814:
2079:
2067:
7890:
158:. Using maximum likelihood to estimate these parameters often breaks down, with one parameter tending to the specific value that causes the likelihood to be infinite, rendering the other parameters inconsistent. The method of maximum spacings, however, being dependent on the difference between points on the cumulative distribution function and not individual likelihood points, does not have this issue, and will return valid results over a much wider array of distributions.
7876:
705:
3537:
7914:
7902:
85:, in that a set of independent random samples derived from any random variable should on average be uniformly distributed with respect to the cumulative distribution function of the random variable. The MPS method chooses the parameter values that make the observed data as uniform as possible, according to a specific quantitative measure of uniformity.
20:
449:
3286:
1470:
3770:
165:
seek to analyze flood alleviation methods, which requires accurate models of river flood effects. The distributions that better model these effects are all three-parameter models, which suffer from the infinite likelihood issue described above, leading to Hall's investigation of the maximum spacing
4616:
is logged at this step, the result is always ⤠0, as the difference between two adjacent points on a cumulative distribution is always ⤠1, and strictly < 1 unless there are only two points at the bookends. Also, in section 4.3, on page 392, calculation shows that it is the
1755:
2659:
1935:
1115:
1224:
3635:
700:{\displaystyle {\hat {\theta }}={\underset {\theta \in \Theta }{\operatorname {arg\,max} }}\;S_{n}(\theta ),\quad {\text{where }}\ S_{n}(\theta )=\ln \!\!{\sqrt{D_{1}D_{2}\cdots D_{n+1}}}={\frac {1}{n+1}}\sum _{i=1}^{n+1}\ln {D_{i}}(\theta ).}
423:
4264:
1486:
3532:{\displaystyle {\begin{aligned}\mu _{M}&\approx (n+1)(\ln(n+1)+\gamma )-{\frac {1}{2}}-{\frac {1}{12(n+1)}},\\\sigma _{M}^{2}&\approx (n+1)\left({\frac {\pi ^{2}}{6}}-1\right)-{\frac {1}{2}}-{\frac {1}{6(n+1)}},\end{aligned}}}
3204:
2449:
5003:
inside the logged summation. The extra factors will make a difference in terms of the expected mean and variance of the statistic. For consistency, this article will continue to use the Cheng & Amin/Wong & Li form. --
4969:
2370:
4791:
3022:
4157:
825:
for the simplistic example under both likelihood and spacing estimation. The values for which both likelihood and spacing are maximized, the maximum likelihood and maximum spacing estimates, are identified.
3984:
170:, when comparing the method to maximum likelihood, use various data sets ranging from a set on the oldest ages at death in Sweden between 1905 and 1958 to a set containing annual maximum wind speeds.
3640:
3291:
2899:
2836:
2253:
4653:
1044:
2374:
When the ties are due to multiple observations, the repeated spacings (those that would otherwise be zero) should be replaced by the corresponding likelihood. That is, one should substitute
2042:
1992:
4684:
3630:
4829:
3277:
4435:
285:
4162:
92:(MLE), can break down in various cases, such as involving certain mixtures of continuous distributions. In these cases the method of maximum spacing estimation may be successful.
4376:
1771:
4077:
2444:
2408:
3802:
2705:
2139:
estimators. In particular, in cases where the underlying distribution is J-shaped, maximum likelihood will fail where MSE succeeds. An example of a J-shaped density is the
3911:
3088:
2773:
3235:
4560:
2258:
4034:
5001:
4614:
4587:
4531:
4504:
4011:
3856:
7011:
2908:
7516:
4284:
3876:
3826:
3568:
4159:
has the same asymptotic mean and variance as in the known case. However, the test statistic to be used requires the addition of a bias correction term and is:
1465:{\displaystyle D_{1}={\frac {x_{(1)}-a}{b-a}},\ \ D_{i}={\frac {x_{(i)}-x_{(i-1)}}{b-a}}\ {\text{for }}i=2,\ldots ,n,\ \ D_{n+1}={\frac {b-x_{(n)}}{b-a}}\ \ }
127:
at the true parameter, the âspacingâ between each observation should be uniformly distributed. This would imply that the difference between the values of the
3765:{\displaystyle {\begin{aligned}C_{1}&=\mu _{M}-{\sqrt {\frac {\sigma _{M}^{2}n}{2}}},\\C_{2}&={\sqrt {\frac {\sigma _{M}^{2}}{2n}}},\\\end{aligned}}}
710:
7666:
7290:
120:
4842:
5931:
3026:
The MSE method is also sensitive to secondary clustering. One example of this phenomenon is when a set of observations is thought to come from a single
1750:{\displaystyle S_{n}(a,b)={\tfrac {\ln(x_{(1)}-a)}{n+1}}+{\tfrac {\sum _{i=2}^{n}\ln(x_{(i)}-x_{(i-1)})}{n+1}}+{\tfrac {\ln(b-x_{(n)})}{n+1}}-\ln(b-a)}
7947:
4705:
7064:
5229:
7503:
5563:. Institute of Mathematical Statistics Lecture Notes â Monograph Series. Beachwood, Ohio: Institute of Mathematical Statistic. pp. 272â283.
95:
Apart from its use in pure mathematics and statistics, the trial applications of the method have been reported using data from fields such as
5586:
2654:{\displaystyle \lim _{x_{i}\to x_{i-1}}{\frac {\int _{x_{i-1}}^{x_{i}}f(t;\theta )\,dt}{x_{i}-x_{i-1}}}=f(x_{i-1},\theta )=f(x_{i},\theta ),}
2135:, as the sample size increases to infinity. The consistency of maximum spacing estimation holds under much more general conditions than for
784:
factor in front of the sum and add the âââ sign in order to turn the maximization into minimization. As these are constants with respect to
5926:
5626:
4479:
There appear to be some minor typographical errors in the paper. For example, in section 4.2, equation (4.1), the rounding replacement for
6530:
5678:
2177:
1941:
7918:
4086:
7313:
7205:
2902:
1163:
5490:
5334:
135:
of such spacings, so solving for the parameters that maximize the geometric mean would achieve the âbestâ fit as defined this way.
3916:
3573:
1765:. Differentiating with respect to those parameters and solving the resulting linear system, the maximum spacing estimates will be
7491:
7365:
128:
75:
4450:
140:
7549:
7210:
6955:
6326:
5916:
3828:
6540:
7600:
6812:
6619:
6508:
6466:
4390:
124:
82:
5705:
5020:
3544:
1944:(UMVU) estimators for the continuous uniform distribution. In comparison, the maximum likelihood estimates for this problem
5559:
Wong, T.S.T; Li, W.K. (2006). "A note on the estimation of extreme value distributions using maximum product of spacings".
7843:
6802:
5275:
Cheng, R.C.H.; Amin, N.A.K. (1983). "Estimating parameters in continuous univariate distributions with a shifted origin".
144:
6852:
2841:
2778:
7394:
7343:
7328:
7318:
7187:
7059:
7026:
6807:
6637:
5526:
7463:
6764:
4620:
7942:
7738:
7539:
6518:
6187:
5651:
4037:
2078:
7623:
7590:
1997:
1947:
7595:
7338:
7097:
7003:
6983:
6891:
6602:
6420:
5903:
5775:
4658:
2125:
104:
6769:
6535:
6393:
5312:
Cheng, R.C.H; Stephens, M. A. (1989). "A goodness-of-fit test using Moran's statistic with estimated parameters".
7355:
7123:
6844:
6698:
6627:
6547:
6405:
6386:
6094:
5815:
5460:
Ranneby, Bo (1984). "The maximum spacing method. An estimation method related to the maximum likelihood method".
5169:
4796:
3244:
7468:
257:
sample, that is the result of sorting of all observations from smallest to largest. For convenience also denote
7838:
7605:
7153:
7118:
7082:
6867:
6309:
6218:
6177:
6089:
5780:
5619:
4460:
4014:
3805:
3070:
3035:
844:
6875:
6859:
4396:
3210:, and that a chi-squared approximation exists for small samples. In the case where we know the true parameter
2166:
as maximum likelihood estimators, where the latter exist. However, MSEs may exist in cases where MLEs do not.
813:
7747:
7360:
7300:
7237:
6597:
6459:
6449:
6299:
6213:
2163:
1041:
is actually e, it has to be greater than zero but less than one. Therefore, the only acceptable solution is
1001:
that maximizes the geometric mean of the âdifferenceâ column. Using the convention that ignores taking the (
206:
7508:
7445:
7785:
7715:
7200:
7087:
6084:
5981:
5888:
5767:
5666:
5184:
4310:
further expanded the method to investigate properties of estimators using higher order spacings, where an
1930:{\displaystyle {\hat {a}}={\frac {nx_{(1)}-x_{(n)}}{n-1}},\ \ {\hat {b}}={\frac {nx_{(n)}-x_{(1)}}{n-1}}.}
7906:
6784:
5252:
4317:
3050:) would indicate this secondary clustering effect, and suggesting a closer look at the data is required.
1110:{\displaystyle \mu =0.6\quad \Rightarrow \quad \lambda _{\text{MSE}}={\frac {\ln 0.6}{-2}}\approx 0.255,}
7810:
7752:
7695:
7521:
7414:
7323:
7049:
6933:
6792:
6674:
6666:
6481:
6377:
6355:
6314:
6279:
6246:
6192:
6167:
6122:
6061:
6021:
5823:
5646:
3031:
1005:+1)st root, this turns into the maximization of the following product: (1 â e) ¡ (e â e) ¡ (e). Letting
161:
The distributions that tend to have likelihood issues are often those used to model physical phenomena.
88:
One of the most common methods for estimating the parameters of a distribution from data, the method of
7889:
6779:
1129:â 3.915. For comparison, the maximum likelihood estimate of Îť is the inverse of the sample mean, 3, so
4053:
7733:
7308:
7257:
7233:
7195:
7113:
7092:
7044:
6923:
6901:
6870:
6656:
6607:
6525:
6498:
6454:
6410:
6172:
5948:
5828:
2144:
2140:
2121:
2091:
5189:
2413:
2377:
2174:
Maximum spacing estimators are sensitive to closely spaced observations, and especially ties. Given
7880:
7805:
7728:
7409:
7173:
7166:
7128:
7036:
7016:
6988:
6721:
6587:
6582:
6572:
6564:
6382:
6343:
6233:
6223:
6132:
5911:
5867:
5785:
5710:
5612:
3775:
3280:
3027:
2664:
2110:
approaches 10, rendering the estimates of the other parameters inconsistent. Note that there is no
7455:
3881:
2752:
7894:
7705:
7559:
7404:
7280:
7177:
7161:
7138:
6915:
6649:
6632:
6592:
6503:
6398:
6360:
6331:
6291:
6251:
6197:
6114:
5800:
5795:
5592:
5564:
5477:
5448:
5300:
5202:
4455:
3213:
3039:
2156:
2136:
2103:
2045:
418:{\displaystyle D_{i}(\theta )=F(x_{(i)};\,\theta )-F(x_{(i-1)};\,\theta ),\quad i=1,\ldots ,n+1.}
116:
89:
4536:
4698:
The literature refers to related statistics as Moran or Moran-Darling statistics. For example,
7800:
7770:
7762:
7582:
7573:
7498:
7429:
7285:
7270:
7245:
7133:
7074:
6940:
6928:
6554:
6471:
6415:
6338:
6182:
6104:
5883:
5757:
5582:
5502:
5469:
5440:
5405:
5392:
5346:
5292:
5244:
4259:{\displaystyle T({\hat {\theta }})={\frac {M({\hat {\theta }})+{\frac {k}{2}}-C_{1}}{C_{2}}},}
818:
439:
5371:"The construction of confidence intervals for frequency analysis using resampling techniques"
4019:
7825:
7780:
7544:
7531:
7424:
7399:
7333:
7265:
7143:
6751:
6644:
6577:
6490:
6437:
6256:
6127:
5921:
5720:
5687:
5574:
5541:
5432:
5411:
5382:
5321:
5284:
5194:
2111:
63:
4974:
4592:
4565:
4509:
4482:
3989:
3834:
3034:
normals with different means. A second example is when the data is thought to come from an
7742:
7486:
7348:
7275:
6950:
6824:
6797:
6774:
6743:
6370:
6365:
6319:
6049:
5700:
3082:
2746:
2148:
2099:
2095:
282:
as the âgapsâ between the values of the distribution function at adjacent ordered points:
254:
155:
5370:
23:
The maximum spacing method tries to find a distribution function such that the spacings,
5510:
5354:
7691:
7686:
6149:
6079:
5725:
5436:
5288:
4438:
4269:
3861:
3811:
3553:
443:
150:
There are certain distributions, especially those with three or more parameters, whose
132:
67:
35:
7936:
7848:
7815:
7678:
7639:
7450:
7419:
6883:
6837:
6442:
6144:
5971:
5735:
5730:
5206:
3199:{\displaystyle S_{n}(\theta )=M_{n}(\theta )=-\sum _{j=1}^{n+1}\ln {D_{j}(\theta )},}
182:
6001:
5596:
3042:. In the latter case, smaller spacings may occur in the lower tail. A high value of
7790:
7723:
7700:
7615:
6945:
6241:
6139:
6074:
6016:
5938:
5893:
4964:{\displaystyle \scriptstyle M_{n}=-\sum _{j=0}^{n}\ln {((n+1)(X_{n,i+1}-X_{n,i}))}}
4303:
100:
2365:{\displaystyle D_{i+k}(\theta )=D_{i+k-1}(\theta )=\cdots =D_{i+1}(\theta )=0.\,}
805:
This section presents two examples of calculating the maximum spacing estimator.
131:
at consecutive observations should be equal. This is the case that maximizes the
7833:
7795:
7478:
7379:
7241:
7054:
7021:
6513:
6430:
6425:
6069:
6026:
6006:
5986:
5976:
5745:
5410:. IEEE International Conference on Image Processing. Paris. pp. 1743â1747.
5578:
5545:
4786:{\displaystyle \scriptstyle M(\theta )=-\sum _{j=1}^{n+1}\log {D_{i}(\theta )}}
2066:
6679:
6159:
5859:
5790:
5740:
5715:
5635:
5415:
5325:
5198:
232:
151:
43:
5506:
5473:
5444:
5396:
5350:
5296:
5248:
788:, the modifications do not alter the location of the maximum of the function
115:
The MSE method was derived independently by Russel Cheng and Nik Amin at the
34:, are all approximately of the same length. This is done by maximizing their
6832:
6684:
6304:
6099:
6011:
5996:
5991:
5956:
5387:
3207:
3017:{\displaystyle D_{j}={\frac {y_{U}-y_{L}}{r-1}}\quad (j=i+1,\ldots ,i+r-1).}
2090:
Plot of a âJ-shapedâ density function and its corresponding distribution. A
96:
5369:
Hall, M.J.; van den
Boogaard, H.F.P.; Fernando, R.C.; Mynett, A.E. (2004).
866:> 0. In order to construct the MSE we have to first find the spacings:
6348:
5966:
5843:
5838:
5833:
5805:
4437:, they discuss two alternative approaches: a geometric approach based on
731:
4441:
and a probabilistic approach based on a ânearest neighbor ballâ metric.
1474:
Calculating the geometric mean and then taking the logarithm, statistic
7853:
7554:
5569:
5481:
5452:
5407:
The maximum spacing noise estimation in single-coil background MRI data
5304:
5228:
Beirlant, J.; Dudewicz, E.J.; GyĂśrfi, L.; van der Meulen, E.C. (1997).
2775:. The corresponding points on the distribution should now fall between
5210:
7775:
6756:
6730:
6710:
5961:
5752:
139:
justified the method by demonstrating that it is an estimator of the
19:
2901:. Cheng and Stephens suggest assuming that the rounded values are
5525:
Ranneby, Bo; Jammalamadakab, S. Rao; Teterukovskiy, Alex (2005).
74:
in the data, which are the differences between the values of the
5695:
1117:
which corresponds to an exponential distribution with a mean of
147:, but with more robust properties for some classes of problems.
7664:
7231:
6978:
6277:
6047:
5664:
5608:
4152:{\displaystyle S_{n}({\hat {\theta }})=M_{n}({\hat {\theta }})}
5527:"The maximum spacing estimation for multivariate observations"
5237:
International
Journal of Mathematical and Statistical Sciences
4506:, should not have the log term. In section 1, equation (1.2),
179:
5604:
3979:{\displaystyle T(\theta )={\frac {M(\theta )-C_{1}}{C_{2}}}}
62:, is a method for estimating the parameters of a univariate
5561:
Time series and related topics: in memory of Ching-Zong Wei
4655:
which has MPS estimate of 6.87, not the standard deviation
2159:
rendering estimates of the other parameters inconsistent.
2106:
of 10. The density asymptotically approaches infinity as
5170:"An alternative to maximum likelihood based on spacings"
3085:. It has been shown that the statistic, when defined as
2749:. All of the true values should then fall in the range
730:+1), and thus the maximum has to exist at least in the
117:
University of Wales
Institute of Science and Technology
5491:"Maximum spacing estimates based on different metrics"
4846:
4800:
4709:
4662:
4624:
3248:
2001:
1951:
1671:
1575:
1519:
4977:
4845:
4799:
4708:
4661:
4623:
4595:
4568:
4539:
4512:
4485:
4399:
4320:
4272:
4165:
4089:
4056:
4022:
3992:
3919:
3884:
3864:
3837:
3814:
3778:
3638:
3576:
3556:
3550:
The distribution can also be approximated by that of
3289:
3247:
3216:
3091:
2911:
2844:
2781:
2755:
2667:
2452:
2416:
2380:
2261:
2180:
2000:
1950:
1774:
1489:
1227:
1047:
452:
288:
16:
Method of estimating a statistical model's parameters
7517:
Autoregressive conditional heteroskedasticity (ARCH)
2714:
suggest another method to remove the effects. Given
7824:
7761:
7714:
7677:
7632:
7614:
7581:
7572:
7530:
7477:
7438:
7387:
7378:
7299:
7256:
7186:
7152:
7106:
7073:
7035:
7002:
6914:
6823:
6742:
6697:
6665:
6618:
6563:
6489:
6480:
6290:
6232:
6206:
6158:
6113:
6060:
5947:
5902:
5876:
5858:
5814:
5766:
5686:
5677:
2894:{\displaystyle y_{U}=F(x+\delta ,{\hat {\theta }})}
2831:{\displaystyle y_{L}=F(x-\delta ,{\hat {\theta }})}
5425:Journal of the Royal Statistical Society, Series B
5277:Journal of the Royal Statistical Society, Series B
4995:
4963:
4823:
4785:
4678:
4647:
4608:
4581:
4554:
4525:
4498:
4429:
4370:
4278:
4258:
4151:
4071:
4028:
4005:
3978:
3905:
3870:
3850:
3820:
3796:
3764:
3624:
3562:
3531:
3271:
3229:
3198:
3016:
2893:
2830:
2767:
2699:
2653:
2438:
2402:
2364:
2248:{\displaystyle X_{i+k}=X_{i+k-1}=\cdots =X_{i},\,}
2247:
2151:less than 1. The density will tend to infinity as
2036:
1986:
1929:
1749:
1464:
1109:
699:
417:
81:The concept underlying the method is based on the
4648:{\displaystyle \textstyle {\tilde {\sigma ^{2}}}}
562:
561:
209:with continuous cumulative distribution function
5099:
4389:discuss extended maximum spacing methods to the
4302:generalized the MSE method to approximate other
2454:
2162:Maximum spacing estimators are also at least as
1009:= e, the problem becomes finding the maximum of
7065:Multivariate adaptive regression splines (MARS)
5230:"Nonparametric entropy estimation: an overview"
5168:Anatolyev, Stanislav; Kosenok, Grigory (2005).
2037:{\displaystyle \scriptstyle {\hat {b}}=x_{(n)}}
1987:{\displaystyle \scriptstyle {\hat {a}}=x_{(1)}}
1757:Here only three terms depend on the parameters
1221:â. Therefore, individual spacings are given by
1037:= 0. This equation has roots 0, 0.6, and 1. As
154:may become infinite along certain paths in the
5269:Note: linked paper is an updated 2001 version.
4299:
5620:
5534:Journal of Statistical Planning and Inference
5495:University of UmeĂĽ, Department of Mathematics
5339:University of UmeĂĽ, Department of Mathematics
5152:
4699:
4679:{\displaystyle \textstyle {\tilde {\sigma }}}
4286:is the number of parameters in the estimate.
4080:
4040:of the appropriate chi-squared distribution.
3238:
2711:
768:
8:
3625:{\displaystyle A=C_{1}+C_{2}\chi _{n}^{2}\,}
711:inequality of arithmetic and geometric means
4836:
4824:{\displaystyle \scriptstyle D_{i}(\theta )}
3272:{\displaystyle \scriptstyle M_{n}(\theta )}
737:Note that some authors define the function
121:Swedish University of Agricultural Sciences
60:maximum product of spacing estimation (MPS)
7674:
7661:
7578:
7384:
7253:
7228:
6999:
6975:
6703:
6486:
6287:
6274:
6057:
6044:
5683:
5674:
5661:
5627:
5613:
5605:
4386:
1185:. The cumulative distribution function is
504:
66:. The method requires maximization of the
5568:
5386:
5188:
5061:
4976:
4941:
4916:
4890:
4878:
4867:
4851:
4844:
4805:
4798:
4766:
4761:
4743:
4732:
4707:
4664:
4663:
4660:
4632:
4626:
4625:
4622:
4600:
4594:
4573:
4567:
4538:
4533:is defined to be the spacing itself, and
4517:
4511:
4490:
4484:
4406:
4402:
4401:
4398:
4359:
4331:
4319:
4271:
4245:
4234:
4217:
4200:
4199:
4190:
4173:
4172:
4164:
4135:
4134:
4125:
4104:
4103:
4094:
4088:
4058:
4057:
4055:
4021:
3997:
3991:
3968:
3957:
3935:
3918:
3883:
3863:
3842:
3836:
3813:
3788:
3783:
3777:
3738:
3733:
3726:
3713:
3686:
3681:
3673:
3664:
3647:
3639:
3637:
3621:
3615:
3610:
3600:
3587:
3575:
3555:
3495:
3482:
3457:
3451:
3418:
3413:
3375:
3362:
3298:
3290:
3288:
3253:
3246:
3221:
3215:
3177:
3172:
3154:
3143:
3118:
3096:
3090:
2945:
2932:
2925:
2916:
2910:
2877:
2876:
2849:
2843:
2814:
2813:
2786:
2780:
2754:
2685:
2672:
2666:
2633:
2599:
2571:
2558:
2545:
2519:
2514:
2501:
2496:
2489:
2475:
2462:
2457:
2451:
2421:
2415:
2385:
2379:
2361:
2334:
2294:
2266:
2260:
2244:
2235:
2204:
2185:
2179:
2021:
2003:
2002:
1999:
1971:
1953:
1952:
1949:
1898:
1879:
1869:
1855:
1854:
1819:
1800:
1790:
1776:
1775:
1773:
1693:
1670:
1631:
1612:
1593:
1582:
1574:
1535:
1518:
1494:
1488:
1430:
1417:
1402:
1363:
1328:
1309:
1302:
1293:
1248:
1241:
1232:
1226:
1075:
1066:
1046:
678:
673:
655:
644:
622:
606:
593:
580:
570:
563:
540:
528:
509:
480:
470:
468:
454:
453:
451:
438:is defined as a value that maximizes the
377:
356:
336:
321:
293:
287:
4430:{\displaystyle \mathbb {R} ^{k}(k>1)}
868:
812:
123:. The authors explained that due to the
18:
5335:"Generalized maximum spacing estimates"
5110:
5088:
5077:
5043:
4472:
4393:case. As there is no natural order for
4307:
751:
750:) somewhat differently. In particular,
162:
136:
7591:KaplanâMeier estimator (product limit)
5148:
5146:
5144:
5142:
5140:
5121:
5057:
5055:
5053:
5051:
5049:
5047:
5016:
4832:
4306:besides the KullbackâLeibler measure.
167:
5489:Ranneby, Bo; EkstrĂśm, Magnus (1997).
5073:
5071:
5069:
2710:When ties are due to rounding error,
997:The process continues by finding the
7:
7901:
7601:Accelerated failure time (AFT) model
5132:
4835:use the same form as well. However,
3831:. Therefore, to test the hypothesis
2128:to the true value of the parameter,
7913:
7196:Analysis of variance (ANOVA, anova)
5375:Hydrology and Earth System Sciences
4562:is the negative sum of the logs of
4371:{\displaystyle F(X_{j+m})-F(X_{j})}
4314:-order spacing would be defined as
3878:values comes from the distribution
2120:The maximum spacing estimator is a
1942:uniformly minimum variance unbiased
7291:CochranâMantelâHaenszel statistics
5917:Pearson product-moment correlation
5462:Scandinavian Journal of Statistics
5437:10.1111/j.2517-6161.1965.tb00602.x
5289:10.1111/j.2517-6161.1983.tb01268.x
498:
487:
484:
481:
477:
474:
471:
231:â Î is an unknown parameter to be
14:
5423:Pyke, Ronald (1965). "Spacings".
4036:if the value is greater than the
2114:in the graph of the distribution.
7948:Probability distribution fitting
7912:
7900:
7888:
7875:
7874:
4971:, with the additional factor of
4072:{\displaystyle {\hat {\theta }}}
3547:which is approximately 0.57722.
2077:
2065:
129:cumulative distribution function
76:cumulative distribution function
7550:Least-squares spectral analysis
4295:Alternate measures and spacings
2965:
1162:} is the ordered sample from a
1061:
1057:
726:) is bounded from above by âln(
527:
387:
6531:Mean-unbiased minimum-variance
5100:Anatolyev & Kosenok (2004)
4990:
4978:
4956:
4953:
4909:
4906:
4894:
4891:
4817:
4811:
4778:
4772:
4719:
4713:
4669:
4638:
4549:
4543:
4424:
4412:
4365:
4352:
4343:
4324:
4211:
4205:
4196:
4184:
4178:
4169:
4146:
4140:
4131:
4115:
4109:
4100:
4063:
3947:
3941:
3929:
3923:
3900:
3888:
3516:
3504:
3443:
3431:
3396:
3384:
3356:
3347:
3335:
3326:
3323:
3311:
3265:
3259:
3189:
3183:
3130:
3124:
3108:
3102:
3008:
2966:
2905:in this interval, by defining
2888:
2882:
2861:
2825:
2819:
2798:
2645:
2626:
2617:
2592:
2542:
2530:
2468:
2439:{\displaystyle D_{i}(\theta )}
2433:
2427:
2403:{\displaystyle f_{i}(\theta )}
2397:
2391:
2352:
2346:
2318:
2312:
2284:
2278:
2028:
2022:
2008:
1978:
1972:
1958:
1905:
1899:
1886:
1880:
1860:
1826:
1820:
1807:
1801:
1781:
1744:
1732:
1705:
1700:
1694:
1680:
1649:
1644:
1632:
1619:
1613:
1605:
1553:
1542:
1536:
1528:
1512:
1500:
1437:
1431:
1341:
1329:
1316:
1310:
1255:
1249:
1058:
691:
685:
552:
546:
521:
515:
459:
381:
369:
357:
349:
340:
328:
322:
314:
305:
299:
125:probability integral transform
83:probability integral transform
1:
7844:Geographic information system
7060:Simultaneous equations models
3797:{\displaystyle \chi _{n}^{2}}
3081:), which can be used to test
2700:{\displaystyle x_{i}=x_{i-1}}
145:maximum likelihood estimation
78:at neighbouring data points.
7027:Coefficient of determination
6638:Uniformly most powerful test
4300:Ranneby & EkstrĂśm (1997)
3906:{\displaystyle F(x,\theta )}
3073:or Moran-Darling statistic,
3038:, but actually comes from a
2768:{\displaystyle x\pm \delta }
7596:Proportional hazards models
7540:Spectral density estimation
7522:Vector autoregression (VAR)
6956:Maximum posterior estimator
6188:Randomized controlled trial
5153:Cheng & Stephens (1989)
5023:from their description. --
4700:Cheng & Stephens (1989)
4451:KullbackâLeibler divergence
4290:Generalized maximum spacing
4081:Cheng & Stephens (1989)
3239:Cheng & Stephens (1989)
3230:{\displaystyle \theta ^{0}}
3030:, but in fact comes from a
2712:Cheng & Stephens (1989)
2044:are biased and have higher
862:⼠0 with unknown parameter
769:Cheng & Stephens (1989)
141:KullbackâLeibler divergence
7964:
7356:Multivariate distributions
5776:Average absolute deviation
5579:10.1214/074921706000001102
5546:10.1016/j.jspi.2004.06.059
4555:{\displaystyle M(\theta )}
4382:Multivariate distributions
2057:Consistency and efficiency
1940:These are known to be the
843:= 4 were sampled from the
105:magnetic resonance imaging
48:maximum spacing estimation
7870:
7673:
7660:
7344:Structural equation model
7252:
7227:
6998:
6974:
6706:
6680:Score/Lagrange multiplier
6286:
6273:
6095:Sample size determination
6056:
6043:
5673:
5660:
5642:
5416:10.1109/icip.2014.7025349
5199:10.1017/S0266466605050255
5021:EulerâMascheroni constant
4837:Beirlant & al. (2001)
3545:EulerâMascheroni constant
1177:) with unknown endpoints
429:maximum spacing estimator
7839:Environmental statistics
7361:Elliptical distributions
7154:Generalized linear model
7083:Simple linear regression
6853:HodgesâLehmann estimator
6310:Probability distribution
6219:Stochastic approximation
5781:Coefficient of variation
5404:Pieciak, Tomasz (2014).
5333:EkstrĂśm, Magnus (1997).
4461:Probability distribution
4387:Ranneby & al. (2005)
4013:should be rejected with
3986:can be calculated. Then
3858:that a random sample of
3806:chi-squared distribution
3241:show that the statistic
3036:exponential distribution
2164:asymptotically efficient
2126:converges in probability
845:exponential distribution
119:, and Bo Ranneby at the
7499:Cross-correlation (XCF)
7107:Non-standard predictors
6541:LehmannâScheffĂŠ theorem
6214:Adaptive clinical trial
5388:10.5194/hess-8-235-2004
5326:10.1093/biomet/76.2.385
5062:Cheng & Amin (1983)
4029:{\displaystyle \alpha }
2718:tied observations from
1021:. Differentiating, the
253:} be the corresponding
207:univariate distribution
7895:Mathematics portal
7716:Engineering statistics
7624:NelsonâAalen estimator
7201:Analysis of covariance
7088:Ordinary least squares
7012:Pearson product-moment
6416:Statistical functional
6327:Empirical distribution
6160:Controlled experiments
5889:Frequency distribution
5667:Descriptive statistics
4997:
4965:
4883:
4825:
4787:
4754:
4680:
4649:
4610:
4583:
4556:
4527:
4500:
4431:
4372:
4280:
4260:
4153:
4073:
4050:is being estimated by
4030:
4007:
3980:
3907:
3872:
3852:
3822:
3798:
3766:
3626:
3564:
3533:
3273:
3231:
3200:
3165:
3018:
2895:
2832:
2769:
2701:
2655:
2440:
2404:
2366:
2249:
2038:
1988:
1931:
1751:
1598:
1466:
1111:
826:
701:
666:
419:
39:
7811:Population statistics
7753:System identification
7487:Autocorrelation (ACF)
7415:Exponential smoothing
7329:Discriminant analysis
7324:Canonical correlation
7188:Partition of variance
7050:Regression validation
6894:(JonckheereâTerpstra)
6793:Likelihood-ratio test
6482:Frequentist inference
6394:Locationâscale family
6315:Sampling distribution
6280:Statistical inference
6247:Cross-sectional study
6234:Observational studies
6193:Randomized experiment
6022:Stem-and-leaf display
5824:Central limit theorem
5089:Hall & al. (2004)
4998:
4996:{\displaystyle (n+1)}
4966:
4863:
4831:is defined as above.
4826:
4788:
4728:
4681:
4650:
4611:
4609:{\displaystyle D_{j}}
4584:
4582:{\displaystyle D_{j}}
4557:
4528:
4526:{\displaystyle D_{j}}
4501:
4499:{\displaystyle D_{j}}
4432:
4373:
4281:
4261:
4154:
4074:
4031:
4008:
4006:{\displaystyle H_{0}}
3981:
3908:
3873:
3853:
3851:{\displaystyle H_{0}}
3823:
3799:
3767:
3627:
3565:
3534:
3274:
3232:
3208:asymptotically normal
3201:
3139:
3019:
2896:
2833:
2770:
2702:
2656:
2441:
2405:
2367:
2250:
2039:
1989:
1932:
1752:
1578:
1467:
1112:
816:
702:
640:
420:
163:Hall & al. (2004)
22:
7734:Probabilistic design
7319:Principal components
7162:Exponential families
7114:Nonlinear regression
7093:General linear model
7055:Mixed effects models
7045:Errors and residuals
7022:Confounding variable
6924:Bayesian probability
6902:Van der Waerden test
6892:Ordered alternative
6657:Multiple comparisons
6536:RaoâBlackwellization
6499:Estimating equations
6455:Statistical distance
6173:Factorial experiment
5706:Arithmetic-Geometric
5513:on February 14, 2007
5357:on February 14, 2007
5122:Wong & Li (2006)
5017:Wong & Li (2006)
4975:
4843:
4833:Wong & Li (2006)
4797:
4706:
4659:
4621:
4593:
4566:
4537:
4510:
4483:
4397:
4318:
4270:
4163:
4087:
4054:
4020:
3990:
3917:
3882:
3862:
3835:
3812:
3776:
3636:
3574:
3554:
3287:
3245:
3214:
3089:
3069:) is also a form of
2909:
2842:
2779:
2753:
2665:
2450:
2414:
2378:
2259:
2178:
2141:Weibull distribution
2122:consistent estimator
1998:
1948:
1772:
1487:
1225:
1164:uniform distribution
1045:
450:
446:of sample spacings:
286:
168:Wong & Li (2006)
7806:Official statistics
7729:Methods engineering
7410:Seasonal adjustment
7178:Poisson regressions
7098:Bayesian regression
7037:Regression analysis
7017:Partial correlation
6989:Regression analysis
6588:Prediction interval
6583:Likelihood interval
6573:Confidence interval
6565:Interval estimation
6526:Unbiased estimators
6344:Model specification
6224:Up-and-down designs
5912:Partial correlation
5868:Index of dispersion
5786:Interquartile range
3793:
3743:
3691:
3620:
3423:
3281:normal distribution
3028:normal distribution
2526:
829:Suppose two values
7943:Estimation methods
7826:Spatial statistics
7706:Medical statistics
7606:First hitting time
7560:Whittle likelihood
7211:Degrees of freedom
7206:Multivariate ANOVA
7139:Heteroscedasticity
6951:Bayesian estimator
6916:Bayesian inference
6765:KolmogorovâSmirnov
6650:Randomization test
6620:Testing hypotheses
6593:Tolerance interval
6504:Maximum likelihood
6399:Exponential family
6332:Density estimation
6292:Statistical theory
6252:Natural experiment
6198:Scientific control
6115:Survey methodology
5801:Standard deviation
5177:Econometric Theory
4993:
4961:
4960:
4821:
4820:
4783:
4782:
4676:
4675:
4645:
4644:
4606:
4579:
4552:
4523:
4496:
4456:Maximum likelihood
4427:
4368:
4276:
4256:
4149:
4069:
4026:
4003:
3976:
3903:
3868:
3848:
3829:degrees of freedom
3818:
3794:
3779:
3762:
3760:
3729:
3677:
3622:
3606:
3560:
3529:
3527:
3409:
3269:
3268:
3227:
3196:
3040:gamma distribution
3014:
2891:
2828:
2765:
2697:
2651:
2492:
2488:
2436:
2400:
2362:
2245:
2157:location parameter
2137:maximum likelihood
2104:location parameter
2046:mean-squared error
2034:
2033:
1984:
1983:
1927:
1747:
1721:
1665:
1569:
1462:
1107:
827:
697:
502:
415:
90:maximum likelihood
40:
7928:
7927:
7866:
7865:
7862:
7861:
7801:National accounts
7771:Actuarial science
7763:Social statistics
7656:
7655:
7652:
7651:
7648:
7647:
7583:Survival function
7568:
7567:
7430:Granger causality
7271:Contingency table
7246:Survival analysis
7223:
7222:
7219:
7218:
7075:Linear regression
6970:
6969:
6966:
6965:
6941:Credible interval
6910:
6909:
6693:
6692:
6509:Method of moments
6378:Parametric family
6339:Statistical model
6269:
6268:
6265:
6264:
6183:Random assignment
6105:Statistical power
6039:
6038:
6035:
6034:
5884:Contingency table
5854:
5853:
5721:Generalized/power
5588:978-0-940600-68-3
5271:
4702:analyze the form
4672:
4641:
4279:{\displaystyle k}
4251:
4225:
4208:
4181:
4143:
4112:
4066:
3974:
3871:{\displaystyle n}
3821:{\displaystyle n}
3753:
3752:
3700:
3699:
3563:{\displaystyle A}
3520:
3490:
3466:
3400:
3370:
2963:
2885:
2822:
2584:
2453:
2143:, specifically a
2011:
1961:
1922:
1863:
1853:
1850:
1843:
1784:
1720:
1664:
1568:
1483:will be equal to
1461:
1458:
1454:
1397:
1394:
1366:
1362:
1358:
1288:
1285:
1278:
1096:
1069:
995:
994:
638:
617:
535:
531:
469:
462:
111:History and usage
64:statistical model
7955:
7916:
7915:
7904:
7903:
7893:
7892:
7878:
7877:
7781:Crime statistics
7675:
7662:
7579:
7545:Fourier analysis
7532:Frequency domain
7512:
7459:
7425:Structural break
7385:
7334:Cluster analysis
7281:Log-linear model
7254:
7229:
7170:
7144:Homoscedasticity
7000:
6976:
6895:
6887:
6879:
6878:(KruskalâWallis)
6863:
6848:
6803:Cross validation
6788:
6770:AndersonâDarling
6717:
6704:
6675:Likelihood-ratio
6667:Parametric tests
6645:Permutation test
6628:1- & 2-tails
6519:Minimum distance
6491:Point estimation
6487:
6438:Optimal decision
6389:
6288:
6275:
6257:Quasi-experiment
6207:Adaptive designs
6058:
6045:
5922:Rank correlation
5684:
5675:
5662:
5629:
5622:
5615:
5606:
5600:
5572:
5555:
5553:
5552:
5540:(1â2): 427â446.
5531:
5521:
5519:
5518:
5509:. Archived from
5485:
5456:
5419:
5400:
5390:
5365:
5363:
5362:
5353:. Archived from
5329:
5308:
5267:
5266:
5264:
5263:
5257:
5251:. Archived from
5234:
5224:
5222:
5221:
5215:
5209:. Archived from
5192:
5174:
5155:
5150:
5135:
5130:
5124:
5119:
5113:
5108:
5102:
5097:
5091:
5086:
5080:
5075:
5064:
5059:
5027:
5014:
5008:
5002:
5000:
4999:
4994:
4970:
4968:
4967:
4962:
4959:
4952:
4951:
4933:
4932:
4882:
4877:
4856:
4855:
4830:
4828:
4827:
4822:
4810:
4809:
4792:
4790:
4789:
4784:
4781:
4771:
4770:
4753:
4742:
4696:
4690:
4685:
4683:
4682:
4677:
4674:
4673:
4665:
4654:
4652:
4651:
4646:
4643:
4642:
4637:
4636:
4627:
4615:
4613:
4612:
4607:
4605:
4604:
4588:
4586:
4585:
4580:
4578:
4577:
4561:
4559:
4558:
4553:
4532:
4530:
4529:
4524:
4522:
4521:
4505:
4503:
4502:
4497:
4495:
4494:
4477:
4436:
4434:
4433:
4428:
4411:
4410:
4405:
4377:
4375:
4374:
4369:
4364:
4363:
4342:
4341:
4285:
4283:
4282:
4277:
4265:
4263:
4262:
4257:
4252:
4250:
4249:
4240:
4239:
4238:
4226:
4218:
4210:
4209:
4201:
4191:
4183:
4182:
4174:
4158:
4156:
4155:
4150:
4145:
4144:
4136:
4130:
4129:
4114:
4113:
4105:
4099:
4098:
4078:
4076:
4075:
4070:
4068:
4067:
4059:
4035:
4033:
4032:
4027:
4012:
4010:
4009:
4004:
4002:
4001:
3985:
3983:
3982:
3977:
3975:
3973:
3972:
3963:
3962:
3961:
3936:
3913:, the statistic
3912:
3910:
3909:
3904:
3877:
3875:
3874:
3869:
3857:
3855:
3854:
3849:
3847:
3846:
3827:
3825:
3824:
3819:
3803:
3801:
3800:
3795:
3792:
3787:
3771:
3769:
3768:
3763:
3761:
3754:
3751:
3742:
3737:
3728:
3727:
3718:
3717:
3701:
3695:
3690:
3685:
3675:
3674:
3669:
3668:
3652:
3651:
3631:
3629:
3628:
3623:
3619:
3614:
3605:
3604:
3592:
3591:
3569:
3567:
3566:
3561:
3538:
3536:
3535:
3530:
3528:
3521:
3519:
3496:
3491:
3483:
3478:
3474:
3467:
3462:
3461:
3452:
3422:
3417:
3401:
3399:
3376:
3371:
3363:
3303:
3302:
3278:
3276:
3275:
3270:
3258:
3257:
3236:
3234:
3233:
3228:
3226:
3225:
3205:
3203:
3202:
3197:
3192:
3182:
3181:
3164:
3153:
3123:
3122:
3101:
3100:
3023:
3021:
3020:
3015:
2964:
2962:
2951:
2950:
2949:
2937:
2936:
2926:
2921:
2920:
2903:uniformly spaced
2900:
2898:
2897:
2892:
2887:
2886:
2878:
2854:
2853:
2837:
2835:
2834:
2829:
2824:
2823:
2815:
2791:
2790:
2774:
2772:
2771:
2766:
2706:
2704:
2703:
2698:
2696:
2695:
2677:
2676:
2660:
2658:
2657:
2652:
2638:
2637:
2610:
2609:
2585:
2583:
2582:
2581:
2563:
2562:
2552:
2525:
2524:
2523:
2513:
2512:
2511:
2490:
2487:
2486:
2485:
2467:
2466:
2445:
2443:
2442:
2437:
2426:
2425:
2409:
2407:
2406:
2401:
2390:
2389:
2371:
2369:
2368:
2363:
2345:
2344:
2311:
2310:
2277:
2276:
2254:
2252:
2251:
2246:
2240:
2239:
2221:
2220:
2196:
2195:
2112:inflection point
2081:
2069:
2043:
2041:
2040:
2035:
2032:
2031:
2013:
2012:
2004:
1993:
1991:
1990:
1985:
1982:
1981:
1963:
1962:
1954:
1936:
1934:
1933:
1928:
1923:
1921:
1910:
1909:
1908:
1890:
1889:
1870:
1865:
1864:
1856:
1851:
1848:
1844:
1842:
1831:
1830:
1829:
1811:
1810:
1791:
1786:
1785:
1777:
1756:
1754:
1753:
1748:
1722:
1719:
1708:
1704:
1703:
1672:
1666:
1663:
1652:
1648:
1647:
1623:
1622:
1597:
1592:
1576:
1570:
1567:
1556:
1546:
1545:
1520:
1499:
1498:
1471:
1469:
1468:
1463:
1459:
1456:
1455:
1453:
1442:
1441:
1440:
1418:
1413:
1412:
1395:
1392:
1367:
1364:
1360:
1359:
1357:
1346:
1345:
1344:
1320:
1319:
1303:
1298:
1297:
1286:
1283:
1279:
1277:
1266:
1259:
1258:
1242:
1237:
1236:
1128:
1127:
1121:
1116:
1114:
1113:
1108:
1097:
1095:
1087:
1076:
1071:
1070:
1067:
1025:has to satisfy 5
869:
783:
782:
775:
763:by a factor of (
754:multiplies each
706:
704:
703:
698:
684:
683:
682:
665:
654:
639:
637:
623:
618:
616:
605:
604:
603:
585:
584:
575:
574:
564:
545:
544:
533:
532:
529:
514:
513:
503:
501:
490:
464:
463:
455:
424:
422:
421:
416:
373:
372:
332:
331:
298:
297:
7963:
7962:
7958:
7957:
7956:
7954:
7953:
7952:
7933:
7932:
7929:
7924:
7887:
7858:
7820:
7757:
7743:quality control
7710:
7692:Clinical trials
7669:
7644:
7628:
7616:Hazard function
7610:
7564:
7526:
7510:
7473:
7469:BreuschâGodfrey
7457:
7434:
7374:
7349:Factor analysis
7295:
7276:Graphical model
7248:
7215:
7182:
7168:
7148:
7102:
7069:
7031:
6994:
6993:
6962:
6906:
6893:
6885:
6877:
6861:
6846:
6825:Rank statistics
6819:
6798:Model selection
6786:
6744:Goodness of fit
6738:
6715:
6689:
6661:
6614:
6559:
6548:Median unbiased
6476:
6387:
6320:Order statistic
6282:
6261:
6228:
6202:
6154:
6109:
6052:
6050:Data collection
6031:
5943:
5898:
5872:
5850:
5810:
5762:
5679:Continuous data
5669:
5656:
5638:
5633:
5603:
5589:
5558:
5550:
5548:
5529:
5524:
5516:
5514:
5488:
5459:
5422:
5403:
5368:
5360:
5358:
5332:
5311:
5274:
5261:
5259:
5255:
5232:
5227:
5219:
5217:
5213:
5190:10.1.1.494.7340
5172:
5167:
5163:
5158:
5151:
5138:
5131:
5127:
5120:
5116:
5109:
5105:
5098:
5094:
5087:
5083:
5076:
5067:
5060:
5045:
5041:
5036:
5031:
5030:
5015:
5011:
4973:
4972:
4937:
4912:
4847:
4841:
4840:
4801:
4795:
4794:
4762:
4704:
4703:
4697:
4693:
4657:
4656:
4628:
4619:
4618:
4596:
4591:
4590:
4569:
4564:
4563:
4535:
4534:
4513:
4508:
4507:
4486:
4481:
4480:
4478:
4474:
4469:
4447:
4439:Dirichlet cells
4400:
4395:
4394:
4384:
4355:
4327:
4316:
4315:
4297:
4292:
4268:
4267:
4241:
4230:
4192:
4161:
4160:
4121:
4090:
4085:
4084:
4052:
4051:
4049:
4018:
4017:
3993:
3988:
3987:
3964:
3953:
3937:
3915:
3914:
3880:
3879:
3860:
3859:
3838:
3833:
3832:
3810:
3809:
3774:
3773:
3759:
3758:
3744:
3719:
3709:
3706:
3705:
3676:
3660:
3653:
3643:
3634:
3633:
3596:
3583:
3572:
3571:
3552:
3551:
3526:
3525:
3500:
3453:
3450:
3446:
3424:
3406:
3405:
3380:
3304:
3294:
3285:
3284:
3249:
3243:
3242:
3217:
3212:
3211:
3173:
3114:
3092:
3087:
3086:
3083:goodness of fit
3063:
3056:
2952:
2941:
2928:
2927:
2912:
2907:
2906:
2845:
2840:
2839:
2782:
2777:
2776:
2751:
2750:
2747:round-off error
2740:
2726:
2681:
2668:
2663:
2662:
2629:
2595:
2567:
2554:
2553:
2515:
2497:
2491:
2471:
2458:
2448:
2447:
2417:
2412:
2411:
2381:
2376:
2375:
2330:
2290:
2262:
2257:
2256:
2231:
2200:
2181:
2176:
2175:
2172:
2155:approaches the
2149:shape parameter
2145:shifted Weibull
2134:
2118:
2117:
2116:
2115:
2100:shape parameter
2096:scale parameter
2092:shifted Weibull
2087:
2086:
2085:
2082:
2074:
2073:
2070:
2059:
2054:
2017:
1996:
1995:
1967:
1946:
1945:
1911:
1894:
1875:
1871:
1832:
1815:
1796:
1792:
1770:
1769:
1709:
1689:
1673:
1653:
1627:
1608:
1577:
1557:
1531:
1521:
1490:
1485:
1484:
1482:
1443:
1426:
1419:
1398:
1347:
1324:
1305:
1304:
1289:
1267:
1244:
1243:
1228:
1223:
1222:
1161:
1150:
1142:
1135:
1123:
1119:
1118:
1088:
1077:
1062:
1043:
1042:
948:
933:
918:
907:
890:
842:
835:
811:
803:
796:
777:
773:
772:
762:
745:
721:
674:
627:
589:
576:
566:
565:
536:
505:
491:
448:
447:
437:
352:
317:
289:
284:
283:
274:
263:
252:
241:
230:
223:
200:
191:
176:
156:parameter space
113:
33:
17:
12:
11:
5:
7961:
7959:
7951:
7950:
7945:
7935:
7934:
7926:
7925:
7923:
7922:
7910:
7898:
7884:
7871:
7868:
7867:
7864:
7863:
7860:
7859:
7857:
7856:
7851:
7846:
7841:
7836:
7830:
7828:
7822:
7821:
7819:
7818:
7813:
7808:
7803:
7798:
7793:
7788:
7783:
7778:
7773:
7767:
7765:
7759:
7758:
7756:
7755:
7750:
7745:
7736:
7731:
7726:
7720:
7718:
7712:
7711:
7709:
7708:
7703:
7698:
7689:
7687:Bioinformatics
7683:
7681:
7671:
7670:
7665:
7658:
7657:
7654:
7653:
7650:
7649:
7646:
7645:
7643:
7642:
7636:
7634:
7630:
7629:
7627:
7626:
7620:
7618:
7612:
7611:
7609:
7608:
7603:
7598:
7593:
7587:
7585:
7576:
7570:
7569:
7566:
7565:
7563:
7562:
7557:
7552:
7547:
7542:
7536:
7534:
7528:
7527:
7525:
7524:
7519:
7514:
7506:
7501:
7496:
7495:
7494:
7492:partial (PACF)
7483:
7481:
7475:
7474:
7472:
7471:
7466:
7461:
7453:
7448:
7442:
7440:
7439:Specific tests
7436:
7435:
7433:
7432:
7427:
7422:
7417:
7412:
7407:
7402:
7397:
7391:
7389:
7382:
7376:
7375:
7373:
7372:
7371:
7370:
7369:
7368:
7353:
7352:
7351:
7341:
7339:Classification
7336:
7331:
7326:
7321:
7316:
7311:
7305:
7303:
7297:
7296:
7294:
7293:
7288:
7286:McNemar's test
7283:
7278:
7273:
7268:
7262:
7260:
7250:
7249:
7232:
7225:
7224:
7221:
7220:
7217:
7216:
7214:
7213:
7208:
7203:
7198:
7192:
7190:
7184:
7183:
7181:
7180:
7164:
7158:
7156:
7150:
7149:
7147:
7146:
7141:
7136:
7131:
7126:
7124:Semiparametric
7121:
7116:
7110:
7108:
7104:
7103:
7101:
7100:
7095:
7090:
7085:
7079:
7077:
7071:
7070:
7068:
7067:
7062:
7057:
7052:
7047:
7041:
7039:
7033:
7032:
7030:
7029:
7024:
7019:
7014:
7008:
7006:
6996:
6995:
6992:
6991:
6986:
6980:
6979:
6972:
6971:
6968:
6967:
6964:
6963:
6961:
6960:
6959:
6958:
6948:
6943:
6938:
6937:
6936:
6931:
6920:
6918:
6912:
6911:
6908:
6907:
6905:
6904:
6899:
6898:
6897:
6889:
6881:
6865:
6862:(MannâWhitney)
6857:
6856:
6855:
6842:
6841:
6840:
6829:
6827:
6821:
6820:
6818:
6817:
6816:
6815:
6810:
6805:
6795:
6790:
6787:(ShapiroâWilk)
6782:
6777:
6772:
6767:
6762:
6754:
6748:
6746:
6740:
6739:
6737:
6736:
6728:
6719:
6707:
6701:
6699:Specific tests
6695:
6694:
6691:
6690:
6688:
6687:
6682:
6677:
6671:
6669:
6663:
6662:
6660:
6659:
6654:
6653:
6652:
6642:
6641:
6640:
6630:
6624:
6622:
6616:
6615:
6613:
6612:
6611:
6610:
6605:
6595:
6590:
6585:
6580:
6575:
6569:
6567:
6561:
6560:
6558:
6557:
6552:
6551:
6550:
6545:
6544:
6543:
6538:
6523:
6522:
6521:
6516:
6511:
6506:
6495:
6493:
6484:
6478:
6477:
6475:
6474:
6469:
6464:
6463:
6462:
6452:
6447:
6446:
6445:
6435:
6434:
6433:
6428:
6423:
6413:
6408:
6403:
6402:
6401:
6396:
6391:
6375:
6374:
6373:
6368:
6363:
6353:
6352:
6351:
6346:
6336:
6335:
6334:
6324:
6323:
6322:
6312:
6307:
6302:
6296:
6294:
6284:
6283:
6278:
6271:
6270:
6267:
6266:
6263:
6262:
6260:
6259:
6254:
6249:
6244:
6238:
6236:
6230:
6229:
6227:
6226:
6221:
6216:
6210:
6208:
6204:
6203:
6201:
6200:
6195:
6190:
6185:
6180:
6175:
6170:
6164:
6162:
6156:
6155:
6153:
6152:
6150:Standard error
6147:
6142:
6137:
6136:
6135:
6130:
6119:
6117:
6111:
6110:
6108:
6107:
6102:
6097:
6092:
6087:
6082:
6080:Optimal design
6077:
6072:
6066:
6064:
6054:
6053:
6048:
6041:
6040:
6037:
6036:
6033:
6032:
6030:
6029:
6024:
6019:
6014:
6009:
6004:
5999:
5994:
5989:
5984:
5979:
5974:
5969:
5964:
5959:
5953:
5951:
5945:
5944:
5942:
5941:
5936:
5935:
5934:
5929:
5919:
5914:
5908:
5906:
5900:
5899:
5897:
5896:
5891:
5886:
5880:
5878:
5877:Summary tables
5874:
5873:
5871:
5870:
5864:
5862:
5856:
5855:
5852:
5851:
5849:
5848:
5847:
5846:
5841:
5836:
5826:
5820:
5818:
5812:
5811:
5809:
5808:
5803:
5798:
5793:
5788:
5783:
5778:
5772:
5770:
5764:
5763:
5761:
5760:
5755:
5750:
5749:
5748:
5743:
5738:
5733:
5728:
5723:
5718:
5713:
5711:Contraharmonic
5708:
5703:
5692:
5690:
5681:
5671:
5670:
5665:
5658:
5657:
5655:
5654:
5649:
5643:
5640:
5639:
5634:
5632:
5631:
5624:
5617:
5609:
5602:
5601:
5587:
5570:math/0702830v1
5556:
5522:
5486:
5457:
5431:(3): 395â449.
5420:
5401:
5381:(2): 235â246.
5366:
5330:
5320:(2): 386â392.
5309:
5283:(3): 394â403.
5272:
5258:on May 5, 2005
5225:
5183:(2): 472â476.
5164:
5162:
5159:
5157:
5156:
5136:
5125:
5114:
5111:Pieciak (2014)
5103:
5092:
5081:
5078:Ranneby (1984)
5065:
5042:
5040:
5037:
5035:
5032:
5029:
5028:
5019:leave out the
5009:
4992:
4989:
4986:
4983:
4980:
4958:
4955:
4950:
4947:
4944:
4940:
4936:
4931:
4928:
4925:
4922:
4919:
4915:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4889:
4886:
4881:
4876:
4873:
4870:
4866:
4862:
4859:
4854:
4850:
4839:uses the form
4819:
4816:
4813:
4808:
4804:
4780:
4777:
4774:
4769:
4765:
4760:
4757:
4752:
4749:
4746:
4741:
4738:
4735:
4731:
4727:
4724:
4721:
4718:
4715:
4712:
4691:
4671:
4668:
4640:
4635:
4631:
4603:
4599:
4576:
4572:
4551:
4548:
4545:
4542:
4520:
4516:
4493:
4489:
4471:
4470:
4468:
4465:
4464:
4463:
4458:
4453:
4446:
4443:
4426:
4423:
4420:
4417:
4414:
4409:
4404:
4383:
4380:
4367:
4362:
4358:
4354:
4351:
4348:
4345:
4340:
4337:
4334:
4330:
4326:
4323:
4308:EkstrĂśm (1997)
4296:
4293:
4291:
4288:
4275:
4255:
4248:
4244:
4237:
4233:
4229:
4224:
4221:
4216:
4213:
4207:
4204:
4198:
4195:
4189:
4186:
4180:
4177:
4171:
4168:
4148:
4142:
4139:
4133:
4128:
4124:
4120:
4117:
4111:
4108:
4102:
4097:
4093:
4065:
4062:
4047:
4038:critical value
4025:
4000:
3996:
3971:
3967:
3960:
3956:
3952:
3949:
3946:
3943:
3940:
3934:
3931:
3928:
3925:
3922:
3902:
3899:
3896:
3893:
3890:
3887:
3867:
3845:
3841:
3817:
3791:
3786:
3782:
3757:
3750:
3747:
3741:
3736:
3732:
3725:
3722:
3720:
3716:
3712:
3708:
3707:
3704:
3698:
3694:
3689:
3684:
3680:
3672:
3667:
3663:
3659:
3656:
3654:
3650:
3646:
3642:
3641:
3618:
3613:
3609:
3603:
3599:
3595:
3590:
3586:
3582:
3579:
3559:
3524:
3518:
3515:
3512:
3509:
3506:
3503:
3499:
3494:
3489:
3486:
3481:
3477:
3473:
3470:
3465:
3460:
3456:
3449:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3425:
3421:
3416:
3412:
3408:
3407:
3404:
3398:
3395:
3392:
3389:
3386:
3383:
3379:
3374:
3369:
3366:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3305:
3301:
3297:
3293:
3292:
3267:
3264:
3261:
3256:
3252:
3224:
3220:
3195:
3191:
3188:
3185:
3180:
3176:
3171:
3168:
3163:
3160:
3157:
3152:
3149:
3146:
3142:
3138:
3135:
3132:
3129:
3126:
3121:
3117:
3113:
3110:
3107:
3104:
3099:
3095:
3061:
3058:The statistic
3055:
3052:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2961:
2958:
2955:
2948:
2944:
2940:
2935:
2931:
2924:
2919:
2915:
2890:
2884:
2881:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2852:
2848:
2827:
2821:
2818:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2789:
2785:
2764:
2761:
2758:
2745:represent the
2731:
2722:
2694:
2691:
2688:
2684:
2680:
2675:
2671:
2650:
2647:
2644:
2641:
2636:
2632:
2628:
2625:
2622:
2619:
2616:
2613:
2608:
2605:
2602:
2598:
2594:
2591:
2588:
2580:
2577:
2574:
2570:
2566:
2561:
2557:
2551:
2548:
2544:
2541:
2538:
2535:
2532:
2529:
2522:
2518:
2510:
2507:
2504:
2500:
2495:
2484:
2481:
2478:
2474:
2470:
2465:
2461:
2456:
2435:
2432:
2429:
2424:
2420:
2399:
2396:
2393:
2388:
2384:
2360:
2357:
2354:
2351:
2348:
2343:
2340:
2337:
2333:
2329:
2326:
2323:
2320:
2317:
2314:
2309:
2306:
2303:
2300:
2297:
2293:
2289:
2286:
2283:
2280:
2275:
2272:
2269:
2265:
2243:
2238:
2234:
2230:
2227:
2224:
2219:
2216:
2213:
2210:
2207:
2203:
2199:
2194:
2191:
2188:
2184:
2171:
2168:
2132:
2102:of 0.5, and a
2089:
2088:
2083:
2076:
2075:
2071:
2064:
2063:
2062:
2061:
2060:
2058:
2055:
2053:
2050:
2030:
2027:
2024:
2020:
2016:
2010:
2007:
1980:
1977:
1974:
1970:
1966:
1960:
1957:
1938:
1937:
1926:
1920:
1917:
1914:
1907:
1904:
1901:
1897:
1893:
1888:
1885:
1882:
1878:
1874:
1868:
1862:
1859:
1847:
1841:
1838:
1835:
1828:
1825:
1822:
1818:
1814:
1809:
1806:
1803:
1799:
1795:
1789:
1783:
1780:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1718:
1715:
1712:
1707:
1702:
1699:
1696:
1692:
1688:
1685:
1682:
1679:
1676:
1669:
1662:
1659:
1656:
1651:
1646:
1643:
1640:
1637:
1634:
1630:
1626:
1621:
1618:
1615:
1611:
1607:
1604:
1601:
1596:
1591:
1588:
1585:
1581:
1573:
1566:
1563:
1560:
1555:
1552:
1549:
1544:
1541:
1538:
1534:
1530:
1527:
1524:
1517:
1514:
1511:
1508:
1505:
1502:
1497:
1493:
1478:
1452:
1449:
1446:
1439:
1436:
1433:
1429:
1425:
1422:
1416:
1411:
1408:
1405:
1401:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1356:
1353:
1350:
1343:
1340:
1337:
1334:
1331:
1327:
1323:
1318:
1315:
1312:
1308:
1301:
1296:
1292:
1282:
1276:
1273:
1270:
1265:
1262:
1257:
1254:
1251:
1247:
1240:
1235:
1231:
1155:
1148:
1141:
1138:
1133:
1106:
1103:
1100:
1094:
1091:
1086:
1083:
1080:
1074:
1065:
1060:
1056:
1053:
1050:
993:
992:
989:
986:
983:
979:
978:
975:
972:
969:
965:
964:
961:
958:
955:
951:
950:
942:
927:
914:
909:
901:
892:
884:
875:
840:
833:
810:
807:
802:
799:
792:
758:
752:Ranneby (1984)
741:
717:
696:
693:
690:
687:
681:
677:
672:
669:
664:
661:
658:
653:
650:
647:
643:
636:
633:
630:
626:
621:
615:
612:
609:
602:
599:
596:
592:
588:
583:
579:
573:
569:
560:
557:
554:
551:
548:
543:
539:
526:
523:
520:
517:
512:
508:
500:
497:
494:
489:
486:
483:
479:
476:
473:
467:
461:
458:
444:geometric mean
435:
414:
411:
408:
405:
402:
399:
396:
393:
390:
386:
383:
380:
376:
371:
368:
365:
362:
359:
355:
351:
348:
345:
342:
339:
335:
330:
327:
324:
320:
316:
313:
310:
307:
304:
301:
296:
292:
268:
261:
246:
239:
228:
221:
196:
189:
175:
172:
137:Ranneby (1984)
133:geometric mean
112:
109:
107:, and others.
68:geometric mean
36:geometric mean
27:
15:
13:
10:
9:
6:
4:
3:
2:
7960:
7949:
7946:
7944:
7941:
7940:
7938:
7931:
7921:
7920:
7911:
7909:
7908:
7899:
7897:
7896:
7891:
7885:
7883:
7882:
7873:
7872:
7869:
7855:
7852:
7850:
7849:Geostatistics
7847:
7845:
7842:
7840:
7837:
7835:
7832:
7831:
7829:
7827:
7823:
7817:
7816:Psychometrics
7814:
7812:
7809:
7807:
7804:
7802:
7799:
7797:
7794:
7792:
7789:
7787:
7784:
7782:
7779:
7777:
7774:
7772:
7769:
7768:
7766:
7764:
7760:
7754:
7751:
7749:
7746:
7744:
7740:
7737:
7735:
7732:
7730:
7727:
7725:
7722:
7721:
7719:
7717:
7713:
7707:
7704:
7702:
7699:
7697:
7693:
7690:
7688:
7685:
7684:
7682:
7680:
7679:Biostatistics
7676:
7672:
7668:
7663:
7659:
7641:
7640:Log-rank test
7638:
7637:
7635:
7631:
7625:
7622:
7621:
7619:
7617:
7613:
7607:
7604:
7602:
7599:
7597:
7594:
7592:
7589:
7588:
7586:
7584:
7580:
7577:
7575:
7571:
7561:
7558:
7556:
7553:
7551:
7548:
7546:
7543:
7541:
7538:
7537:
7535:
7533:
7529:
7523:
7520:
7518:
7515:
7513:
7511:(BoxâJenkins)
7507:
7505:
7502:
7500:
7497:
7493:
7490:
7489:
7488:
7485:
7484:
7482:
7480:
7476:
7470:
7467:
7465:
7464:DurbinâWatson
7462:
7460:
7454:
7452:
7449:
7447:
7446:DickeyâFuller
7444:
7443:
7441:
7437:
7431:
7428:
7426:
7423:
7421:
7420:Cointegration
7418:
7416:
7413:
7411:
7408:
7406:
7403:
7401:
7398:
7396:
7395:Decomposition
7393:
7392:
7390:
7386:
7383:
7381:
7377:
7367:
7364:
7363:
7362:
7359:
7358:
7357:
7354:
7350:
7347:
7346:
7345:
7342:
7340:
7337:
7335:
7332:
7330:
7327:
7325:
7322:
7320:
7317:
7315:
7312:
7310:
7307:
7306:
7304:
7302:
7298:
7292:
7289:
7287:
7284:
7282:
7279:
7277:
7274:
7272:
7269:
7267:
7266:Cohen's kappa
7264:
7263:
7261:
7259:
7255:
7251:
7247:
7243:
7239:
7235:
7230:
7226:
7212:
7209:
7207:
7204:
7202:
7199:
7197:
7194:
7193:
7191:
7189:
7185:
7179:
7175:
7171:
7165:
7163:
7160:
7159:
7157:
7155:
7151:
7145:
7142:
7140:
7137:
7135:
7132:
7130:
7127:
7125:
7122:
7120:
7119:Nonparametric
7117:
7115:
7112:
7111:
7109:
7105:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7080:
7078:
7076:
7072:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7042:
7040:
7038:
7034:
7028:
7025:
7023:
7020:
7018:
7015:
7013:
7010:
7009:
7007:
7005:
7001:
6997:
6990:
6987:
6985:
6982:
6981:
6977:
6973:
6957:
6954:
6953:
6952:
6949:
6947:
6944:
6942:
6939:
6935:
6932:
6930:
6927:
6926:
6925:
6922:
6921:
6919:
6917:
6913:
6903:
6900:
6896:
6890:
6888:
6882:
6880:
6874:
6873:
6872:
6869:
6868:Nonparametric
6866:
6864:
6858:
6854:
6851:
6850:
6849:
6843:
6839:
6838:Sample median
6836:
6835:
6834:
6831:
6830:
6828:
6826:
6822:
6814:
6811:
6809:
6806:
6804:
6801:
6800:
6799:
6796:
6794:
6791:
6789:
6783:
6781:
6778:
6776:
6773:
6771:
6768:
6766:
6763:
6761:
6759:
6755:
6753:
6750:
6749:
6747:
6745:
6741:
6735:
6733:
6729:
6727:
6725:
6720:
6718:
6713:
6709:
6708:
6705:
6702:
6700:
6696:
6686:
6683:
6681:
6678:
6676:
6673:
6672:
6670:
6668:
6664:
6658:
6655:
6651:
6648:
6647:
6646:
6643:
6639:
6636:
6635:
6634:
6631:
6629:
6626:
6625:
6623:
6621:
6617:
6609:
6606:
6604:
6601:
6600:
6599:
6596:
6594:
6591:
6589:
6586:
6584:
6581:
6579:
6576:
6574:
6571:
6570:
6568:
6566:
6562:
6556:
6553:
6549:
6546:
6542:
6539:
6537:
6534:
6533:
6532:
6529:
6528:
6527:
6524:
6520:
6517:
6515:
6512:
6510:
6507:
6505:
6502:
6501:
6500:
6497:
6496:
6494:
6492:
6488:
6485:
6483:
6479:
6473:
6470:
6468:
6465:
6461:
6458:
6457:
6456:
6453:
6451:
6448:
6444:
6443:loss function
6441:
6440:
6439:
6436:
6432:
6429:
6427:
6424:
6422:
6419:
6418:
6417:
6414:
6412:
6409:
6407:
6404:
6400:
6397:
6395:
6392:
6390:
6384:
6381:
6380:
6379:
6376:
6372:
6369:
6367:
6364:
6362:
6359:
6358:
6357:
6354:
6350:
6347:
6345:
6342:
6341:
6340:
6337:
6333:
6330:
6329:
6328:
6325:
6321:
6318:
6317:
6316:
6313:
6311:
6308:
6306:
6303:
6301:
6298:
6297:
6295:
6293:
6289:
6285:
6281:
6276:
6272:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6240:
6239:
6237:
6235:
6231:
6225:
6222:
6220:
6217:
6215:
6212:
6211:
6209:
6205:
6199:
6196:
6194:
6191:
6189:
6186:
6184:
6181:
6179:
6176:
6174:
6171:
6169:
6166:
6165:
6163:
6161:
6157:
6151:
6148:
6146:
6145:Questionnaire
6143:
6141:
6138:
6134:
6131:
6129:
6126:
6125:
6124:
6121:
6120:
6118:
6116:
6112:
6106:
6103:
6101:
6098:
6096:
6093:
6091:
6088:
6086:
6083:
6081:
6078:
6076:
6073:
6071:
6068:
6067:
6065:
6063:
6059:
6055:
6051:
6046:
6042:
6028:
6025:
6023:
6020:
6018:
6015:
6013:
6010:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5980:
5978:
5975:
5973:
5972:Control chart
5970:
5968:
5965:
5963:
5960:
5958:
5955:
5954:
5952:
5950:
5946:
5940:
5937:
5933:
5930:
5928:
5925:
5924:
5923:
5920:
5918:
5915:
5913:
5910:
5909:
5907:
5905:
5901:
5895:
5892:
5890:
5887:
5885:
5882:
5881:
5879:
5875:
5869:
5866:
5865:
5863:
5861:
5857:
5845:
5842:
5840:
5837:
5835:
5832:
5831:
5830:
5827:
5825:
5822:
5821:
5819:
5817:
5813:
5807:
5804:
5802:
5799:
5797:
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5774:
5773:
5771:
5769:
5765:
5759:
5756:
5754:
5751:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5709:
5707:
5704:
5702:
5699:
5698:
5697:
5694:
5693:
5691:
5689:
5685:
5682:
5680:
5676:
5672:
5668:
5663:
5659:
5653:
5650:
5648:
5645:
5644:
5641:
5637:
5630:
5625:
5623:
5618:
5616:
5611:
5610:
5607:
5598:
5594:
5590:
5584:
5580:
5576:
5571:
5566:
5562:
5557:
5547:
5543:
5539:
5535:
5528:
5523:
5512:
5508:
5504:
5500:
5496:
5492:
5487:
5483:
5479:
5475:
5471:
5468:(2): 93â112.
5467:
5463:
5458:
5454:
5450:
5446:
5442:
5438:
5434:
5430:
5426:
5421:
5417:
5413:
5409:
5408:
5402:
5398:
5394:
5389:
5384:
5380:
5376:
5372:
5367:
5356:
5352:
5348:
5344:
5340:
5336:
5331:
5327:
5323:
5319:
5315:
5310:
5306:
5302:
5298:
5294:
5290:
5286:
5282:
5278:
5273:
5270:
5254:
5250:
5246:
5242:
5238:
5231:
5226:
5216:on 2011-08-16
5212:
5208:
5204:
5200:
5196:
5191:
5186:
5182:
5178:
5171:
5166:
5165:
5160:
5154:
5149:
5147:
5145:
5143:
5141:
5137:
5134:
5129:
5126:
5123:
5118:
5115:
5112:
5107:
5104:
5101:
5096:
5093:
5090:
5085:
5082:
5079:
5074:
5072:
5070:
5066:
5063:
5058:
5056:
5054:
5052:
5050:
5048:
5044:
5038:
5033:
5026:
5022:
5018:
5013:
5010:
5007:
4987:
4984:
4981:
4948:
4945:
4942:
4938:
4934:
4929:
4926:
4923:
4920:
4917:
4913:
4903:
4900:
4897:
4887:
4884:
4879:
4874:
4871:
4868:
4864:
4860:
4857:
4852:
4848:
4838:
4834:
4814:
4806:
4802:
4775:
4767:
4763:
4758:
4755:
4750:
4747:
4744:
4739:
4736:
4733:
4729:
4725:
4722:
4716:
4710:
4701:
4695:
4692:
4689:
4666:
4633:
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2101:
2097:
2093:
2080:
2068:
2056:
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2049:
2047:
2025:
2018:
2014:
2005:
1975:
1968:
1964:
1955:
1943:
1924:
1918:
1915:
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1902:
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1697:
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1184:
1180:
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1165:
1159:
1154:
1147:
1139:
1137:
1136:= â
â 0.333.
1132:
1126:
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1101:
1098:
1092:
1089:
1084:
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1078:
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1063:
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867:
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849:
846:
839:
832:
824:
820:
817:Plots of the
815:
808:
806:
800:
798:
795:
791:
787:
780:
770:
767:+1), whereas
766:
761:
757:
753:
749:
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733:
729:
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720:
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712:
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227:
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208:
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199:
195:
188:
184:
183:random sample
181:
173:
171:
169:
164:
159:
157:
153:
148:
146:
143:, similar to
142:
138:
134:
130:
126:
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118:
110:
108:
106:
102:
98:
93:
91:
86:
84:
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69:
65:
61:
57:
53:
49:
45:
37:
31:
26:
21:
7930:
7917:
7905:
7886:
7879:
7791:Econometrics
7741: /
7724:Chemometrics
7701:Epidemiology
7694: /
7667:Applications
7509:ARIMA model
7456:Q-statistic
7405:Stationarity
7301:Multivariate
7244: /
7240: /
7238:Multivariate
7236: /
7176: /
7172: /
6946:Bayes factor
6845:Signed rank
6757:
6731:
6723:
6711:
6406:Completeness
6242:Cohort study
6140:Opinion poll
6075:Missing data
6062:Study design
6017:Scatter plot
5939:Scatter plot
5932:Spearman's Ď
5894:Grouped data
5560:
5549:. Retrieved
5537:
5533:
5515:. Retrieved
5511:the original
5498:
5494:
5465:
5461:
5428:
5424:
5406:
5378:
5374:
5359:. Retrieved
5355:the original
5342:
5338:
5317:
5313:
5280:
5276:
5268:
5260:. Retrieved
5253:the original
5243:(1): 17â40.
5240:
5236:
5218:. Retrieved
5211:the original
5180:
5176:
5128:
5117:
5106:
5095:
5084:
5024:
5012:
5005:
4694:
4687:
4475:
4391:multivariate
4385:
4311:
4298:
4083:showed that
4044:
4042:
4015:significance
3549:
3540:
3078:
3074:
3066:
3059:
3057:
3047:
3043:
3025:
2742:
2736:
2732:
2728:
2723:
2719:
2715:
2709:
2373:
2173:
2161:
2152:
2129:
2119:
2107:
2084:Distribution
1939:
1762:
1758:
1479:
1475:
1473:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1157:
1152:
1145:
1143:
1130:
1124:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
996:
944:
939:
935:
929:
924:
920:
915:
911:
903:
898:
894:
886:
881:
877:
872:
863:
859:
855:
851:
847:
837:
830:
828:
822:
804:
793:
789:
785:
778:
764:
759:
755:
747:
742:
738:
736:
727:
723:
718:
714:
708:
432:
428:
426:
279:
277:
270:
265:
258:
248:
243:
236:
225:
218:
214:
210:
202:
197:
193:
186:
177:
160:
149:
114:
101:econometrics
94:
87:
80:
71:
59:
55:
51:
47:
41:
29:
24:
7919:WikiProject
7834:Cartography
7796:Jurimetrics
7748:Reliability
7479:Time domain
7458:(LjungâBox)
7380:Time-series
7258:Categorical
7242:Time-series
7234:Categorical
7169:(Bernoulli)
7004:Correlation
6984:Correlation
6780:JarqueâBera
6752:Chi-squared
6514:M-estimator
6467:Asymptotics
6411:Sufficiency
6178:Interaction
6090:Replication
6070:Effect size
6027:Violin plot
6007:Radar chart
5987:Forest plot
5977:Correlogram
5927:Kendall's Ď
5161:Works cited
5133:Pyke (1965)
3632:, in which
2170:Sensitivity
2124:in that it
858:) = 1 â e,
713:, function
530:where
278:Define the
166:procedure.
152:likelihoods
7937:Categories
7786:Demography
7504:ARMA model
7309:Regression
6886:(Friedman)
6847:(Wilcoxon)
6785:Normality
6775:Lilliefors
6722:Student's
6598:Resampling
6472:Robustness
6460:divergence
6450:Efficiency
6388:(monotone)
6383:Likelihood
6300:Population
6133:Stratified
6085:Population
5904:Dependence
5860:Count data
5791:Percentile
5768:Dispersion
5701:Arithmetic
5636:Statistics
5551:2008-12-31
5517:2008-12-30
5361:2008-12-30
5314:Biometrika
5262:2008-12-31
5220:2009-01-21
5034:References
3804:follows a
3772:and where
3054:Moran test
2052:Properties
201:} of size
174:Definition
44:statistics
7167:Logistic
6934:posterior
6860:Rank sum
6608:Jackknife
6603:Bootstrap
6421:Bootstrap
6356:Parameter
6305:Statistic
6100:Statistic
6012:Run chart
5997:Pie chart
5992:Histogram
5982:Fan chart
5957:Bar chart
5839:L-moments
5726:Geometric
5507:0345-3928
5474:0303-6898
5445:0035-9246
5397:1027-5606
5351:0345-3928
5297:0035-9246
5249:1055-7490
5207:123004317
5185:CiteSeerX
5039:Citations
4935:−
4888:
4865:∑
4861:−
4815:θ
4776:θ
4759:
4730:∑
4726:−
4717:θ
4670:~
4667:σ
4639:~
4630:σ
4617:variance
4547:θ
4347:−
4228:−
4206:^
4203:θ
4179:^
4176:θ
4141:^
4138:θ
4110:^
4107:θ
4064:^
4061:θ
4024:α
3951:−
3945:θ
3927:θ
3898:θ
3781:χ
3731:σ
3679:σ
3671:−
3662:μ
3608:χ
3493:−
3480:−
3469:−
3455:π
3429:≈
3411:σ
3373:−
3360:−
3354:γ
3333:
3309:≈
3296:μ
3263:θ
3219:θ
3187:θ
3170:
3141:∑
3137:−
3128:θ
3106:θ
3003:−
2988:…
2957:−
2939:−
2883:^
2880:θ
2871:δ
2820:^
2817:θ
2808:δ
2805:−
2763:δ
2760:±
2690:−
2643:θ
2615:θ
2604:−
2576:−
2565:−
2540:θ
2506:−
2494:∫
2480:−
2469:→
2431:θ
2395:θ
2350:θ
2325:⋯
2316:θ
2305:−
2282:θ
2226:⋯
2215:−
2147:, with a
2098:of 15, a
2009:^
1959:^
1916:−
1892:−
1861:^
1837:−
1813:−
1782:^
1739:−
1730:
1724:−
1687:−
1678:
1639:−
1625:−
1603:
1580:∑
1548:−
1526:
1448:−
1424:−
1381:…
1365:for
1352:−
1336:−
1322:−
1272:−
1261:−
1144:Suppose {
1140:Example 2
1099:≈
1090:−
1082:
1064:λ
1059:⇒
1049:μ
821:value of
809:Example 1
771:omit the
689:θ
671:
642:∑
587:⋯
550:θ
519:θ
499:Θ
496:∈
493:θ
460:^
457:θ
440:logarithm
427:Then the
401:…
379:θ
364:−
344:−
338:θ
303:θ
264:= ââ and
233:estimated
224:), where
178:Given an
97:hydrology
7881:Category
7574:Survival
7451:Johansen
7174:Binomial
7129:Isotonic
6716:(normal)
6361:location
6168:Blocking
6123:Sampling
6002:QâQ plot
5967:Box plot
5949:Graphics
5844:Skewness
5834:Kurtosis
5806:Variance
5736:Heronian
5731:Harmonic
5597:88516426
4445:See also
4304:measures
3570:, where
801:Examples
732:supremum
280:spacings
72:spacings
7907:Commons
7854:Kriging
7739:Process
7696:studies
7555:Wavelet
7388:General
6555:Plug-in
6349:L space
6128:Cluster
5829:Moments
5647:Outline
5482:4615946
5453:2345793
5305:2345411
3543:is the
3032:mixture
2255:we get
2094:with a
2072:Density
1217:) when
1151:, ...,
1122:⁄
776:⁄
734:sense.
709:By the
442:of the
255:ordered
242:, ...,
235:, let {
205:from a
192:, ...,
7776:Census
7366:Normal
7314:Manova
7134:Robust
6884:2-way
6876:1-way
6714:-test
6385:
5962:Biplot
5753:Median
5746:Lehmer
5688:Center
5595:
5585:
5505:
5480:
5472:
5451:
5443:
5395:
5349:
5303:
5295:
5247:
5205:
5187:
5025:Editor
5006:Editor
4793:where
4688:Editor
4266:where
4043:Where
3539:where
3279:has a
2741:, let
2661:since
1852:
1849:
1460:
1457:
1396:
1393:
1361:
1287:
1284:
977:e â e
963:1 â e
534:
275:= +â.
58:), or
7400:Trend
6929:prior
6871:anova
6760:-test
6734:-test
6726:-test
6633:Power
6578:Pivot
6371:shape
6366:scale
5816:Shape
5796:Range
5741:Heinz
5716:Cubic
5652:Index
5593:S2CID
5565:arXiv
5530:(PDF)
5478:JSTOR
5449:JSTOR
5301:JSTOR
5256:(PDF)
5233:(PDF)
5214:(PDF)
5203:S2CID
5173:(PDF)
4589:. If
4467:Notes
3808:with
3283:with
3071:Moran
2446:, as
1201:) = (
1102:0.255
988:1 â e
974:1 â e
971:1 â e
957:1 â e
836:= 2,
7633:Test
6833:Sign
6685:Wald
5758:Mode
5696:Mean
5583:ISBN
5503:ISSN
5470:ISSN
5441:ISSN
5393:ISSN
5347:ISSN
5293:ISSN
5245:ISSN
4686:. â
4419:>
2838:and
2410:for
1994:and
1761:and
1181:and
934:) â
6813:BIC
6808:AIC
5575:doi
5542:doi
5538:129
5433:doi
5412:doi
5383:doi
5322:doi
5285:doi
5195:doi
4756:log
3206:is
2727:to
2455:lim
1209:)/(
1149:(1)
1134:MLE
1085:0.6
1068:MSE
1055:0.6
947:â1)
906:â1)
841:(2)
834:(1)
819:log
431:of
273:+1)
262:(0)
240:(1)
180:iid
70:of
56:MSP
54:or
52:MSE
42:In
7939::
5591:.
5581:.
5573:.
5536:.
5532:.
5501:.
5497:.
5493:.
5476:.
5466:11
5464:.
5447:.
5439:.
5429:27
5427:.
5391:.
5377:.
5373:.
5345:.
5341:.
5337:.
5318:76
5316:.
5299:.
5291:.
5281:45
5279:.
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