(4): 422–434.
6959:
6946:(4): 467–468.
6926:
6907:(1): 359–364.
6891:
6878:(4): 334–344.
6858:
6839:
6798:
6787:(2): 616–620.
6771:
6729:
6718:(3): 290–301.
6702:
6661:
6602:
6597:10.1086/160554
6583:(1): 835–853.
6567:
6514:
6465:
6454:(4): 491–505.
6438:
6387:
6378:
6326:
6287:
6239:
6220:(3): 325–341.
6204:
6191:10.1086/117248
6151:
6132:(1): 131–139.
6113:
6071:
6026:
6015:(4): 357–363.
5999:
5972:
5953:
5946:
5928:
5909:(3): 223–229.
5882:
5823:
5780:
5731:
5693:
5671:
5636:
5623:(3): 224–240.
5603:
5584:
5565:(5): 947–960.
5537:
5526:
5507:(2): 271–289.
5487:
5460:
5453:
5434:
5432:
5429:
5428:
5427:
5422:
5416:
5414:Overdispersion
5409:
5406:
5403:
5389:
5386:
5357:
5356:
5342:A package for
5340:
5339:
5334:
5331:
5323:
5320:
5316:
5315:
5301:
5300:
5292:
5289:
5284:
5281:
5275:
5272:
5237:bandwidth test
5232:
5229:
5222:
5221:
5213:
5212:Antimode tests
5210:
5183:
5176:
5170:
5169:
5158:
5155:
5150:
5146:
5142:
5137:
5133:
5113:
5110:
5097:
5075:
5074:
5060:
5055:
5051:
5047:
5042:
5038:
5034:
5031:
5024:
5020:
5016:
5011:
5007:
5000:
4995:
4992:
4989:
4986:
4973:
4972:
4958:
4953:
4949:
4945:
4940:
4936:
4932:
4929:
4922:
4918:
4914:
4911:
4906:
4902:
4898:
4893:
4889:
4882:
4876:
4871:
4867:
4863:
4858:
4854:
4850:
4847:
4840:
4836:
4832:
4829:
4824:
4820:
4816:
4811:
4807:
4800:
4795:
4792:
4789:
4786:
4773:
4772:
4759:
4753:
4749:
4745:
4740:
4736:
4729:
4724:
4718:
4714:
4710:
4705:
4701:
4694:
4689:
4686:
4683:
4680:
4677:
4674:
4661:
4660:
4647:
4641:
4637:
4633:
4628:
4624:
4620:
4615:
4611:
4604:
4599:
4596:
4593:
4590:
4574:
4573:
4552:
4549:
4544:
4541:
4518:
4505:
4485:
4481:
4477:
4470:
4465:
4461:
4455:
4451:
4447:
4442:
4437:
4433:
4427:
4423:
4405:
4404:
4384:
4377:
4370:
4363:
4342:
4338:
4332:
4328:
4321:
4315:
4311:
4307:
4302:
4298:
4293:
4289:
4286:
4266:
4265:
4232:
4219:
4213:
4212:
4196:
4192:
4186:
4181:
4178:
4175:
4171:
4163:
4159:
4153:
4149:
4143:
4138:
4135:
4132:
4128:
4121:
4116:
4112:
4093:
4092:
4080:
4074:
4070:
4066:
4063:
4059:
4055:
4052:
4034:
4033:
4007:
4006:
3994:
3988:
3983:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3953:
3948:
3944:
3939:
3934:
3929:
3924:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3894:
3889:
3885:
3880:
3874:
3868:
3865:
3860:
3857:
3836:
3835:
3824:
3823:
3810:
3807:
3804:
3801:
3798:
3795:
3790:
3787:
3784:
3781:
3762:
3755:
3749:
3748:
3735:
3730:
3724:
3720:
3716:
3711:
3707:
3702:
3695:
3692:
3674:
3673:
3668:
3665:
3646:
3639:
3632:
3626:
3625:
3612:
3608:
3604:
3596:
3592:
3586:
3582:
3575:
3572:
3553:
3550:
3545:
3538:
3532:
3531:
3516:
3512:
3506:
3502:
3496:
3493:
3478:
3475:
3470:
3461:
3454:
3448:
3447:
3432:
3428:
3421:
3418:
3414:
3410:
3405:
3401:
3394:
3389:
3385:
3369:
3366:
3330:is the sample
3312:
3311:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3260:
3256:
3252:
3249:
3246:
3243:
3240:
3234:
3231:
3226:
3223:
3218:
3214:
3207:
3204:
3166:
3165:
3152:
3148:
3145:
3140:
3136:
3129:
3126:
3107:
3104:
3103:
3102:
3088:
3083:
3079:
3075:
3070:
3066:
3062:
3059:
3052:
3048:
3044:
3039:
3035:
3028:
3025:
3010:
3003:
2996:
2989:
2982:
2979:
2960:
2959:
2946:
2942:
2936:
2932:
2926:
2922:
2918:
2909:
2893:
2886:
2876:
2875:
2872:
2869:
2847:
2846:
2834:
2827:
2824:
2821:
2816:
2813:
2810:
2804:
2801:
2797:
2793:
2790:
2787:
2760:
2757:
2745:
2738:
2731:
2724:
2718:
2717:
2703:
2698:
2693:
2689:
2685:
2680:
2675:
2671:
2667:
2664:
2659:
2653:
2649:
2645:
2640:
2636:
2631:
2625:
2622:
2607:
2604:
2582:
2579:
2578:
2577:
2566:
2559:
2554:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2516:
2509:
2506:
2501:
2498:
2495:
2491:
2485:
2481:
2477:
2472:
2468:
2463:
2442:
2439:
2428:
2417:
2414:
2409:
2405:
2401:
2396:
2392:
2388:
2385:
2382:
2379:
2375:
2369:
2365:
2361:
2356:
2352:
2347:
2335:
2324:
2320:
2314:
2310:
2306:
2301:
2297:
2292:
2288:
2285:
2281:
2275:
2271:
2267:
2262:
2258:
2253:
2228:
2222:
2217:
2212:
2209:
2206:
2201:
2192:
2188:
2182:
2178:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2151:
2147:
2143:
2140:
2135:
2131:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2103:
2100:
2076:
2069:
2064:
2060:
2054:
2049:
2045:
2039:
2036:
2017:
2009:
2002:
1987:
1981:
1978:
1973:
1967:
1963:
1959:
1954:
1950:
1945:
1939:
1936:
1912:
1907:
1904:
1899:
1895:
1889:
1886:
1883:
1880:
1875:
1872:
1867:
1863:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1835:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1791:
1770:
1767:
1764:
1754:if and only if
1748:
1745:
1709:
1706:
1704:distribution.
1687:
1678:
1672:
1671:
1660:
1657:
1653:
1650:
1647:
1644:
1639:
1635:
1631:
1628:
1625:
1622:
1619:
1616:
1611:
1607:
1595:
1594:
1583:
1580:
1575:
1571:
1567:
1562:
1558:
1546:
1545:
1534:
1531:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1492:
1491:
1480:
1475:
1470:
1466:
1462:
1457:
1452:
1448:
1442:
1437:
1433:
1429:
1426:
1421:
1416:
1412:
1406:
1402:
1396:
1392:
1388:
1385:
1380:
1375:
1371:
1365:
1361:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1331:
1326:
1322:
1318:
1313:
1308:
1304:
1298:
1293:
1289:
1285:
1282:
1277:
1272:
1268:
1262:
1258:
1252:
1248:
1244:
1241:
1236:
1231:
1227:
1221:
1217:
1213:
1210:
1207:
1202:
1198:
1186:
1185:
1174:
1169:
1164:
1160:
1156:
1151:
1146:
1142:
1136:
1132:
1128:
1125:
1120:
1115:
1111:
1105:
1101:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1071:
1066:
1062:
1058:
1053:
1048:
1044:
1038:
1034:
1030:
1027:
1022:
1017:
1013:
1007:
1003:
999:
996:
993:
988:
984:
972:
971:
960:
955:
950:
946:
942:
937:
932:
928:
924:
921:
918:
915:
912:
909:
906:
903:
898:
893:
889:
885:
880:
875:
871:
867:
864:
861:
856:
852:
840:
839:
826:
822:
818:
815:
812:
809:
806:
803:
798:
794:
790:
787:
784:
752:
746:
745:
733:
730:
727:
722:
718:
714:
711:
708:
705:
702:
699:
696:
693:
690:
685:
681:
677:
674:
671:
668:
665:
662:
647:
644:
630:
627:
600:
599:
596:
593:
590:
587:
579:
576:
568:
567:
560:
557:
554:
545:
542:
525:
522:
519:
516:
513:
485:
482:
479:
476:
473:
470:
450:
430:
410:
394:
391:
386:Main article:
383:
380:
371:
368:
330:
327:
277:
276:
262:
259:
256:
251:
248:
245:
239:
236:
198:
195:
190:
187:
183:
182:
179:
176:
169:
168:
165:
162:
159:
151:
148:
127:
124:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9204:
9193:
9190:
9189:
9187:
9172:
9164:
9162:
9154:
9153:
9150:
9144:
9141:
9139:
9136:
9134:
9131:
9129:
9126:
9124:
9121:
9119:
9116:
9114:
9111:
9109:
9106:
9104:
9101:
9099:
9096:
9094:
9091:
9090:
9088:
9084:
9078:
9075:
9072:
9068:
9066:
9063:
9060:
9056:
9055:
9053:
9051:
9046:
9042:
9036:
9033:
9031:
9028:
9025:
9021:
9019:
9016:
9013:
9009:
9007:
9004:
9001:
8997:
8995:
8992:
8990:
8987:
8985:
8982:
8980:
8977:
8975:
8972:
8970:
8967:
8965:
8962:
8959:
8958:
8952:
8951:
8949:
8947:
8943:
8935:
8932:
8930:
8927:
8925:
8922:
8920:
8917:
8916:
8915:
8912:
8908:
8905:
8904:
8903:
8900:
8898:
8897:
8892:
8890:
8889:Matrix normal
8887:
8885:
8882:
8879:
8878:
8873:
8869:
8866:
8865:
8864:
8861:
8859:
8858:
8855:Multivariate
8853:
8851:
8848:
8846:
8843:
8841:
8838:
8834:
8831:
8830:
8829:
8826:
8823:
8819:
8815:
8812:
8810:
8807:
8806:
8805:
8802:
8800:
8797:
8794:
8790:
8789:
8787:
8785:
8782:Multivariate
8779:
8769:
8766:
8765:
8763:
8757:
8754:
8748:
8738:
8735:
8733:
8730:
8728:
8726:
8722:
8720:
8718:
8714:
8712:
8710:
8706:
8704:
8702:
8697:
8695:
8693:
8688:
8686:
8684:
8679:
8677:
8675:
8670:
8668:
8666:
8661:
8659:
8656:
8654:
8651:
8649:
8646:
8644:
8641:
8640:
8638:
8634:with support
8632:
8626:
8623:
8621:
8618:
8616:
8613:
8611:
8610:
8605:
8603:
8600:
8598:
8595:
8593:
8590:
8588:
8585:
8583:
8580:
8578:
8577:
8572:
8570:
8567:
8563:
8560:
8559:
8558:
8555:
8553:
8550:
8548:
8547:
8539:
8537:
8534:
8532:
8529:
8527:
8524:
8522:
8519:
8517:
8514:
8512:
8509:
8507:
8506:
8501:
8499:
8496:
8494:
8493:
8488:
8486:
8483:
8481:
8478:
8477:
8475:
8471:on the whole
8467:
8461:
8458:
8454:
8451:
8450:
8449:
8446:
8444:
8443:type-2 Gumbel
8441:
8439:
8436:
8434:
8431:
8429:
8426:
8424:
8421:
8419:
8416:
8414:
8411:
8409:
8406:
8404:
8401:
8399:
8396:
8394:
8391:
8389:
8386:
8384:
8381:
8379:
8376:
8374:
8371:
8369:
8366:
8364:
8361:
8359:
8356:
8354:
8351:
8349:
8346:
8344:
8341:
8337:
8334:
8333:
8332:
8329:
8327:
8325:
8320:
8318:
8315:
8313:
8312:Half-logistic
8310:
8306:
8303:
8302:
8301:
8298:
8296:
8293:
8289:
8286:
8284:
8281:
8280:
8279:
8276:
8274:
8271:
8269:
8268:Folded normal
8266:
8262:
8259:
8258:
8257:
8256:
8252:
8248:
8245:
8243:
8240:
8238:
8235:
8234:
8233:
8230:
8226:
8223:
8222:
8221:
8218:
8216:
8213:
8211:
8208:
8202:
8199:
8198:
8197:
8194:
8192:
8189:
8188:
8187:
8184:
8182:
8179:
8177:
8174:
8172:
8169:
8167:
8164:
8162:
8159:
8157:
8154:
8153:
8151:
8143:
8137:
8134:
8132:
8129:
8127:
8124:
8122:
8119:
8117:
8114:
8112:
8111:Raised cosine
8109:
8107:
8104:
8102:
8099:
8097:
8094:
8092:
8089:
8087:
8084:
8082:
8079:
8077:
8074:
8072:
8069:
8067:
8064:
8062:
8059:
8057:
8054:
8052:
8049:
8048:
8046:
8040:
8037:
8031:
8021:
8018:
8016:
8013:
8011:
8008:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7986:
7985:Mixed Poisson
7983:
7981:
7978:
7976:
7973:
7971:
7968:
7966:
7963:
7961:
7958:
7956:
7953:
7951:
7948:
7946:
7943:
7941:
7938:
7936:
7933:
7932:
7930:
7924:
7918:
7915:
7913:
7910:
7908:
7905:
7903:
7900:
7898:
7895:
7893:
7890:
7886:
7883:
7882:
7881:
7878:
7876:
7873:
7871:
7868:
7866:
7865:Beta-binomial
7863:
7861:
7858:
7856:
7853:
7852:
7850:
7844:
7841:
7835:
7830:
7826:
7819:
7814:
7812:
7807:
7805:
7800:
7799:
7796:
7788:
7783:
7780:
7768:
7764:
7758:
7755:
7750:
7743:
7740:
7735:
7728:
7725:
7720:
7713:
7710:
7705:
7698:
7695:
7683:
7676:
7675:"discrimARTs"
7670:
7667:
7662:
7655:
7652:
7647:
7640:
7637:
7632:
7628:
7622:
7619:
7614:
7608:
7594:on 2013-11-03
7590:
7583:
7577:
7574:
7566:
7559:
7558:
7550:
7547:
7542:
7538:
7533:
7528:
7524:
7520:
7516:
7512:
7508:
7501:
7498:
7493:
7489:
7485:
7481:
7477:
7473:
7466:
7463:
7458:
7454:
7450:
7446:
7441:
7436:
7432:
7428:
7421:
7414:
7411:
7406:
7399:
7396:
7391:
7389:0-444-70404-3
7385:
7381:
7374:
7371:
7366:
7362:
7355:
7352:
7347:
7343:
7339:
7335:
7331:
7327:
7320:
7317:
7311:
7306:
7302:
7298:
7294:
7287:
7284:
7279:
7275:
7271:
7267:
7263:
7259:
7252:
7249:
7244:
7240:
7236:
7232:
7228:
7224:
7217:
7214:
7208:
7203:
7199:
7195:
7191:
7184:
7181:
7176:
7172:
7168:
7164:
7160:
7156:
7152:
7148:
7141:
7139:
7135:
7130:
7128:3-540-67731-3
7124:
7120:
7119:Data Analysis
7116:
7109:
7106:
7100:
7095:
7090:
7085:
7081:
7077:
7073:
7066:
7063:
7057:
7052:
7048:
7044:
7040:
7036:
7029:
7026:
7021:
7017:
7013:
7009:
7005:
7001:
6994:
6991:
6986:
6982:
6978:
6974:
6970:
6963:
6960:
6954:
6949:
6945:
6941:
6937:
6930:
6927:
6922:
6918:
6914:
6910:
6906:
6902:
6895:
6892:
6886:
6881:
6877:
6873:
6869:
6862:
6859:
6854:
6850:
6843:
6840:
6834:
6829:
6825:
6821:
6817:
6813:
6809:
6802:
6799:
6794:
6790:
6786:
6782:
6775:
6772:
6767:
6763:
6759:
6755:
6751:
6747:
6743:
6736:
6734:
6730:
6725:
6721:
6717:
6713:
6706:
6703:
6697:
6692:
6688:
6684:
6680:
6676:
6672:
6665:
6662:
6657:
6653:
6648:
6643:
6638:
6633:
6629:
6625:
6622:(3): e91195.
6621:
6617:
6613:
6606:
6603:
6598:
6594:
6590:
6586:
6582:
6578:
6571:
6568:
6563:
6559:
6555:
6551:
6547:
6543:
6538:
6533:
6529:
6525:
6524:Solar Physics
6518:
6515:
6510:
6506:
6501:
6496:
6492:
6488:
6484:
6480:
6476:
6469:
6466:
6461:
6457:
6453:
6449:
6442:
6439:
6434:
6430:
6425:
6420:
6415:
6410:
6406:
6402:
6398:
6391:
6388:
6382:
6379:
6374:
6370:
6365:
6360:
6356:
6352:
6348:
6344:
6340:
6333:
6331:
6327:
6322:
6318:
6314:
6310:
6306:
6302:
6298:
6291:
6288:
6283:
6279:
6275:
6271:
6267:
6263:
6259:
6255:
6248:
6246:
6244:
6240:
6235:
6231:
6227:
6223:
6219:
6215:
6208:
6205:
6200:
6196:
6192:
6188:
6184:
6180:
6175:
6170:
6167:: 2348–2361.
6166:
6162:
6155:
6152:
6147:
6143:
6139:
6135:
6131:
6127:
6126:Technometrics
6120:
6118:
6114:
6109:
6105:
6101:
6097:
6093:
6089:
6085:
6078:
6076:
6072:
6067:
6063:
6059:
6055:
6050:
6045:
6041:
6037:
6030:
6027:
6022:
6018:
6014:
6010:
6009:Technometrics
6003:
6000:
5995:
5991:
5987:
5983:
5976:
5973:
5968:
5964:
5957:
5954:
5949:
5947:0-201-04854-X
5943:
5939:
5932:
5929:
5924:
5920:
5916:
5912:
5908:
5904:
5903:
5898:
5891:
5889:
5887:
5883:
5878:
5874:
5869:
5864:
5859:
5854:
5850:
5846:
5842:
5838:
5834:
5827:
5824:
5819:
5815:
5811:
5807:
5803:
5799:
5795:
5791:
5784:
5781:
5776:
5772:
5767:
5762:
5758:
5754:
5750:
5746:
5742:
5735:
5732:
5727:
5723:
5719:
5715:
5708:
5706:
5697:
5694:
5689:
5682:
5675:
5672:
5667:
5663:
5659:
5655:
5651:
5647:
5640:
5637:
5631:
5626:
5622:
5618:
5614:
5607:
5604:
5600:(2): 379–396.
5599:
5595:
5588:
5585:
5580:
5576:
5572:
5568:
5564:
5560:
5559:
5551:
5547:
5541:
5538:
5535:
5530:
5527:
5522:
5518:
5514:
5510:
5506:
5502:
5498:
5491:
5488:
5483:
5479:
5475:
5471:
5464:
5461:
5456:
5454:0-04-300017-7
5450:
5446:
5439:
5436:
5430:
5426:
5423:
5420:
5419:Mixture model
5417:
5415:
5412:
5411:
5407:
5402:
5400:
5396:
5387:
5385:
5383:
5374:
5370:
5368:
5363:
5360:
5354:
5353:
5352:
5348:
5345:
5337:
5336:
5332:
5330:
5327:
5321:
5319:
5313:
5312:
5311:
5309:
5304:
5298:
5297:
5296:
5291:Special cases
5290:
5288:
5282:
5280:
5271:
5269:
5264:
5262:
5258:
5254:
5250:
5246:
5242:
5238:
5231:General tests
5230:
5228:
5226:
5225:Otsu's method
5220:Otsu's method
5219:
5218:
5217:
5211:
5209:
5206:
5204:
5198:
5195:
5193:
5189:
5182:
5175:
5156:
5153:
5148:
5144:
5140:
5135:
5131:
5123:
5122:
5121:
5119:
5111:
5109:
5107:
5103:
5096:
5092:
5088:
5084:
5081:is the mean,
5080:
5053:
5049:
5045:
5040:
5036:
5029:
5022:
5018:
5014:
5009:
5005:
4998:
4975:
4974:
4951:
4947:
4943:
4938:
4934:
4927:
4920:
4916:
4912:
4909:
4904:
4900:
4896:
4891:
4887:
4880:
4869:
4865:
4861:
4856:
4852:
4845:
4838:
4834:
4830:
4827:
4822:
4818:
4814:
4809:
4805:
4798:
4775:
4774:
4757:
4751:
4747:
4743:
4738:
4734:
4727:
4722:
4716:
4712:
4708:
4703:
4699:
4692:
4663:
4662:
4645:
4639:
4635:
4631:
4626:
4622:
4618:
4613:
4609:
4602:
4579:
4578:
4577:
4571:
4570:
4569:
4565:
4563:
4559:
4550:
4548:
4542:
4540:
4538:
4534:
4530:
4526:
4521:
4517:
4513:
4508:
4504:
4483:
4479:
4475:
4468:
4463:
4459:
4453:
4449:
4445:
4440:
4435:
4431:
4425:
4421:
4409:
4408:Otsu's method
4403:Otsu's method
4402:
4401:
4400:
4396:
4394:
4390:
4383:
4376:
4369:
4362:
4357:
4340:
4336:
4330:
4326:
4313:
4309:
4305:
4300:
4296:
4287:
4284:
4276:
4275:
4271:
4263:
4262:
4261:
4259:
4255:
4251:
4246:
4244:
4240:
4235:
4231:
4227:
4222:
4218:
4194:
4190:
4184:
4179:
4176:
4173:
4169:
4161:
4157:
4151:
4147:
4141:
4136:
4133:
4130:
4126:
4119:
4114:
4110:
4102:
4101:
4100:
4098:
4072:
4068:
4064:
4061:
4053:
4050:
4043:
4042:
4041:
4039:
4031:
4030:
4029:
4027:
4022:
4020:
4016:
4012:
3992:
3986:
3981:
3974:
3971:
3968:
3965:
3959:
3956:
3951:
3946:
3942:
3937:
3932:
3927:
3922:
3915:
3912:
3909:
3906:
3900:
3897:
3892:
3887:
3883:
3878:
3872:
3866:
3863:
3858:
3855:
3848:
3847:
3846:
3844:
3839:
3833:
3832:
3831:
3829:
3805:
3802:
3799:
3793:
3788:
3785:
3782:
3779:
3772:
3771:
3770:
3768:
3761:
3754:
3733:
3722:
3718:
3714:
3709:
3705:
3693:
3690:
3683:
3682:
3681:
3679:
3671:
3670:
3666:
3664:
3660:
3658:
3654:
3649:
3645:
3638:
3631:
3610:
3606:
3602:
3594:
3590:
3584:
3580:
3573:
3570:
3563:
3562:
3561:
3559:
3551:
3549:
3544:
3537:
3514:
3510:
3504:
3500:
3494:
3491:
3484:
3483:
3482:
3477:Bimodal ratio
3476:
3474:
3469:
3465:
3460:
3453:
3430:
3426:
3419:
3416:
3412:
3408:
3403:
3399:
3392:
3387:
3383:
3375:
3374:
3373:
3367:
3365:
3363:
3358:
3356:
3352:
3348:
3344:
3340:
3337:The value of
3335:
3333:
3329:
3325:
3321:
3317:
3289:
3286:
3283:
3274:
3271:
3268:
3258:
3250:
3247:
3244:
3238:
3232:
3229:
3224:
3221:
3216:
3212:
3205:
3202:
3195:
3194:
3193:
3190:
3187:
3183:
3179:
3175:
3171:
3150:
3146:
3143:
3138:
3134:
3127:
3124:
3117:
3116:
3115:
3113:
3105:
3081:
3077:
3073:
3068:
3064:
3057:
3050:
3046:
3042:
3037:
3033:
3026:
3023:
3016:
3015:
3014:
3009:
3002:
2995:
2988:
2980:
2978:
2975:
2973:
2969:
2965:
2944:
2940:
2934:
2930:
2924:
2920:
2916:
2907:
2899:
2898:
2897:
2892:
2885:
2881:
2873:
2870:
2867:
2866:
2865:
2862:
2860:
2856:
2852:
2832:
2825:
2822:
2819:
2814:
2811:
2808:
2802:
2799:
2795:
2791:
2788:
2785:
2778:
2777:
2776:
2774:
2770:
2766:
2756:
2754:
2749:
2744:
2737:
2730:
2723:
2696:
2691:
2687:
2683:
2678:
2673:
2669:
2662:
2657:
2651:
2647:
2643:
2638:
2634:
2629:
2623:
2620:
2613:
2612:
2611:
2603:
2599:
2596:
2592:
2588:
2580:
2564:
2557:
2545:
2542:
2539:
2533:
2530:
2527:
2524:
2521:
2518:
2507:
2504:
2499:
2496:
2493:
2483:
2479:
2475:
2470:
2466:
2440:
2437:
2429:
2415:
2407:
2403:
2399:
2394:
2390:
2380:
2377:
2367:
2363:
2359:
2354:
2350:
2336:
2322:
2312:
2308:
2304:
2299:
2295:
2286:
2283:
2273:
2269:
2265:
2260:
2256:
2242:
2226:
2215:
2210:
2207:
2199:
2190:
2180:
2176:
2172:
2169:
2166:
2163:
2157:
2154:
2149:
2145:
2141:
2138:
2133:
2129:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2101:
2098:
2090:
2074:
2067:
2062:
2058:
2052:
2047:
2043:
2037:
2034:
2026:
2022:
2018:
2015:
2008:
2001:
1985:
1979:
1976:
1971:
1965:
1961:
1957:
1952:
1948:
1943:
1937:
1934:
1926:
1910:
1905:
1902:
1897:
1893:
1887:
1884:
1881:
1873:
1870:
1865:
1861:
1855:
1852:
1846:
1843:
1840:
1837:
1833:
1826:
1820:
1817:
1814:
1808:
1805:
1802:
1796:
1793:
1789:
1768:
1765:
1762:
1755:
1751:
1750:
1746:
1744:
1741:
1738:
1735:
1733:
1728:
1726:
1725:homoscedastic
1722:
1718:
1713:
1707:
1705:
1703:
1699:
1695:
1690:
1686:
1681:
1677:
1658:
1655:
1648:
1642:
1637:
1629:
1626:
1623:
1617:
1614:
1609:
1605:
1597:
1596:
1581:
1578:
1573:
1569:
1565:
1560:
1556:
1548:
1547:
1532:
1529:
1522:
1516:
1513:
1510:
1507:
1504:
1497:
1496:
1495:
1473:
1468:
1464:
1460:
1455:
1450:
1446:
1440:
1435:
1431:
1427:
1424:
1419:
1414:
1410:
1404:
1400:
1394:
1390:
1386:
1383:
1378:
1373:
1369:
1363:
1359:
1349:
1346:
1343:
1337:
1329:
1324:
1320:
1316:
1311:
1306:
1302:
1296:
1291:
1287:
1283:
1280:
1275:
1270:
1266:
1260:
1256:
1250:
1246:
1242:
1239:
1234:
1229:
1225:
1219:
1215:
1208:
1205:
1200:
1196:
1188:
1187:
1167:
1162:
1158:
1154:
1149:
1144:
1140:
1134:
1130:
1126:
1123:
1118:
1113:
1109:
1103:
1099:
1089:
1086:
1083:
1077:
1069:
1064:
1060:
1056:
1051:
1046:
1042:
1036:
1032:
1028:
1025:
1020:
1015:
1011:
1005:
1001:
994:
991:
986:
982:
974:
973:
953:
948:
944:
940:
935:
930:
926:
916:
913:
910:
904:
896:
891:
887:
883:
878:
873:
869:
862:
859:
854:
850:
842:
841:
824:
820:
813:
810:
807:
801:
796:
792:
788:
785:
782:
775:
774:
773:
771:
767:
762:
760:
755:
751:
728:
720:
716:
709:
706:
703:
697:
691:
683:
679:
675:
672:
666:
660:
653:
652:
651:
645:
643:
639:
637:
628:
626:
624:
620:
616:
612:
607:
605:
597:
594:
591:
588:
585:
584:
583:
577:
575:
573:
565:
562:A mixture of
561:
558:
555:
552:
551:
550:
543:
541:
537:
523:
520:
517:
514:
511:
503:
499:
483:
477:
474:
471:
448:
428:
408:
400:
392:
389:
381:
379:
377:
369:
367:
364:
360:
356:
352:
348:
344:
340:
336:
328:
326:
324:
320:
315:
313:
309:
304:
302:
298:
294:
290:
286:
282:
260:
257:
254:
249:
246:
243:
237:
234:
227:
226:
225:
222:
220:
216:
212:
208:
204:
196:
194:
188:
186:
180:
177:
174:
173:
172:
166:
163:
160:
157:
156:
155:
147:
145:
141:
137:
133:
125:
123:
121:
117:
113:
109:
106:
102:
94:
93:
88:
84:
77:
73:
66:
62:
55:
51:
47:
43:
39:
33:
19:
9070:
9058:
9024:Multivariate
9023:
9011:
8999:
8994:Wrapped Lévy
8954:
8902:Matrix gamma
8895:
8875:
8863:Normal-gamma
8856:
8822:Continuous:
8821:
8792:
8737:Tukey lambda
8724:
8716:
8711:-exponential
8708:
8700:
8691:
8682:
8673:
8667:-exponential
8664:
8608:
8575:
8542:
8504:
8491:
8418:Poly-Weibull
8363:Log-logistic
8323:
8322:Hotelling's
8254:
8096:Logit-normal
7970:Gauss–Kuzmin
7965:Flory–Schulz
7846:with finite
7782:
7770:. Retrieved
7766:
7757:
7742:
7727:
7712:
7697:
7685:. Retrieved
7681:
7669:
7654:
7639:
7630:
7621:
7596:. Retrieved
7589:the original
7576:
7565:the original
7556:
7549:
7514:
7510:
7500:
7475:
7471:
7465:
7433:(1): 83–97.
7430:
7426:
7413:
7398:
7379:
7373:
7364:
7360:
7354:
7329:
7325:
7319:
7300:
7296:
7286:
7261:
7257:
7251:
7226:
7222:
7216:
7200:(1): 70–84.
7197:
7193:
7183:
7153:(1): 97–99.
7150:
7146:
7118:
7108:
7079:
7075:
7065:
7056:10400.3/1408
7038:
7034:
7028:
7003:
6999:
6993:
6976:
6972:
6968:
6962:
6943:
6939:
6929:
6904:
6900:
6894:
6875:
6871:
6861:
6852:
6848:
6842:
6815:
6811:
6801:
6784:
6780:
6774:
6749:
6745:
6715:
6711:
6705:
6678:
6674:
6664:
6619:
6615:
6605:
6580:
6576:
6570:
6527:
6523:
6517:
6482:
6478:
6468:
6451:
6447:
6441:
6404:
6400:
6390:
6381:
6346:
6342:
6304:
6300:
6296:
6290:
6257:
6253:
6217:
6213:
6207:
6164:
6160:
6154:
6129:
6125:
6094:(1): 57–69.
6091:
6087:
6049:math/0602238
6039:
6035:
6029:
6012:
6008:
6002:
5985:
5981:
5975:
5961:Kim, T.-H.;
5956:
5937:
5931:
5906:
5900:
5840:
5836:
5826:
5796:(8): 610–8.
5793:
5789:
5783:
5748:
5744:
5734:
5717:
5713:
5704:
5696:
5687:
5674:
5649:
5645:
5639:
5620:
5616:
5606:
5597:
5593:
5587:
5562:
5556:
5540:
5529:
5504:
5500:
5490:
5473:
5469:
5463:
5444:
5438:
5391:
5382:Scikit-learn
5379:
5364:
5361:
5358:
5349:
5341:
5328:
5325:
5317:
5305:
5302:
5294:
5286:
5277:
5265:
5234:
5223:
5215:
5207:
5199:
5196:
5180:
5173:
5171:
5115:
5105:
5101:
5094:
5090:
5086:
5082:
5078:
5076:
4575:
4566:
4554:
4546:
4532:
4528:
4524:
4519:
4515:
4511:
4506:
4502:
4406:
4397:
4392:
4388:
4381:
4374:
4367:
4360:
4358:
4277:
4273:
4269:
4267:
4257:
4253:
4249:
4247:
4242:
4238:
4233:
4229:
4225:
4220:
4216:
4214:
4096:
4094:
4037:
4035:
4023:
4018:
4014:
4010:
4008:
3842:
3841:This index (
3840:
3837:
3827:
3825:
3766:
3759:
3752:
3750:
3677:
3675:
3672:Wang's index
3661:
3656:
3652:
3647:
3643:
3636:
3629:
3627:
3557:
3555:
3542:
3535:
3533:
3480:
3467:
3466:
3458:
3451:
3449:
3371:
3361:
3359:
3338:
3336:
3327:
3319:
3315:
3313:
3191:
3185:
3177:
3169:
3167:
3111:
3109:
3007:
3000:
2993:
2986:
2984:
2976:
2971:
2967:
2961:
2890:
2883:
2879:
2877:
2863:
2858:
2854:
2850:
2848:
2764:
2762:
2752:
2750:
2742:
2735:
2728:
2721:
2719:
2609:
2600:
2584:
2240:
2088:
2024:
2020:
2013:
2006:
1999:
1924:
1742:
1739:
1736:
1729:
1714:
1711:
1701:
1688:
1684:
1679:
1675:
1673:
1493:
769:
765:
763:
758:
753:
749:
747:
649:
640:
632:
608:
601:
581:
569:
547:
538:
501:
497:
396:
393:Mathematical
373:
370:Econometrics
332:
325:is bimodal.
318:
316:
311:
305:
296:
292:
288:
284:
280:
278:
223:
214:
210:
200:
192:
184:
170:
153:
129:
108:distribution
107:
104:
98:
90:
86:
75:
64:
41:
9108:Exponential
8957:directional
8946:Directional
8833:Generalized
8804:Multinomial
8759:continuous-
8699:Kaniadakis
8690:Kaniadakis
8681:Kaniadakis
8672:Kaniadakis
8663:Kaniadakis
8615:Tracy–Widom
8592:Skew normal
8574:Noncentral
8358:Log-Laplace
8336:Generalized
8317:Half-normal
8283:Generalized
8247:Logarithmic
8232:Exponential
8186:Chi-squared
8126:U-quadratic
8091:Kumaraswamy
8033:Continuous
7980:Logarithmic
7875:Categorical
7772:30 November
7264:(1): 5–36.
7082:(1): 1–19.
6752:(1): 3–26.
6530:(1): 1–10.
6485:: 199–216.
5261:saddle test
4026:periodogram
2773:unimodality
1732:unimodality
1719:with equal
376:econometric
363:crepuscular
343:weaver ants
126:Terminology
9103:Elliptical
9059:Degenerate
9045:Degenerate
8793:Discrete:
8752:univariate
8607:Student's
8562:Asymmetric
8541:Johnson's
8469:supported
8413:Phase-type
8368:Log-normal
8353:Log-Cauchy
8343:Kolmogorov
8261:Noncentral
8191:Noncentral
8171:Beta prime
8121:Triangular
8116:Reciprocal
8086:Irwin–Hall
8035:univariate
8015:Yule–Simon
7897:Rademacher
7839:univariate
7598:2013-11-01
6969:Pteranodon
6855:: 370–375.
6849:Biometrika
5705:Oecophylla
5470:Biometrika
5431:References
5259:, and the
5118:polynomial
4572:Statistics
3362:vide infra
2769:bimodality
2606:Ashman's D
1998:and where
604:weaver ant
357:, and the
308:reciprocal
105:multimodal
101:statistics
32:Bimodality
8828:Dirichlet
8809:Dirichlet
8719:-Gaussian
8694:-Logistic
8531:Holtsmark
8503:Gaussian
8490:Fisher's
8473:real line
7975:Geometric
7955:Delaporte
7860:Bernoulli
7837:Discrete
7492:121113601
7367:(1): 1–6.
7346:121960832
7332:: 63–70.
7278:118500771
7084:CiteSeerX
6818:: 71–90.
6562:118389173
6537:0711.0216
6234:189822180
5963:White, H.
5666:124087015
5308:astronomy
5257:span test
5253:runt test
5154:≥
5141:−
5050:ϕ
5046:−
5037:ϕ
5019:ϕ
5015:−
5006:ϕ
4948:ϕ
4944:−
4935:ϕ
4917:ϕ
4910:−
4901:ϕ
4888:ϕ
4866:ϕ
4862:−
4853:ϕ
4835:ϕ
4828:−
4819:ϕ
4806:ϕ
4748:ϕ
4744:−
4735:ϕ
4713:ϕ
4709:−
4700:ϕ
4636:ϕ
4623:ϕ
4610:ϕ
4558:logarithm
4480:σ
4460:σ
4432:σ
4310:ϕ
4306:−
4297:ϕ
4170:∑
4127:∑
4111:μ
4069:μ
4065:−
4062:μ
3975:γ
3969:π
3960:
3943:∑
3916:γ
3910:π
3901:
3884:∑
3803:−
3789:δ
3734:σ
3719:μ
3715:−
3706:μ
3691:δ
3603:∑
3409:−
3287:−
3272:−
3248:−
3151:κ
3135:γ
3125:β
3078:σ
3065:σ
3047:μ
3043:−
3034:μ
2921:∑
2823:−
2812:−
2803:−
2688:σ
2670:σ
2648:μ
2644:−
2635:μ
2543:−
2534:
2528:−
2522:
2500:σ
2494:≤
2480:μ
2476:−
2467:μ
2438:σ
2404:σ
2391:σ
2378:≤
2364:μ
2360:−
2351:μ
2309:σ
2296:σ
2270:μ
2266:−
2257:μ
2167:−
2139:−
2108:−
2059:σ
2044:σ
1980:σ
1962:μ
1958:−
1949:μ
1903:−
1871:−
1856:−
1847:
1838:≥
1821:
1815:−
1806:−
1797:
1766:≤
1630:μ
1627:−
1618:∫
1606:ν
1582:μ
1579:−
1570:μ
1557:δ
1511:∫
1505:μ
1465:δ
1447:σ
1432:δ
1411:σ
1401:δ
1370:σ
1347:−
1321:δ
1303:σ
1288:δ
1267:σ
1257:δ
1226:σ
1197:ν
1159:δ
1141:σ
1131:δ
1110:σ
1087:−
1061:δ
1043:σ
1033:δ
1012:σ
983:ν
945:δ
927:σ
914:−
888:δ
870:σ
851:ν
821:μ
811:−
793:μ
783:μ
707:−
623:mutations
518:α
478:α
475:−
429:α
361:of those
351:isoniazid
144:batiphase
140:acrophase
136:amplitude
87:Figure 4.
76:Figure 3.
65:Figure 2.
42:Figure 1.
9186:Category
9161:Category
9093:Circular
9086:Families
9071:Singular
9050:singular
8814:Negative
8761:discrete
8727:-Weibull
8685:-Weibull
8569:Logistic
8453:Discrete
8423:Rayleigh
8403:Nakagami
8326:-squared
8300:Gompertz
8149:interval
7885:Negative
7870:Binomial
7687:22 March
7607:cite web
7457:14500508
7449:22806703
7020:22912914
6921:14953132
6656:24663432
6616:PLOS ONE
6509:19718451
6433:24109465
6282:17153773
6199:13464256
6108:14470055
6066:36234163
5965:(2003).
5923:53495657
5877:21464309
5818:10868777
5810:17637733
5775:20478892
5720:: 7–10.
5579:16775883
5548:(2006).
5408:See also
5333:Software
5241:dip test
4562:Krumbein
4387:are the
4241:bin and
4013:= 2 and
3341:for the
3182:kurtosis
3174:skewness
1698:kurtosis
1694:skewness
1692:are the
399:unimodal
205:and the
189:Examples
132:antimode
92:unimodal
9171:Commons
9143:Wrapped
9138:Tweedie
9133:Pearson
9128:Mixture
9035:Bingham
8934:Complex
8924:Inverse
8914:Wishart
8907:Inverse
8894:Matrix
8868:Inverse
8784:(joint)
8703:-Erlang
8557:Laplace
8448:Weibull
8305:Shifted
8288:Inverse
8273:Fréchet
8196:Inverse
8131:Uniform
8051:Arcsine
8010:Skellam
8005:Poisson
7928:support
7902:Soliton
7855:Benford
7848:support
7734:"agrmt"
7541:2611088
7532:1380036
7243:2290406
7175:2985156
7155:Bibcode
6820:Bibcode
6754:Bibcode
6683:Bibcode
6647:3963849
6624:Bibcode
6585:Bibcode
6542:Bibcode
6500:2730180
6424:3791391
6407:: 700.
6351:Bibcode
6321:2444163
6262:Bibcode
6179:Bibcode
6146:1267357
5868:3093508
5845:Bibcode
5766:2880115
5104:at the
3322:is the
3180:is the
3172:is the
2912:overall
1727:case).
1700:of the
1494:where
615:genomes
578:Biology
382:Origins
335:geysers
48:of two
46:mixture
9077:Cantor
8919:Normal
8750:Mixed
8676:-Gamma
8602:Stable
8552:Landau
8526:Gumbel
8480:Cauchy
8408:Pareto
8220:Erlang
8201:Scaled
8156:Benini
7995:Panjer
7539:
7529:
7490:
7455:
7447:
7386:
7344:
7276:
7241:
7173:
7125:
7086:
7018:
6919:
6654:
6644:
6560:
6507:
6497:
6431:
6421:
6371:
6319:
6280:
6232:
6197:
6144:
6106:
6064:
5944:
5921:
5875:
5865:
5816:
5808:
5773:
5763:
5664:
5577:
5521:363740
5519:
5451:
5397:as in
5255:, the
5251:, the
5243:, the
5239:, the
5172:where
5083:StdDev
5077:where
4501:where
4359:where
4215:where
4095:where
3826:where
3751:where
3628:where
3534:where
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