3340:
2686:
3335:{\displaystyle {\begin{pmatrix}X_{1}\\\vdots \\X_{n}\\Y_{1}\\\vdots \\Y_{n}\\Z_{1}\\\vdots \\Z_{n}\end{pmatrix}}={\bar {M}}{\begin{pmatrix}1\\\vdots \\1\\1\\\vdots \\1\\1\\\vdots \\1\end{pmatrix}}+{\begin{pmatrix}{\bar {X}}-{\bar {M}}\\\vdots \\{\bar {X}}-{\bar {M}}\\{\bar {Y}}-{\bar {M}}\\\vdots \\{\bar {Y}}-{\bar {M}}\\{\bar {Z}}-{\bar {M}}\\\vdots \\{\bar {Z}}-{\bar {M}}\end{pmatrix}}+{\begin{pmatrix}X_{1}-{\bar {X}}\\\vdots \\X_{n}-{\bar {X}}\\Y_{1}-{\bar {Y}}\\\vdots \\Y_{n}-{\bar {Y}}\\Z_{1}-{\bar {Z}}\\\vdots \\Z_{n}-{\bar {Z}}\end{pmatrix}}.}
8514:
5515:
8500:
1245:
factor analysis with 4 items, there are 10 knowns (the six unique covariances among the four items and the four item variances) and 8 unknowns (4 factor loadings and 4 error variances) for 2 degrees of freedom. Degrees of freedom are important to the understanding of model fit if for no other reason than that, all else being equal, the fewer degrees of freedom, the better indices such as
2524:
8538:
8526:
606:
5108:
1268:
parameter of the population from which that sample is drawn. For example, if we have two observations, when calculating the mean we have two independent observations; however, when calculating the variance, we have only one independent observation, since the two observations are equally distant from the sample mean.
2199:
1252:
It has been shown that degrees of freedom can be used by readers of papers that contain SEMs to determine if the authors of those papers are in fact reporting the correct model fit statistics. In the organizational sciences, for example, nearly half of papers published in top journals report degrees
1244:
Degrees of freedom in SEM are computed as a difference between the number of unique pieces of information that are used as input into the analysis, sometimes called knowns, and the number of parameters that are uniquely estimated, sometimes called unknowns. For example, in a one-factor confirmatory
3907:
designs, the sums-of-squares no longer have scaled chi-squared distributions. Comparison of sum-of-squares with degrees-of-freedom is no longer meaningful, and software may report certain fractional 'degrees of freedom' in these cases. Such numbers have no genuine degrees-of-freedom interpretation,
3888:
Under the null hypothesis of no difference between population means (and assuming that standard ANOVA regularity assumptions are satisfied) the sums of squares have scaled chi-squared distributions, with the corresponding degrees of freedom. The F-test statistic is the ratio, after scaling by the
1267:
A common way to think of degrees of freedom is as the number of independent pieces of information available to estimate another piece of information. More concretely, the number of degrees of freedom is the number of independent observations in a sample of data that are available to estimate a
56:
can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent
3876:
3493:
In statistical testing problems, one usually is not interested in the component vectors themselves, but rather in their squared lengths, or Sum of
Squares. The degrees of freedom associated with a sum-of-squares is the degrees-of-freedom of the corresponding component vectors.
3885:−1) degrees of freedom. Of course, introductory books on ANOVA usually state formulae without showing the vectors, but it is this underlying geometry that gives rise to SS formulae, and shows how to unambiguously determine the degrees of freedom in any given situation.
934:
4249:
and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures. However, these procedures are still linear in the observations, and the fitted values of the regression can be expressed in the form
416:
1181:
4652:, used to mitigate data noise. In contrast to a simple linear or polynomial fit, computing the effective degrees of freedom of the smoothing function is not straightforward. In these cases, it is important to estimate the Degrees of Freedom permitted by the
4906:
3662:
4636:
5216:
4917:
3941:. This terminology simply reflects that in many applications where these distributions occur, the parameter corresponds to the degrees of freedom of an underlying random vector, as in the preceding ANOVA example. Another simple example is: if
2519:{\displaystyle {\begin{aligned}X_{i}&={\bar {M}}+({\bar {X}}-{\bar {M}})+(X_{i}-{\bar {X}})\\Y_{i}&={\bar {M}}+({\bar {Y}}-{\bar {M}})+(Y_{i}-{\bar {Y}})\\Z_{i}&={\bar {M}}+({\bar {Z}}-{\bar {M}})+(Z_{i}-{\bar {Z}})\end{aligned}}}
156:
While introductory textbooks may introduce degrees of freedom as distribution parameters or through hypothesis testing, it is the underlying geometry that defines degrees of freedom, and is critical to a proper understanding of the concept.
4423:
does not correspond to an ordinary least-squares fit (i.e. is not an orthogonal projection), these sums-of-squares no longer have (scaled, non-central) chi-squared distributions, and dimensionally defined degrees-of-freedom are not useful.
4192:
values. The underlying families of distributions allow fractional values for the degrees-of-freedom parameters, which can arise in more sophisticated uses. One set of examples is problems where chi-squared approximations based on
1812:
4124:
179:
article "The
Probable Error of a Mean", published under the pen name "Student". While Gosset did not actually use the term 'degrees of freedom', he explained the concept in the course of developing what became known as
1982:
3673:
1427:
361:
1891:
2674:
1621:
228:
Geometrically, the degrees of freedom can be interpreted as the dimension of certain vector subspaces. As a starting point, suppose that we have a sample of independent normally distributed observations,
1271:
In fitting statistical models to data, the vectors of residuals are constrained to lie in a space of smaller dimension than the number of components in the vector. That smaller dimension is the number of
5778:
Cortina, J. M., Green, J. P., Keeler, K. R., & Vandenberg, R. J. (2017). Degrees of freedom in SEM: Are we testing the models that we claim to test?. Organizational
Research Methods, 20(3), 350-378.
773:
2037:
and chi-squared distributions for one-sample problems above is the simplest example where degrees-of-freedom arise. However, similar geometry and vector decompositions underlie much of the theory of
5336:
601:{\displaystyle {\begin{pmatrix}X_{1}\\\vdots \\X_{n}\end{pmatrix}}={\bar {X}}{\begin{pmatrix}1\\\vdots \\1\end{pmatrix}}+{\begin{pmatrix}X_{1}-{\bar {X}}\\\vdots \\X_{n}-{\bar {X}}\end{pmatrix}}.}
2204:
2586:
716:
3475:
1044:
3507:
5418:
The more general formulation of effective degree of freedom would result in a more realistic estimate for, e.g., the error variance Ď, which in its turn scales the unknown parameters'
4641:
the regression (not residual) degrees of freedom in linear models are "the sum of the sensitivities of the fitted values with respect to the observed response values", i.e. the sum of
4524:
1480:
4785:
3435:
3391:
280:
4183:
3990:
2188:
2142:
2096:
5277:
5116:
5103:{\displaystyle {\hat {\sigma }}^{2}={\frac {\|{\hat {r}}\|^{2}}{n-\operatorname {tr} (2H-HH')}}={\frac {\|{\hat {r}}\|^{2}}{n-2\operatorname {tr} (H)+\operatorname {tr} (HH')}}}
4029:
3349:
degrees of freedom. On the right-hand side, the first vector has one degree of freedom (or dimension) for the overall mean. The second vector depends on three random variables,
1335:
4777:
4292:
1706:
1677:
1028: â 1 degrees of freedom. The degrees-of-freedom, here a parameter of the distribution, can still be interpreted as the dimension of an underlying vector subspace.
5497:
4417:
1022:
991:
4375:
4324:
638:
404:
4697:
1219:
2049:. An explicit example based on comparison of three means is presented here; the geometry of linear models is discussed in more complete detail by Christensen (2002).
6063:
5886:
964:
1229:
When the results of structural equation models (SEM) are presented, they generally include one or more indices of overall model fit, the most common of which is a
763:
In statistical testing applications, often one is not directly interested in the component vectors, but rather in their squared lengths. In the example above, the
7635:
8140:
4670:
3477:. However, these must sum to 0 and so are constrained; the vector therefore must lie in a 2-dimensional subspace, and has 2 degrees of freedom. The remaining 3
149:. The degrees of freedom are also commonly associated with the squared lengths (or "sum of squares" of the coordinates) of such vectors, and the parameters of
113:
1233:
statistic. This forms the basis for other indices that are commonly reported. Although it is these other statistics that are most commonly interpreted, the
83:
8290:
7914:
6555:
5392:, would lead to over-estimation of the residuals degree of freedom, as if each observation were independent. More realistically, though, the hat matrix
5237:). In general the numerator would be the objective function being minimized; e.g., if the hat matrix includes an observation covariance matrix, ÎŁ, then
748:
by the vector of 1's. The 1 degree of freedom is the dimension of this subspace. The second residual vector is the least-squares projection onto the (
6000:
7688:
5346:
Note that unlike in the original case, non-integer degrees of freedom are allowed, though the value must usually still be constrained between 0 and
61:
that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself. For example, if the
8127:
1722:
4037:
1897:
3871:{\displaystyle {\text{SSE}}=\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}+\sum _{i=1}^{n}(Y_{i}-{\bar {Y}})^{2}+\sum _{i=1}^{n}(Z_{i}-{\bar {Z}})^{2}}
5918:
5863:
5828:
1253:
of freedom that are inconsistent with the models described in those papers, leaving the reader to wonder which models were actually tested.
6550:
6250:
5500:
4133: â 1 degrees of freedom. Here, the degrees of freedom arises from the residual sum-of-squares in the numerator, and in turn the
1357:
299:
6040:
D. Dong, T. A. Herring and R. W. King (1997), Estimating regional deformation from a combination of space and terrestrial geodetic data,
7154:
6302:
5957:
6204:
5840:
1823:
1490:
3912:
chi-squared distribution for the corresponding sum-of-squares. The details of such approximations are beyond the scope of this page.
2591:
7937:
7829:
6110:
5797:
5533:
1538:
929:{\displaystyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}={\begin{Vmatrix}X_{1}-{\bar {X}}\\\vdots \\X_{n}-{\bar {X}}\end{Vmatrix}}^{2}.}
165:
Although the basic concept of degrees of freedom was recognized as early as 1821 in the work of German astronomer and mathematician
8542:
8115:
7989:
2002:
in the definition of the residuals; that is because the former are hypothesized random variables and the latter are actual data.
1509: â 1 of the residuals, one can thus find the last one. That means they are constrained to lie in a space of dimension
8173:
7579:
6950:
6540:
126:, or essentially the number of "free" components (how many components need to be known before the vector is fully determined).
7164:
8224:
7436:
7243:
7132:
7090:
6329:
611:
The first vector on the right-hand side is constrained to be a multiple of the vector of 1's, and the only free quantity is
8467:
7426:
4436:
89:) minus the number of parameters estimated as intermediate steps (one, namely, the sample mean) and is therefore equal to
7476:
8018:
7967:
7952:
7942:
7811:
7683:
7650:
7431:
7261:
5369:
measured points, the weight of the original value on the linear combination that makes up the predicted value is just 1/
5358:
4483:
4242:
8087:
7388:
5282:
1176:{\displaystyle {\frac {{\sqrt {n}}({\bar {X}}-\mu _{0})}{\sqrt {\sum \limits _{i=1}^{n}(X_{i}-{\bar {X}})^{2}/(n-1)}}}}
8564:
8362:
8163:
7142:
6811:
6275:
5942:
5384:
As another example, consider the existence of nearly duplicated observations. Naive application of classical formula,
2532:
217:
8247:
8214:
5908:
3921:
3440:
646:
181:
2190:. The restriction to three groups and equal sample sizes simplifies notation, but the ideas are easily generalized.
8219:
7962:
7721:
7627:
7607:
7515:
7226:
7044:
6527:
6399:
6223:
3657:{\displaystyle {\text{SST}}=n({\bar {X}}-{\bar {M}})^{2}+n({\bar {Y}}-{\bar {M}})^{2}+n({\bar {Z}}-{\bar {M}})^{2}}
7393:
7159:
7017:
4631:{\displaystyle \operatorname {tr} (H)=\sum _{i}h_{ii}=\sum _{i}{\frac {\partial {\hat {y}}_{i}}{\partial y_{i}}},}
7979:
7747:
7468:
7322:
7251:
7171:
7029:
7010:
6718:
6439:
6157:
5538:
5443:
4246:
4234:
4218:
4198:
58:
8092:
5503:, and the theory associated with this distribution provides an alternative route to the answers provided above.
8462:
8229:
7777:
7742:
7706:
7491:
6933:
6842:
6801:
6713:
6404:
6243:
5543:
4901:{\displaystyle {\hat {\sigma }}^{2}={\frac {\|{\hat {r}}\|^{2}}{\operatorname {tr} \left((I-H)'(I-H)\right)}},}
3928:
1438:
994:
150:
7499:
7483:
220:, the degrees of freedom are typically noted beside the test statistic as either subscript or in parentheses.
5629:
1505:. The sum of the residuals (unlike the sum of the errors) is necessarily 0. If one knows the values of any
8371:
7984:
7924:
7861:
7221:
7083:
7073:
6923:
6837:
4188:
In the application of these distributions to linear models, the degrees of freedom parameters can take only
764:
8132:
8069:
5211:{\displaystyle {\hat {\sigma }}^{2}\approx {\frac {\|{\hat {r}}\|^{2}}{n-1.25\operatorname {tr} (H)+0.5}}.}
3396:
3352:
235:
8409:
8339:
7824:
7711:
6708:
6605:
6512:
6391:
6290:
5528:
5415:
would involve an observation covariance matrix ÎŁ indicating the non-zero correlation among observations.
4460:
4238:
4140:
1262:
8530:
7408:
3944:
2147:
2101:
2055:
8434:
8376:
8319:
8145:
8038:
7947:
7673:
7557:
7416:
7298:
7290:
7105:
7001:
6979:
6938:
6903:
6870:
6816:
6791:
6746:
6685:
6645:
6447:
6270:
5422:
standard deviation; the degree of freedom will also affect the expansion factor necessary to produce an
5240:
4440:
3995:
1294:
753:
53:
8513:
7403:
5514:
4672:
matrix so that the residual degrees of freedom can then be used to estimate statistical tests such as
8357:
7932:
7881:
7857:
7819:
7737:
7716:
7668:
7547:
7525:
7494:
7280:
7231:
7149:
7122:
7078:
7034:
6796:
6572:
6452:
4738:
4432:
2046:
170:
166:
138:
4256:
4209:
interpretation to the distribution parameters, even though the terminology may continue to be used.
1682:
1653:
8504:
8429:
8352:
8033:
7797:
7790:
7752:
7660:
7640:
7612:
7345:
7211:
7206:
7196:
7188:
7006:
6967:
6857:
6847:
6756:
6535:
6491:
6409:
6334:
6236:
8079:
5858:. Quantitative Applications in the Social Sciences. Vol. 130. SAGE Publications. p. 58.
5460:
4380:
2013: â 1 predictors and one mean (=intercept in the regression)), in which case the cost in
8518:
8329:
8183:
8028:
7904:
7801:
7785:
7762:
7539:
7273:
7256:
7216:
7127:
7022:
6984:
6955:
6915:
6875:
6821:
6738:
6424:
6419:
6174:
6143:
6068:
5891:
5722:
5683:
5520:
1000:
969:
742:
738:
31:
4344:
4300:
614:
410:. The random vector can be decomposed as the sum of the sample mean plus a vector of residuals:
380:
5741:
4675:
4326:
is the vector of fitted values at each of the original covariate values from the fitted model,
8424:
8394:
8386:
8206:
8197:
8122:
8053:
7909:
7894:
7869:
7757:
7698:
7564:
7552:
7178:
7095:
7039:
6962:
6806:
6728:
6507:
6381:
6106:
5914:
5859:
5824:
5816:
5793:
5761:
5553:
4230:
2042:
1486:
1221:
is correct. Again, the degrees-of-freedom arises from the residual vector in the denominator.
1197:
134:
85:
independent scores, then the degrees of freedom is equal to the number of independent scores (
17:
5989:
5884:
Ye, J. (1998), "On
Measuring and Correcting the Effects of Data Mining and Model Selection",
5853:
3889:
degrees of freedom. If there is no difference between population means this ratio follows an
8449:
8404:
8168:
8155:
8048:
8023:
7957:
7889:
7767:
7375:
7268:
7201:
7114:
7061:
6880:
6751:
6545:
6344:
6311:
6195:
6166:
6133:
6045:
5922:
5832:
5753:
5714:
5675:
5644:
5427:
4341:
For statistical inference, sums-of-squares can still be formed: the model sum-of-squares is
4222:
205:
4719:. For example, if the goal is to estimate error variance, the redf would be defined as tr((
942:
722: â 1 components of this vector can be anything. However, once you know the first
8366:
8110:
7972:
7899:
7574:
7448:
7421:
7398:
7367:
6994:
6989:
6943:
6673:
6324:
4707:
There are corresponding definitions of residual effective degrees-of-freedom (redf), with
1341:
146:
142:
4459:
For the regression effective degrees of freedom, appropriate definitions can include the
5742:"Reporting statistical methods and outcome of statistical analyses in research articles"
5577:
4205:-distribution may be used as an empirical model. In these cases, there is no particular
8315:
8310:
6773:
6703:
6349:
4655:
4642:
3932:
3890:
1345:
1187:
6217:
6084:
Jones, D.A. (1983) "Statistical analysis of empirical models fitted by optimisation",
6061:
H. Theil (1963), "On the Use of
Incomplete Prior Information in Regression Analysis",
4648:
One way to help to conceptualize this is to consider a simple smoothing matrix like a
3501:. The model, or treatment, sum-of-squares is the squared length of the second vector,
8558:
8472:
8439:
8302:
8263:
8074:
8043:
7507:
7461:
7066:
6768:
6595:
6359:
6354:
6147:
6138:
6121:
5812:
5423:
4649:
3498:
2038:
1521:
290:
185:
130:
123:
92:
6625:
6002:
Nonparametric regression and generalized linear models: a roughness penalty approach
1241:
are essential to understanding model fit as well as the nature of the model itself.
8414:
8347:
8324:
8239:
7569:
6865:
6763:
6698:
6640:
6562:
6517:
6170:
68:
8457:
8419:
8102:
8003:
7865:
7678:
7645:
7137:
7054:
7049:
6693:
6650:
6630:
6610:
6600:
6369:
5548:
4518:, and all these definitions reduce to the usual degrees of freedom. Notice that
745:
407:
201:
5757:
5703:"On the Interpretation of Ď2 from Contingency Tables, and the Calculation of P"
169:, its modern definition and usage was first elaborated by English statistician
153:
and other distributions that arise in associated statistical testing problems.
7303:
6783:
6483:
6414:
6364:
6339:
6259:
6085:
5836:
5603:
5510:
4335:
3904:
175:
38:
27:
Number of values in the final calculation of a statistic that are free to vary
5765:
1807:{\displaystyle {\widehat {e}}_{i}=y_{i}-({\widehat {a}}+{\widehat {b}}x_{i})}
7456:
7308:
6928:
6723:
6635:
6620:
6615:
6580:
5821:
The elements of statistical learning: data mining, inference, and prediction
4226:
4119:{\displaystyle {\frac {\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}}{\sigma ^{2}}}}
119:
46:
6218:
Illustrating degrees of freedom in terms of sample size and dimensionality
6049:
1977:{\displaystyle x_{1}{\widehat {e}}_{1}+\cdots +x_{n}{\widehat {e}}_{n}=0.}
6972:
6590:
6467:
6462:
6457:
6429:
5558:
3481: â 3 degrees of freedom are in the residual vector (made up of
62:
184:. The term itself was popularized by English statistician and biologist
8477:
8178:
6178:
6072:
5895:
5726:
5687:
4189:
2680:
observations. In vector notation this decomposition can be written as
8399:
7380:
7354:
7334:
6585:
6376:
6199:
5648:
3667:
with 2 degrees of freedom. The residual, or error, sum-of-squares is
1817:
are constrained to lie within the space defined by the two equations
1032:
726: â 1 components, the constraint tells you the value of the
5718:
5702:
5679:
5663:
4137: â 1 degrees of freedom of the underlying residual vector
1422:{\displaystyle {\overline {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}
356:{\displaystyle {\begin{pmatrix}X_{1}\\\vdots \\X_{n}\end{pmatrix}}.}
212:
to symbolize degrees of freedom but modern usage typically reserves
5926:
3485: â 1 degrees of freedom within each of the populations).
5365:
nearest measured values to the given point. Then, at each of the
5221:
The last approximation above reduces the computational cost from
2052:
Suppose independent observations are made for three populations,
204:). In text and tables, the abbreviation "d.f." is commonly used.
6319:
1886:{\displaystyle {\widehat {e}}_{1}+\cdots +{\widehat {e}}_{n}=0,}
8288:
7855:
7602:
6901:
6671:
6288:
6232:
5790:
Plane
Answers to Complex Questions: The Theory of Linear Models
2669:{\displaystyle {\bar {M}}=({\bar {X}}+{\bar {Y}}+{\bar {Z}})/3}
145:, and the number of degrees of freedom is the dimension of the
1616:{\displaystyle Y_{i}=a+bx_{i}+e_{i}{\text{ for }}i=1,\dots ,n}
6228:
6016:
Inverse methods for atmospheric sounding: theory and practice
1194: â 1 degrees of freedom when the hypothesized mean
196:
In equations, the typical symbol for degrees of freedom is
4467:), the trace of the quadratic form of the hat matrix, tr(
141:), where certain random vectors are constrained to lie in
6029:
1520:
An example which is only slightly less simple is that of
3920:
Several commonly encountered statistical distributions (
2005:
We can generalise this to multiple regression involving
6027:
Adrian Doicu, Thomas
Trautmann, Franz Schreier (2010),
5855:
Nonparametric Simple
Regression: Smoothing Scatterplots
4431:
of the fit can be defined in various ways to implement
3127:
2905:
2827:
2695:
845:
649:
523:
487:
425:
308:
95:
71:
45:
is the number of values in the final calculation of a
5463:
5285:
5243:
5119:
4920:
4788:
4741:
4678:
4658:
4527:
4383:
4347:
4303:
4259:
4143:
4040:
3998:
3947:
3676:
3510:
3443:
3399:
3355:
2689:
2594:
2535:
2202:
2150:
2104:
2058:
1900:
1826:
1725:
1685:
1656:
1541:
1441:
1360:
1297:
1200:
1047:
1003:
972:
945:
776:
617:
419:
383:
302:
238:
8141:
Autoregressive conditional heteroskedasticity (ARCH)
6155:
Good, I. J. (1973). "What Are
Degrees of Freedom?".
3497:
The three-population example above is an example of
3345:
The observation vector, on the left-hand side, has 3
118:
Mathematically, degrees of freedom is the number of
8448:
8385:
8338:
8301:
8256:
8238:
8205:
8196:
8154:
8101:
8062:
8011:
8002:
7923:
7880:
7810:
7776:
7730:
7697:
7659:
7626:
7538:
7447:
7366:
7321:
7289:
7242:
7187:
7113:
7104:
6914:
6856:
6830:
6782:
6737:
6684:
6571:
6526:
6500:
6482:
6438:
6390:
6310:
6301:
5959:
Generalized additive models: an introduction with R
4498:. In the case of linear regression, the hat matrix
4197:are used. In other applications, such as modelling
3937:) have parameters that are commonly referred to as
5491:
5330:
5271:
5210:
5102:
4900:
4771:
4691:
4664:
4630:
4411:
4369:
4318:
4286:
4177:
4118:
4023:
3984:
3870:
3656:
3469:
3429:
3385:
3334:
2668:
2580:
2518:
2182:
2136:
2090:
1998:is used in specifying the model, while lower-case
1976:
1885:
1806:
1700:
1671:
1615:
1474:
1421:
1329:
1213:
1175:
1016:
985:
966:are normally distributed with mean 0 and variance
958:
928:
710:
632:
600:
398:
355:
274:
107:
77:
5971:David Ruppert, M. P. Wand, R. J. Carroll (2003),
5331:{\displaystyle {\hat {r}}'\Sigma ^{-1}{\hat {r}}}
4194:
3903:In some complicated settings, such as unbalanced
643:The second vector is constrained by the relation
4217:Many non-standard regression methods, including
2581:{\displaystyle {\bar {X}},{\bar {Y}},{\bar {Z}}}
993:, then the residual sum of squares has a scaled
711:{\textstyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})=0}
7689:Multivariate adaptive regression splines (MARS)
6064:Journal of the American Statistical Association
5975:, Cambridge University Press (eq.(3.28), p. 82)
5887:Journal of the American Statistical Association
3470:{\displaystyle {\overline {Z}}-{\overline {M}}}
216:for sample size. When reporting the results of
188:, beginning with his 1922 work on chi squares.
1517: â 1 degrees of freedom for errors.
1288:Perhaps the simplest example is this. Suppose
129:The term is most often used in the context of
6244:
5947:, CRC Press, (p. 54) and (eq.(B.1), p. 305))
5937:
5935:
4443:procedures. Here one can distinguish between
2588:are the means of the individual samples, and
1991: â 2 degrees of freedom for error.
366:Since this random vector can lie anywhere in
8:
6186:Walker, H. W. (1940). "Degrees of Freedom".
5480:
5464:
5260:
5244:
5161:
5145:
5036:
5020:
4962:
4946:
4830:
4814:
4400:
4384:
4358:
4348:
4172:
4144:
1432:be the "sample mean." Then the quantities
65:is to be estimated from a random sample of
8298:
8285:
8202:
8008:
7877:
7852:
7623:
7599:
7327:
7110:
6911:
6898:
6681:
6668:
6307:
6298:
6285:
6251:
6237:
6229:
6137:
5941:Trevor Hastie, Robert Tibshirani (1990),
5483:
5462:
5317:
5316:
5307:
5288:
5287:
5284:
5263:
5248:
5247:
5242:
5164:
5149:
5148:
5142:
5133:
5122:
5121:
5118:
5039:
5024:
5023:
5017:
4965:
4950:
4949:
4943:
4934:
4923:
4922:
4919:
4833:
4818:
4817:
4811:
4802:
4791:
4790:
4787:
4743:
4742:
4740:
4683:
4677:
4657:
4616:
4601:
4590:
4589:
4582:
4576:
4560:
4550:
4526:
4403:
4382:
4361:
4346:
4330:is the original vector of responses, and
4305:
4304:
4302:
4261:
4260:
4258:
4161:
4160:
4151:
4142:
4108:
4097:
4082:
4081:
4072:
4059:
4048:
4041:
4039:
4012:
3997:
3952:
3946:
3862:
3847:
3846:
3837:
3824:
3813:
3800:
3785:
3784:
3775:
3762:
3751:
3738:
3723:
3722:
3713:
3700:
3689:
3677:
3675:
3648:
3633:
3632:
3618:
3617:
3602:
3587:
3586:
3572:
3571:
3556:
3541:
3540:
3526:
3525:
3511:
3509:
3457:
3444:
3442:
3416:
3415:
3401:
3400:
3398:
3372:
3371:
3357:
3356:
3354:
3310:
3309:
3300:
3274:
3273:
3264:
3245:
3244:
3235:
3209:
3208:
3199:
3180:
3179:
3170:
3144:
3143:
3134:
3122:
3100:
3099:
3085:
3084:
3062:
3061:
3047:
3046:
3031:
3030:
3016:
3015:
2993:
2992:
2978:
2977:
2962:
2961:
2947:
2946:
2924:
2923:
2909:
2908:
2900:
2822:
2811:
2810:
2793:
2772:
2758:
2737:
2723:
2702:
2690:
2688:
2658:
2644:
2643:
2629:
2628:
2614:
2613:
2596:
2595:
2593:
2567:
2566:
2552:
2551:
2537:
2536:
2534:
2498:
2497:
2488:
2464:
2463:
2449:
2448:
2431:
2430:
2417:
2395:
2394:
2385:
2361:
2360:
2346:
2345:
2328:
2327:
2314:
2292:
2291:
2282:
2258:
2257:
2243:
2242:
2225:
2224:
2211:
2203:
2201:
2174:
2155:
2149:
2128:
2109:
2103:
2082:
2063:
2057:
1962:
1951:
1950:
1943:
1924:
1913:
1912:
1905:
1899:
1868:
1857:
1856:
1840:
1829:
1828:
1825:
1795:
1780:
1779:
1765:
1764:
1752:
1739:
1728:
1727:
1724:
1687:
1686:
1684:
1658:
1657:
1655:
1587:
1581:
1568:
1546:
1540:
1513: â 1. One says that there are
1475:{\displaystyle X_{i}-{\overline {X}}_{n}}
1466:
1456:
1446:
1440:
1407:
1388:
1381:
1372:
1362:
1359:
1321:
1302:
1296:
1205:
1199:
1150:
1144:
1129:
1128:
1119:
1106:
1095:
1080:
1062:
1061:
1051:
1048:
1046:
1008:
1002:
977:
971:
950:
944:
917:
898:
897:
888:
862:
861:
852:
840:
830:
815:
814:
805:
792:
781:
775:
730:th component. Therefore, this vector has
688:
687:
678:
665:
654:
648:
619:
618:
616:
576:
575:
566:
540:
539:
530:
518:
482:
471:
470:
453:
432:
420:
418:
385:
384:
382:
336:
315:
303:
301:
271:
262:
243:
237:
94:
70:
5999:Peter J. Green, B. W. Silverman (1994),
5707:Journal of the Royal Statistical Society
4129:follows a chi-squared distribution with
737:Mathematically, the first vector is the
640:. It therefore has 1 degree of freedom.
6105:. London: Macmillan. pp. 175â178.
5569:
5373:. Thus, the trace of the hat matrix is
4735:)), and the unbiased estimate is (with
4455:Regression effective degrees of freedom
4445:regression effective degrees of freedom
8215:KaplanâMeier estimator (product limit)
5792:(Third ed.). New York: Springer.
5361:smoother, which is the average of the
2193:The observations can be decomposed as
6018:, World Scientific (eq.(2.56), p. 31)
4703:Residual effective degrees of freedom
4449:residual effective degrees of freedom
4338:or, more generally, smoother matrix.
3430:{\displaystyle {\bar {Y}}-{\bar {M}}}
3386:{\displaystyle {\bar {X}}-{\bar {M}}}
1485:are residuals that may be considered
275:{\displaystyle X_{1},\dots ,X_{n}.\,}
7:
8525:
8225:Accelerated failure time (AFT) model
6205:Transcription by C Olsen with errata
5501:generalized chi-squared distribution
4178:{\displaystyle \{X_{i}-{\bar {X}}\}}
8537:
7820:Analysis of variance (ANOVA, anova)
3985:{\displaystyle X_{i};i=1,\ldots ,n}
3900: â 3 degrees of freedom.
2183:{\displaystyle Z_{1},\ldots ,Z_{n}}
2137:{\displaystyle Y_{1},\ldots ,Y_{n}}
2091:{\displaystyle X_{1},\ldots ,X_{n}}
1092:
760: â 1 degrees of freedom.
734: â 1 degrees of freedom.
7915:CochranâMantelâHaenszel statistics
6541:Pearson product-moment correlation
5986:Richly Parameterized Linear Models
5304:
5272:{\displaystyle \|{\hat {r}}\|^{2}}
4609:
4585:
4024:{\displaystyle (\mu ,\sigma ^{2})}
1708:be the least-squares estimates of
1330:{\displaystyle X_{1},\dots ,X_{n}}
25:
6188:Journal of Educational Psychology
5637:Journal of Educational Psychology
5534:Chi-squared per degree of freedom
5450:in atmospheric studies, and the
4377:; the residual sum-of-squares is
1994:Notationally, the capital letter
8536:
8524:
8512:
8499:
8498:
6139:10.1111/j.1467-9639.2008.00324.x
5962:, CRC Press, (eq.(4,14), p. 172)
5513:
4031:random variables, the statistic
2009:parameters and covariates (e.g.
8174:Least-squares spectral analysis
6031:, Springer (eq.(4.26), p. 114)
5910:Local regression and likelihood
4772:{\displaystyle {\hat {r}}=y-Hy}
3489:In analysis of variance (ANOVA)
7155:Mean-unbiased minimum-variance
6171:10.1080/00031305.1973.10479042
6005:, CRC Press (eq.(3.15), p. 37)
5740:CichoĹ, Mariusz (2020-06-01).
5701:Fisher, R. A. (January 1922).
5664:"The Probable Error of a Mean"
5381:effective degrees of freedom.
5322:
5293:
5253:
5193:
5187:
5154:
5127:
5094:
5080:
5068:
5062:
5029:
5008:
4985:
4955:
4928:
4884:
4872:
4865:
4852:
4823:
4796:
4748:
4595:
4540:
4534:
4310:
4287:{\displaystyle {\hat {y}}=Hy,}
4266:
4166:
4094:
4087:
4065:
4018:
3999:
3859:
3852:
3830:
3797:
3790:
3768:
3735:
3728:
3706:
3645:
3638:
3623:
3614:
3599:
3592:
3577:
3568:
3553:
3546:
3531:
3522:
3421:
3406:
3377:
3362:
3315:
3279:
3250:
3214:
3185:
3149:
3105:
3090:
3067:
3052:
3036:
3021:
2998:
2983:
2967:
2952:
2929:
2914:
2816:
2655:
2649:
2634:
2619:
2610:
2601:
2572:
2557:
2542:
2509:
2503:
2481:
2475:
2469:
2454:
2445:
2436:
2406:
2400:
2378:
2372:
2366:
2351:
2342:
2333:
2303:
2297:
2275:
2269:
2263:
2248:
2239:
2230:
2025:degrees of freedom for errors
1801:
1761:
1701:{\displaystyle {\widehat {b}}}
1672:{\displaystyle {\widehat {a}}}
1167:
1155:
1141:
1134:
1112:
1086:
1067:
1058:
912:
903:
867:
842:
827:
820:
798:
699:
693:
671:
624:
581:
545:
476:
390:
285:This can be represented as an
18:Degree of freedom (statistics)
1:
8468:Geographic information system
7684:Simultaneous equations models
5819:, Jerome H. Friedman (2009),
5582:Glossary of Statistical Terms
5452:non-integer degree of freedom
5440:equivalent degrees of freedom
5353:Consider, as an example, the
2015:degrees of freedom of the fit
1225:In structural equation models
7651:Coefficient of determination
7262:Uniformly most powerful test
5788:Christensen, Ronald (2002).
5628:Walker, H. M. (April 1940).
5492:{\displaystyle \|y-Hy\|^{2}}
5457:The residual sum-of-squares
4429:effective degrees of freedom
4412:{\displaystyle \|y-Hy\|^{2}}
4195:effective degrees of freedom
3916:In probability distributions
3908:but are simply providing an
3499:one-way Analysis of Variance
3462:
3449:
1461:
1367:
1274:degrees of freedom for error
752: â 1)-dimensional
741:of the data vector onto the
8220:Proportional hazards models
8164:Spectral density estimation
8146:Vector autoregression (VAR)
7580:Maximum posterior estimator
6812:Randomized controlled trial
5944:Generalized additive models
5448:degree of freedom of signal
4484:Satterthwaite approximation
4241:projections, but rather on
1278:residual degrees of freedom
1017:{\displaystyle \sigma ^{2}}
986:{\displaystyle \sigma ^{2}}
370:-dimensional space, it has
8581:
7980:Multivariate distributions
6400:Average absolute deviation
6120:Eisenhauer, J. G. (2008).
5758:10.1007/s43440-020-00110-5
4370:{\displaystyle \|Hy\|^{2}}
4319:{\displaystyle {\hat {y}}}
4213:In non-standard regression
1260:
756:of this subspace, and has
633:{\displaystyle {\bar {X}}}
399:{\displaystyle {\bar {X}}}
29:
8494:
8297:
8284:
7968:Structural equation model
7876:
7851:
7622:
7598:
7330:
7304:Score/Lagrange multiplier
6910:
6897:
6719:Sample size determination
6680:
6667:
6297:
6284:
6266:
6158:The American Statistician
6103:Statistics for Economists
6014:Clive D. Rodgers (2000),
5973:Semiparametric Regression
5837:10.1007/978-0-387-84858-7
5539:Pooled degrees of freedom
5444:non-parametric regression
5438:Similar concepts are the
4692:{\displaystyle \chi ^{2}}
4235:semiparametric regression
4219:regularized least squares
2033:The demonstration of the
1031:Likewise, the one-sample
8463:Environmental statistics
7985:Elliptical distributions
7778:Generalized linear model
7707:Simple linear regression
7477:HodgesâLehmann estimator
6934:Probability distribution
6843:Stochastic approximation
6405:Coefficient of variation
5984:James S. Hodges (2014),
5544:Replication (statistics)
5377:. Thus the smooth costs
1987:One says that there are
1214:{\displaystyle \mu _{0}}
995:chi-squared distribution
182:Student's t-distribution
8123:Cross-correlation (XCF)
7731:Non-standard predictors
7165:LehmannâScheffĂŠ theorem
6838:Adaptive clinical trial
5746:Pharmacological Reports
3992:are independent normal
765:residual sum-of-squares
49:that are free to vary.
8519:Mathematics portal
8340:Engineering statistics
8248:NelsonâAalen estimator
7825:Analysis of covariance
7712:Ordinary least squares
7636:Pearson product-moment
7040:Statistical functional
6951:Empirical distribution
6784:Controlled experiments
6513:Frequency distribution
6291:Descriptive statistics
6101:Bowers, David (1982).
5956:Simon N. Wood (2006),
5662:Student (March 1908).
5610:. Statistics Solutions
5493:
5332:
5273:
5212:
5104:
4902:
4773:
4693:
4666:
4632:
4463:of the hat matrix, tr(
4413:
4371:
4320:
4288:
4239:ordinary least squares
4179:
4120:
4064:
4025:
3986:
3872:
3829:
3767:
3705:
3658:
3471:
3431:
3387:
3336:
2670:
2582:
2520:
2184:
2138:
2092:
1978:
1887:
1808:
1702:
1673:
1617:
1476:
1423:
1331:
1263:Residuals (statistics)
1215:
1177:
1111:
1018:
997:(scaled by the factor
987:
960:
930:
797:
712:
670:
634:
602:
400:
357:
276:
109:
79:
54:statistical parameters
8435:Population statistics
8377:System identification
8111:Autocorrelation (ACF)
8039:Exponential smoothing
7953:Discriminant analysis
7948:Canonical correlation
7812:Partition of variance
7674:Regression validation
7518:(JonckheereâTerpstra)
7417:Likelihood-ratio test
7106:Frequentist inference
7018:Locationâscale family
6939:Sampling distribution
6904:Statistical inference
6871:Cross-sectional study
6858:Observational studies
6817:Randomized experiment
6646:Stem-and-leaf display
6448:Central limit theorem
6050:10.1007/s001900050161
5907:Clive Loader (1999),
5890:, 93 (441), 120â131.
5494:
5333:
5274:
5213:
5105:
4903:
4774:
4694:
4667:
4633:
4441:statistical inference
4433:goodness-of-fit tests
4414:
4372:
4321:
4289:
4180:
4121:
4044:
4026:
3987:
3873:
3809:
3747:
3685:
3659:
3472:
3432:
3388:
3337:
2671:
2583:
2521:
2185:
2139:
2093:
1979:
1888:
1809:
1716:. Then the residuals
1703:
1674:
1618:
1477:
1424:
1332:
1261:Further information:
1216:
1178:
1091:
1019:
988:
961:
959:{\displaystyle X_{i}}
931:
777:
754:orthogonal complement
713:
650:
635:
603:
401:
358:
277:
110:
80:
8358:Probabilistic design
7943:Principal components
7786:Exponential families
7738:Nonlinear regression
7717:General linear model
7679:Mixed effects models
7669:Errors and residuals
7646:Confounding variable
7548:Bayesian probability
7526:Van der Waerden test
7516:Ordered alternative
7281:Multiple comparisons
7160:RaoâBlackwellization
7123:Estimating equations
7079:Statistical distance
6797:Factorial experiment
6330:Arithmetic-Geometric
6216:Yu, Chong-ho (1997)
6122:"Degrees of Freedom"
6067:, 58 (302), 401â414
5929:, (eq.(2.18), p. 30)
5630:"Degrees of Freedom"
5604:"Degrees of Freedom"
5578:"Degrees of Freedom"
5461:
5283:
5241:
5117:
4918:
4786:
4739:
4676:
4656:
4525:
4381:
4345:
4301:
4257:
4141:
4038:
3996:
3945:
3674:
3508:
3441:
3397:
3353:
2687:
2676:is the mean of all 3
2592:
2533:
2200:
2148:
2102:
2056:
2047:analysis of variance
1898:
1824:
1723:
1683:
1654:
1539:
1439:
1358:
1295:
1198:
1045:
1001:
970:
943:
774:
647:
615:
417:
381:
374:degrees of freedom.
300:
236:
171:William Sealy Gosset
167:Carl Friedrich Gauss
139:analysis of variance
93:
69:
30:For other uses, see
8430:Official statistics
8353:Methods engineering
8034:Seasonal adjustment
7802:Poisson regressions
7722:Bayesian regression
7661:Regression analysis
7641:Partial correlation
7613:Regression analysis
7212:Prediction interval
7207:Likelihood interval
7197:Confidence interval
7189:Interval estimation
7150:Unbiased estimators
6968:Model specification
6848:Up-and-down designs
6536:Partial correlation
6492:Index of dispersion
6410:Interquartile range
6222:Dallal, GE. (2003)
6126:Teaching Statistics
6044:, 72 (4), 200â214,
5584:. Animated Software
5529:Bessel's correction
4419:. However, because
4237:, are not based on
939:If the data points
122:of the domain of a
8565:Statistical theory
8450:Spatial statistics
8330:Medical statistics
8230:First hitting time
8184:Whittle likelihood
7835:Degrees of freedom
7830:Multivariate ANOVA
7763:Heteroscedasticity
7575:Bayesian estimator
7540:Bayesian inference
7389:KolmogorovâSmirnov
7274:Randomization test
7244:Testing hypotheses
7217:Tolerance interval
7128:Maximum likelihood
7023:Exponential family
6956:Density estimation
6916:Statistical theory
6876:Natural experiment
6822:Scientific control
6739:Survey methodology
6425:Standard deviation
6224:Degrees of Freedom
6075:(eq.(5.19)â(5.20))
5823:, 2nd ed., 746 p.
5521:Mathematics portal
5489:
5434:Other formulations
5328:
5269:
5208:
5100:
4898:
4769:
4689:
4662:
4628:
4581:
4555:
4409:
4367:
4316:
4284:
4207:degrees of freedom
4175:
4116:
4021:
3982:
3939:degrees of freedom
3868:
3654:
3467:
3427:
3383:
3332:
3323:
3113:
2891:
2801:
2666:
2578:
2516:
2514:
2180:
2134:
2088:
1974:
1883:
1804:
1698:
1669:
1613:
1472:
1419:
1327:
1235:degrees of freedom
1211:
1190:distribution with
1173:
1014:
983:
956:
926:
911:
739:oblique projection
708:
630:
598:
589:
509:
461:
396:
353:
344:
272:
105:
75:
43:degrees of freedom
32:Degrees of freedom
8552:
8551:
8490:
8489:
8486:
8485:
8425:National accounts
8395:Actuarial science
8387:Social statistics
8280:
8279:
8276:
8275:
8272:
8271:
8207:Survival function
8192:
8191:
8054:Granger causality
7895:Contingency table
7870:Survival analysis
7847:
7846:
7843:
7842:
7699:Linear regression
7594:
7593:
7590:
7589:
7565:Credible interval
7534:
7533:
7317:
7316:
7133:Method of moments
7002:Parametric family
6963:Statistical model
6893:
6892:
6889:
6888:
6807:Random assignment
6729:Statistical power
6663:
6662:
6659:
6658:
6508:Contingency table
6478:
6477:
6345:Generalized/power
6052:(eq.(27), p. 205)
5919:978-0-387-98775-0
5865:978-0-7619-1585-0
5829:978-0-387-84857-0
5817:Robert Tibshirani
5554:Statistical model
5359:nearest neighbour
5325:
5296:
5256:
5203:
5157:
5130:
5098:
5032:
5012:
4958:
4931:
4893:
4826:
4799:
4751:
4665:{\displaystyle H}
4623:
4598:
4572:
4546:
4313:
4269:
4231:smoothing splines
4169:
4114:
4090:
3855:
3793:
3731:
3680:
3641:
3626:
3595:
3580:
3549:
3534:
3514:
3465:
3452:
3424:
3409:
3380:
3365:
3318:
3282:
3253:
3217:
3188:
3152:
3108:
3093:
3070:
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3024:
3001:
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2652:
2637:
2622:
2604:
2575:
2560:
2545:
2506:
2472:
2457:
2439:
2403:
2369:
2354:
2336:
2300:
2266:
2251:
2233:
2043:linear regression
1959:
1921:
1865:
1837:
1788:
1773:
1736:
1695:
1666:
1590:
1464:
1417:
1370:
1171:
1170:
1137:
1070:
1056:
906:
870:
823:
696:
627:
584:
548:
479:
393:
224:Of random vectors
218:statistical tests
135:linear regression
16:(Redirected from
8572:
8540:
8539:
8528:
8527:
8517:
8516:
8502:
8501:
8405:Crime statistics
8299:
8286:
8203:
8169:Fourier analysis
8156:Frequency domain
8136:
8083:
8049:Structural break
8009:
7958:Cluster analysis
7905:Log-linear model
7878:
7853:
7794:
7768:Homoscedasticity
7624:
7600:
7519:
7511:
7503:
7502:(KruskalâWallis)
7487:
7472:
7427:Cross validation
7412:
7394:AndersonâDarling
7341:
7328:
7299:Likelihood-ratio
7291:Parametric tests
7269:Permutation test
7252:1- & 2-tails
7143:Minimum distance
7115:Point estimation
7111:
7062:Optimal decision
7013:
6912:
6899:
6881:Quasi-experiment
6831:Adaptive designs
6682:
6669:
6546:Rank correlation
6308:
6299:
6286:
6253:
6246:
6239:
6230:
6203:
6200:10.1037/h0054588
6182:
6151:
6141:
6116:
6089:
6082:
6076:
6059:
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6038:
6032:
6025:
6019:
6012:
6006:
5997:
5991:
5982:
5976:
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5954:
5948:
5939:
5930:
5905:
5899:
5882:
5876:
5875:
5873:
5872:
5852:Fox, J. (2000).
5849:
5843:
5810:
5804:
5803:
5785:
5779:
5776:
5770:
5769:
5737:
5731:
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5698:
5692:
5691:
5659:
5653:
5652:
5649:10.1037/h0054588
5634:
5625:
5619:
5618:
5616:
5615:
5608:HyperStat Online
5599:
5593:
5592:
5590:
5589:
5574:
5523:
5518:
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5498:
5496:
5495:
5490:
5488:
5487:
5428:confidence level
5414:
5337:
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5334:
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5326:
5318:
5315:
5314:
5302:
5298:
5297:
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4600:
4599:
4591:
4583:
4580:
4568:
4567:
4554:
4497:
4471:), the form tr(2
4437:cross-validation
4418:
4416:
4415:
4410:
4408:
4407:
4376:
4374:
4373:
4368:
4366:
4365:
4325:
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4322:
4317:
4315:
4314:
4306:
4293:
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4290:
4285:
4271:
4270:
4262:
4227:linear smoothers
4223:ridge regression
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3392:
3390:
3389:
3384:
3382:
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3373:
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3358:
3341:
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3338:
3333:
3328:
3327:
3320:
3319:
3311:
3305:
3304:
3284:
3283:
3275:
3269:
3268:
3255:
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3246:
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3239:
3219:
3218:
3210:
3204:
3203:
3190:
3189:
3181:
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2132:
2114:
2113:
2097:
2095:
2094:
2089:
2087:
2086:
2068:
2067:
2029:In linear models
1983:
1981:
1980:
1975:
1967:
1966:
1961:
1960:
1952:
1948:
1947:
1929:
1928:
1923:
1922:
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1884:
1873:
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1729:
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1704:
1699:
1697:
1696:
1688:
1678:
1676:
1675:
1670:
1668:
1667:
1659:
1650:are random. Let
1622:
1620:
1619:
1614:
1591:
1588:
1586:
1585:
1573:
1572:
1551:
1550:
1481:
1479:
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1471:
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1465:
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1451:
1450:
1428:
1426:
1425:
1420:
1418:
1413:
1412:
1411:
1393:
1392:
1382:
1377:
1376:
1371:
1363:
1342:random variables
1336:
1334:
1333:
1328:
1326:
1325:
1307:
1306:
1220:
1218:
1217:
1212:
1210:
1209:
1182:
1180:
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1174:
1172:
1154:
1149:
1148:
1139:
1138:
1130:
1124:
1123:
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1105:
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1089:
1085:
1084:
1072:
1071:
1063:
1057:
1052:
1049:
1023:
1021:
1020:
1015:
1013:
1012:
992:
990:
989:
984:
982:
981:
965:
963:
962:
957:
955:
954:
935:
933:
932:
927:
922:
921:
916:
915:
908:
907:
899:
893:
892:
872:
871:
863:
857:
856:
835:
834:
825:
824:
816:
810:
809:
796:
791:
717:
715:
714:
709:
698:
697:
689:
683:
682:
669:
664:
639:
637:
636:
631:
629:
628:
620:
607:
605:
604:
599:
594:
593:
586:
585:
577:
571:
570:
550:
549:
541:
535:
534:
514:
513:
481:
480:
472:
466:
465:
458:
457:
437:
436:
405:
403:
402:
397:
395:
394:
386:
362:
360:
359:
354:
349:
348:
341:
340:
320:
319:
281:
279:
278:
273:
267:
266:
248:
247:
143:linear subspaces
114:
112:
111:
108:{\textstyle N-1}
106:
84:
82:
81:
76:
41:, the number of
21:
8580:
8579:
8575:
8574:
8573:
8571:
8570:
8569:
8555:
8554:
8553:
8548:
8511:
8482:
8444:
8381:
8367:quality control
8334:
8316:Clinical trials
8293:
8268:
8252:
8240:Hazard function
8234:
8188:
8150:
8134:
8097:
8093:BreuschâGodfrey
8081:
8058:
7998:
7973:Factor analysis
7919:
7900:Graphical model
7872:
7839:
7806:
7792:
7772:
7726:
7693:
7655:
7618:
7617:
7586:
7530:
7517:
7509:
7501:
7485:
7470:
7449:Rank statistics
7443:
7422:Model selection
7410:
7368:Goodness of fit
7362:
7339:
7313:
7285:
7238:
7183:
7172:Median unbiased
7100:
7011:
6944:Order statistic
6906:
6885:
6852:
6826:
6778:
6733:
6676:
6674:Data collection
6655:
6567:
6522:
6496:
6474:
6434:
6386:
6303:Continuous data
6293:
6280:
6262:
6257:
6213:
6185:
6154:
6119:
6113:
6100:
6097:
6095:Further reading
6092:
6088:, 70 (1), 67â88
6083:
6079:
6060:
6056:
6039:
6035:
6026:
6022:
6013:
6009:
5998:
5994:
5983:
5979:
5970:
5966:
5955:
5951:
5940:
5933:
5906:
5902:
5883:
5879:
5870:
5868:
5866:
5851:
5850:
5846:
5811:
5807:
5800:
5787:
5786:
5782:
5777:
5773:
5739:
5738:
5734:
5719:10.2307/2340521
5700:
5699:
5695:
5680:10.2307/2331554
5661:
5660:
5656:
5632:
5627:
5626:
5622:
5613:
5611:
5602:Lane, David M.
5601:
5600:
5596:
5587:
5585:
5576:
5575:
5571:
5567:
5519:
5512:
5509:
5479:
5459:
5458:
5436:
5393:
5344:
5303:
5286:
5281:
5280:
5259:
5239:
5238:
5171:
5160:
5144:
5120:
5115:
5114:
5086:
5046:
5035:
5019:
5000:
4972:
4961:
4945:
4921:
4916:
4915:
4864:
4851:
4847:
4840:
4829:
4813:
4789:
4784:
4783:
4737:
4736:
4705:
4679:
4674:
4673:
4654:
4653:
4643:leverage scores
4612:
4608:
4588:
4584:
4556:
4523:
4522:
4487:
4457:
4399:
4379:
4378:
4357:
4343:
4342:
4299:
4298:
4255:
4254:
4215:
4147:
4139:
4138:
4104:
4093:
4068:
4043:
4036:
4035:
4008:
3994:
3993:
3948:
3943:
3942:
3918:
3858:
3833:
3796:
3771:
3734:
3709:
3672:
3671:
3644:
3598:
3552:
3506:
3505:
3491:
3439:
3438:
3395:
3394:
3351:
3350:
3322:
3321:
3296:
3293:
3292:
3286:
3285:
3260:
3257:
3256:
3231:
3228:
3227:
3221:
3220:
3195:
3192:
3191:
3166:
3163:
3162:
3156:
3155:
3130:
3123:
3112:
3111:
3081:
3080:
3074:
3073:
3043:
3042:
3012:
3011:
3005:
3004:
2974:
2973:
2943:
2942:
2936:
2935:
2901:
2890:
2889:
2883:
2882:
2876:
2875:
2869:
2868:
2862:
2861:
2855:
2854:
2848:
2847:
2841:
2840:
2834:
2833:
2823:
2800:
2799:
2789:
2786:
2785:
2779:
2778:
2768:
2765:
2764:
2754:
2751:
2750:
2744:
2743:
2733:
2730:
2729:
2719:
2716:
2715:
2709:
2708:
2698:
2691:
2685:
2684:
2590:
2589:
2531:
2530:
2513:
2512:
2484:
2423:
2413:
2410:
2409:
2381:
2320:
2310:
2307:
2306:
2278:
2217:
2207:
2198:
2197:
2170:
2151:
2146:
2145:
2124:
2105:
2100:
2099:
2078:
2059:
2054:
2053:
2031:
1949:
1939:
1911:
1901:
1896:
1895:
1855:
1827:
1822:
1821:
1791:
1748:
1726:
1721:
1720:
1681:
1680:
1652:
1651:
1649:
1640:
1635:is given, but e
1634:
1589: for
1577:
1564:
1542:
1537:
1536:
1500:
1455:
1442:
1437:
1436:
1403:
1384:
1383:
1361:
1356:
1355:
1317:
1298:
1293:
1292:
1286:
1265:
1259:
1227:
1201:
1196:
1195:
1140:
1115:
1076:
1050:
1043:
1042:
1004:
999:
998:
973:
968:
967:
946:
941:
940:
910:
909:
884:
881:
880:
874:
873:
848:
841:
839:
826:
801:
772:
771:
674:
645:
644:
613:
612:
588:
587:
562:
559:
558:
552:
551:
526:
519:
508:
507:
501:
500:
494:
493:
483:
460:
459:
449:
446:
445:
439:
438:
428:
421:
415:
414:
379:
378:
343:
342:
332:
329:
328:
322:
321:
311:
304:
298:
297:
258:
239:
234:
233:
226:
202:Greek letter nu
194:
163:
91:
90:
67:
66:
35:
28:
23:
22:
15:
12:
11:
5:
8578:
8576:
8568:
8567:
8557:
8556:
8550:
8549:
8547:
8546:
8534:
8522:
8508:
8495:
8492:
8491:
8488:
8487:
8484:
8483:
8481:
8480:
8475:
8470:
8465:
8460:
8454:
8452:
8446:
8445:
8443:
8442:
8437:
8432:
8427:
8422:
8417:
8412:
8407:
8402:
8397:
8391:
8389:
8383:
8382:
8380:
8379:
8374:
8369:
8360:
8355:
8350:
8344:
8342:
8336:
8335:
8333:
8332:
8327:
8322:
8313:
8311:Bioinformatics
8307:
8305:
8295:
8294:
8289:
8282:
8281:
8278:
8277:
8274:
8273:
8270:
8269:
8267:
8266:
8260:
8258:
8254:
8253:
8251:
8250:
8244:
8242:
8236:
8235:
8233:
8232:
8227:
8222:
8217:
8211:
8209:
8200:
8194:
8193:
8190:
8189:
8187:
8186:
8181:
8176:
8171:
8166:
8160:
8158:
8152:
8151:
8149:
8148:
8143:
8138:
8130:
8125:
8120:
8119:
8118:
8116:partial (PACF)
8107:
8105:
8099:
8098:
8096:
8095:
8090:
8085:
8077:
8072:
8066:
8064:
8063:Specific tests
8060:
8059:
8057:
8056:
8051:
8046:
8041:
8036:
8031:
8026:
8021:
8015:
8013:
8006:
8000:
7999:
7997:
7996:
7995:
7994:
7993:
7992:
7977:
7976:
7975:
7965:
7963:Classification
7960:
7955:
7950:
7945:
7940:
7935:
7929:
7927:
7921:
7920:
7918:
7917:
7912:
7910:McNemar's test
7907:
7902:
7897:
7892:
7886:
7884:
7874:
7873:
7856:
7849:
7848:
7845:
7844:
7841:
7840:
7838:
7837:
7832:
7827:
7822:
7816:
7814:
7808:
7807:
7805:
7804:
7788:
7782:
7780:
7774:
7773:
7771:
7770:
7765:
7760:
7755:
7750:
7748:Semiparametric
7745:
7740:
7734:
7732:
7728:
7727:
7725:
7724:
7719:
7714:
7709:
7703:
7701:
7695:
7694:
7692:
7691:
7686:
7681:
7676:
7671:
7665:
7663:
7657:
7656:
7654:
7653:
7648:
7643:
7638:
7632:
7630:
7620:
7619:
7616:
7615:
7610:
7604:
7603:
7596:
7595:
7592:
7591:
7588:
7587:
7585:
7584:
7583:
7582:
7572:
7567:
7562:
7561:
7560:
7555:
7544:
7542:
7536:
7535:
7532:
7531:
7529:
7528:
7523:
7522:
7521:
7513:
7505:
7489:
7486:(MannâWhitney)
7481:
7480:
7479:
7466:
7465:
7464:
7453:
7451:
7445:
7444:
7442:
7441:
7440:
7439:
7434:
7429:
7419:
7414:
7411:(ShapiroâWilk)
7406:
7401:
7396:
7391:
7386:
7378:
7372:
7370:
7364:
7363:
7361:
7360:
7352:
7343:
7331:
7325:
7323:Specific tests
7319:
7318:
7315:
7314:
7312:
7311:
7306:
7301:
7295:
7293:
7287:
7286:
7284:
7283:
7278:
7277:
7276:
7266:
7265:
7264:
7254:
7248:
7246:
7240:
7239:
7237:
7236:
7235:
7234:
7229:
7219:
7214:
7209:
7204:
7199:
7193:
7191:
7185:
7184:
7182:
7181:
7176:
7175:
7174:
7169:
7168:
7167:
7162:
7147:
7146:
7145:
7140:
7135:
7130:
7119:
7117:
7108:
7102:
7101:
7099:
7098:
7093:
7088:
7087:
7086:
7076:
7071:
7070:
7069:
7059:
7058:
7057:
7052:
7047:
7037:
7032:
7027:
7026:
7025:
7020:
7015:
6999:
6998:
6997:
6992:
6987:
6977:
6976:
6975:
6970:
6960:
6959:
6958:
6948:
6947:
6946:
6936:
6931:
6926:
6920:
6918:
6908:
6907:
6902:
6895:
6894:
6891:
6890:
6887:
6886:
6884:
6883:
6878:
6873:
6868:
6862:
6860:
6854:
6853:
6851:
6850:
6845:
6840:
6834:
6832:
6828:
6827:
6825:
6824:
6819:
6814:
6809:
6804:
6799:
6794:
6788:
6786:
6780:
6779:
6777:
6776:
6774:Standard error
6771:
6766:
6761:
6760:
6759:
6754:
6743:
6741:
6735:
6734:
6732:
6731:
6726:
6721:
6716:
6711:
6706:
6704:Optimal design
6701:
6696:
6690:
6688:
6678:
6677:
6672:
6665:
6664:
6661:
6660:
6657:
6656:
6654:
6653:
6648:
6643:
6638:
6633:
6628:
6623:
6618:
6613:
6608:
6603:
6598:
6593:
6588:
6583:
6577:
6575:
6569:
6568:
6566:
6565:
6560:
6559:
6558:
6553:
6543:
6538:
6532:
6530:
6524:
6523:
6521:
6520:
6515:
6510:
6504:
6502:
6501:Summary tables
6498:
6497:
6495:
6494:
6488:
6486:
6480:
6479:
6476:
6475:
6473:
6472:
6471:
6470:
6465:
6460:
6450:
6444:
6442:
6436:
6435:
6433:
6432:
6427:
6422:
6417:
6412:
6407:
6402:
6396:
6394:
6388:
6387:
6385:
6384:
6379:
6374:
6373:
6372:
6367:
6362:
6357:
6352:
6347:
6342:
6337:
6335:Contraharmonic
6332:
6327:
6316:
6314:
6305:
6295:
6294:
6289:
6282:
6281:
6279:
6278:
6273:
6267:
6264:
6263:
6258:
6256:
6255:
6248:
6241:
6233:
6227:
6226:
6220:
6212:
6211:External links
6209:
6208:
6207:
6194:(4): 253â269.
6183:
6165:(5): 227â228.
6152:
6117:
6111:
6096:
6093:
6091:
6090:
6077:
6054:
6033:
6020:
6007:
5992:
5977:
5964:
5949:
5931:
5927:10.1007/b98858
5900:
5877:
5864:
5844:
5805:
5798:
5780:
5771:
5752:(3): 481â485.
5732:
5693:
5654:
5643:(4): 253â269.
5620:
5594:
5568:
5566:
5563:
5562:
5561:
5556:
5551:
5546:
5541:
5536:
5531:
5525:
5524:
5508:
5505:
5486:
5482:
5478:
5475:
5472:
5469:
5466:
5435:
5432:
5343:
5340:
5324:
5321:
5313:
5310:
5306:
5301:
5295:
5292:
5266:
5262:
5255:
5252:
5246:
5219:
5218:
5207:
5201:
5198:
5195:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5167:
5163:
5156:
5153:
5147:
5141:
5136:
5129:
5126:
5111:
5110:
5096:
5092:
5089:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5042:
5038:
5031:
5028:
5022:
5016:
5010:
5006:
5003:
4999:
4996:
4993:
4990:
4987:
4984:
4981:
4978:
4975:
4968:
4964:
4957:
4954:
4948:
4942:
4937:
4930:
4927:
4909:
4908:
4897:
4890:
4886:
4883:
4880:
4877:
4874:
4870:
4867:
4863:
4860:
4857:
4854:
4850:
4846:
4843:
4836:
4832:
4825:
4822:
4816:
4810:
4805:
4798:
4795:
4768:
4765:
4762:
4759:
4756:
4750:
4747:
4704:
4701:
4686:
4682:
4661:
4639:
4638:
4627:
4619:
4615:
4611:
4604:
4597:
4594:
4587:
4579:
4575:
4571:
4566:
4563:
4559:
4553:
4549:
4545:
4542:
4539:
4536:
4533:
4530:
4456:
4453:
4406:
4402:
4398:
4395:
4392:
4389:
4386:
4364:
4360:
4356:
4353:
4350:
4312:
4309:
4295:
4294:
4283:
4280:
4277:
4274:
4268:
4265:
4214:
4211:
4174:
4168:
4165:
4159:
4154:
4150:
4146:
4127:
4126:
4111:
4107:
4100:
4096:
4089:
4086:
4080:
4075:
4071:
4067:
4062:
4057:
4054:
4051:
4047:
4020:
4015:
4011:
4007:
4004:
4001:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3955:
3951:
3917:
3914:
3879:
3878:
3865:
3861:
3854:
3851:
3845:
3840:
3836:
3832:
3827:
3822:
3819:
3816:
3812:
3808:
3803:
3799:
3792:
3789:
3783:
3778:
3774:
3770:
3765:
3760:
3757:
3754:
3750:
3746:
3741:
3737:
3730:
3727:
3721:
3716:
3712:
3708:
3703:
3698:
3695:
3692:
3688:
3684:
3665:
3664:
3651:
3647:
3640:
3637:
3631:
3625:
3622:
3616:
3613:
3610:
3605:
3601:
3594:
3591:
3585:
3579:
3576:
3570:
3567:
3564:
3559:
3555:
3548:
3545:
3539:
3533:
3530:
3524:
3521:
3518:
3490:
3487:
3464:
3461:
3456:
3451:
3448:
3423:
3420:
3414:
3408:
3405:
3379:
3376:
3370:
3364:
3361:
3343:
3342:
3331:
3326:
3317:
3314:
3308:
3303:
3299:
3295:
3294:
3291:
3288:
3287:
3281:
3278:
3272:
3267:
3263:
3259:
3258:
3252:
3249:
3243:
3238:
3234:
3230:
3229:
3226:
3223:
3222:
3216:
3213:
3207:
3202:
3198:
3194:
3193:
3187:
3184:
3178:
3173:
3169:
3165:
3164:
3161:
3158:
3157:
3151:
3148:
3142:
3137:
3133:
3129:
3128:
3126:
3121:
3116:
3107:
3104:
3098:
3092:
3089:
3083:
3082:
3079:
3076:
3075:
3069:
3066:
3060:
3054:
3051:
3045:
3044:
3038:
3035:
3029:
3023:
3020:
3014:
3013:
3010:
3007:
3006:
3000:
2997:
2991:
2985:
2982:
2976:
2975:
2969:
2966:
2960:
2954:
2951:
2945:
2944:
2941:
2938:
2937:
2931:
2928:
2922:
2916:
2913:
2907:
2906:
2904:
2899:
2894:
2888:
2885:
2884:
2881:
2878:
2877:
2874:
2871:
2870:
2867:
2864:
2863:
2860:
2857:
2856:
2853:
2850:
2849:
2846:
2843:
2842:
2839:
2836:
2835:
2832:
2829:
2828:
2826:
2818:
2815:
2809:
2804:
2796:
2792:
2788:
2787:
2784:
2781:
2780:
2775:
2771:
2767:
2766:
2761:
2757:
2753:
2752:
2749:
2746:
2745:
2740:
2736:
2732:
2731:
2726:
2722:
2718:
2717:
2714:
2711:
2710:
2705:
2701:
2697:
2696:
2694:
2665:
2661:
2657:
2651:
2648:
2642:
2636:
2633:
2627:
2621:
2618:
2612:
2609:
2603:
2600:
2574:
2571:
2565:
2559:
2556:
2550:
2544:
2541:
2527:
2526:
2511:
2505:
2502:
2496:
2491:
2487:
2483:
2480:
2477:
2471:
2468:
2462:
2456:
2453:
2447:
2444:
2438:
2435:
2429:
2426:
2424:
2420:
2416:
2412:
2411:
2408:
2402:
2399:
2393:
2388:
2384:
2380:
2377:
2374:
2368:
2365:
2359:
2353:
2350:
2344:
2341:
2335:
2332:
2326:
2323:
2321:
2317:
2313:
2309:
2308:
2305:
2299:
2296:
2290:
2285:
2281:
2277:
2274:
2271:
2265:
2262:
2256:
2250:
2247:
2241:
2238:
2232:
2229:
2223:
2220:
2218:
2214:
2210:
2206:
2205:
2177:
2173:
2169:
2166:
2163:
2158:
2154:
2131:
2127:
2123:
2120:
2117:
2112:
2108:
2085:
2081:
2077:
2074:
2071:
2066:
2062:
2030:
2027:
1985:
1984:
1973:
1970:
1965:
1958:
1955:
1946:
1942:
1938:
1935:
1932:
1927:
1920:
1917:
1908:
1904:
1893:
1882:
1879:
1876:
1871:
1864:
1861:
1854:
1851:
1848:
1843:
1836:
1833:
1815:
1814:
1803:
1798:
1794:
1787:
1784:
1778:
1772:
1769:
1763:
1760:
1755:
1751:
1747:
1742:
1735:
1732:
1694:
1691:
1665:
1662:
1645:
1636:
1630:
1624:
1623:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1584:
1580:
1576:
1571:
1567:
1563:
1560:
1557:
1554:
1549:
1545:
1524:estimation of
1496:
1483:
1482:
1469:
1463:
1460:
1454:
1449:
1445:
1430:
1429:
1416:
1410:
1406:
1402:
1399:
1396:
1391:
1387:
1380:
1375:
1369:
1366:
1346:expected value
1338:
1337:
1324:
1320:
1316:
1313:
1310:
1305:
1301:
1285:
1282:
1276:, also called
1258:
1255:
1226:
1223:
1208:
1204:
1184:
1183:
1169:
1166:
1163:
1160:
1157:
1153:
1147:
1143:
1136:
1133:
1127:
1122:
1118:
1114:
1109:
1104:
1101:
1098:
1094:
1088:
1083:
1079:
1075:
1069:
1066:
1060:
1055:
1011:
1007:
980:
976:
953:
949:
937:
936:
925:
920:
914:
905:
902:
896:
891:
887:
883:
882:
879:
876:
875:
869:
866:
860:
855:
851:
847:
846:
844:
838:
833:
829:
822:
819:
813:
808:
804:
800:
795:
790:
787:
784:
780:
707:
704:
701:
695:
692:
686:
681:
677:
673:
668:
663:
660:
657:
653:
626:
623:
609:
608:
597:
592:
583:
580:
574:
569:
565:
561:
560:
557:
554:
553:
547:
544:
538:
533:
529:
525:
524:
522:
517:
512:
506:
503:
502:
499:
496:
495:
492:
489:
488:
486:
478:
475:
469:
464:
456:
452:
448:
447:
444:
441:
440:
435:
431:
427:
426:
424:
392:
389:
364:
363:
352:
347:
339:
335:
331:
330:
327:
324:
323:
318:
314:
310:
309:
307:
283:
282:
270:
265:
261:
257:
254:
251:
246:
242:
225:
222:
193:
190:
162:
159:
104:
101:
98:
78:{\textstyle N}
74:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8577:
8566:
8563:
8562:
8560:
8545:
8544:
8535:
8533:
8532:
8523:
8521:
8520:
8515:
8509:
8507:
8506:
8497:
8496:
8493:
8479:
8476:
8474:
8473:Geostatistics
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8455:
8453:
8451:
8447:
8441:
8440:Psychometrics
8438:
8436:
8433:
8431:
8428:
8426:
8423:
8421:
8418:
8416:
8413:
8411:
8408:
8406:
8403:
8401:
8398:
8396:
8393:
8392:
8390:
8388:
8384:
8378:
8375:
8373:
8370:
8368:
8364:
8361:
8359:
8356:
8354:
8351:
8349:
8346:
8345:
8343:
8341:
8337:
8331:
8328:
8326:
8323:
8321:
8317:
8314:
8312:
8309:
8308:
8306:
8304:
8303:Biostatistics
8300:
8296:
8292:
8287:
8283:
8265:
8264:Log-rank test
8262:
8261:
8259:
8255:
8249:
8246:
8245:
8243:
8241:
8237:
8231:
8228:
8226:
8223:
8221:
8218:
8216:
8213:
8212:
8210:
8208:
8204:
8201:
8199:
8195:
8185:
8182:
8180:
8177:
8175:
8172:
8170:
8167:
8165:
8162:
8161:
8159:
8157:
8153:
8147:
8144:
8142:
8139:
8137:
8135:(BoxâJenkins)
8131:
8129:
8126:
8124:
8121:
8117:
8114:
8113:
8112:
8109:
8108:
8106:
8104:
8100:
8094:
8091:
8089:
8088:DurbinâWatson
8086:
8084:
8078:
8076:
8073:
8071:
8070:DickeyâFuller
8068:
8067:
8065:
8061:
8055:
8052:
8050:
8047:
8045:
8044:Cointegration
8042:
8040:
8037:
8035:
8032:
8030:
8027:
8025:
8022:
8020:
8019:Decomposition
8017:
8016:
8014:
8010:
8007:
8005:
8001:
7991:
7988:
7987:
7986:
7983:
7982:
7981:
7978:
7974:
7971:
7970:
7969:
7966:
7964:
7961:
7959:
7956:
7954:
7951:
7949:
7946:
7944:
7941:
7939:
7936:
7934:
7931:
7930:
7928:
7926:
7922:
7916:
7913:
7911:
7908:
7906:
7903:
7901:
7898:
7896:
7893:
7891:
7890:Cohen's kappa
7888:
7887:
7885:
7883:
7879:
7875:
7871:
7867:
7863:
7859:
7854:
7850:
7836:
7833:
7831:
7828:
7826:
7823:
7821:
7818:
7817:
7815:
7813:
7809:
7803:
7799:
7795:
7789:
7787:
7784:
7783:
7781:
7779:
7775:
7769:
7766:
7764:
7761:
7759:
7756:
7754:
7751:
7749:
7746:
7744:
7743:Nonparametric
7741:
7739:
7736:
7735:
7733:
7729:
7723:
7720:
7718:
7715:
7713:
7710:
7708:
7705:
7704:
7702:
7700:
7696:
7690:
7687:
7685:
7682:
7680:
7677:
7675:
7672:
7670:
7667:
7666:
7664:
7662:
7658:
7652:
7649:
7647:
7644:
7642:
7639:
7637:
7634:
7633:
7631:
7629:
7625:
7621:
7614:
7611:
7609:
7606:
7605:
7601:
7597:
7581:
7578:
7577:
7576:
7573:
7571:
7568:
7566:
7563:
7559:
7556:
7554:
7551:
7550:
7549:
7546:
7545:
7543:
7541:
7537:
7527:
7524:
7520:
7514:
7512:
7506:
7504:
7498:
7497:
7496:
7493:
7492:Nonparametric
7490:
7488:
7482:
7478:
7475:
7474:
7473:
7467:
7463:
7462:Sample median
7460:
7459:
7458:
7455:
7454:
7452:
7450:
7446:
7438:
7435:
7433:
7430:
7428:
7425:
7424:
7423:
7420:
7418:
7415:
7413:
7407:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7383:
7379:
7377:
7374:
7373:
7371:
7369:
7365:
7359:
7357:
7353:
7351:
7349:
7344:
7342:
7337:
7333:
7332:
7329:
7326:
7324:
7320:
7310:
7307:
7305:
7302:
7300:
7297:
7296:
7294:
7292:
7288:
7282:
7279:
7275:
7272:
7271:
7270:
7267:
7263:
7260:
7259:
7258:
7255:
7253:
7250:
7249:
7247:
7245:
7241:
7233:
7230:
7228:
7225:
7224:
7223:
7220:
7218:
7215:
7213:
7210:
7208:
7205:
7203:
7200:
7198:
7195:
7194:
7192:
7190:
7186:
7180:
7177:
7173:
7170:
7166:
7163:
7161:
7158:
7157:
7156:
7153:
7152:
7151:
7148:
7144:
7141:
7139:
7136:
7134:
7131:
7129:
7126:
7125:
7124:
7121:
7120:
7118:
7116:
7112:
7109:
7107:
7103:
7097:
7094:
7092:
7089:
7085:
7082:
7081:
7080:
7077:
7075:
7072:
7068:
7067:loss function
7065:
7064:
7063:
7060:
7056:
7053:
7051:
7048:
7046:
7043:
7042:
7041:
7038:
7036:
7033:
7031:
7028:
7024:
7021:
7019:
7016:
7014:
7008:
7005:
7004:
7003:
7000:
6996:
6993:
6991:
6988:
6986:
6983:
6982:
6981:
6978:
6974:
6971:
6969:
6966:
6965:
6964:
6961:
6957:
6954:
6953:
6952:
6949:
6945:
6942:
6941:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6921:
6919:
6917:
6913:
6909:
6905:
6900:
6896:
6882:
6879:
6877:
6874:
6872:
6869:
6867:
6864:
6863:
6861:
6859:
6855:
6849:
6846:
6844:
6841:
6839:
6836:
6835:
6833:
6829:
6823:
6820:
6818:
6815:
6813:
6810:
6808:
6805:
6803:
6800:
6798:
6795:
6793:
6790:
6789:
6787:
6785:
6781:
6775:
6772:
6770:
6769:Questionnaire
6767:
6765:
6762:
6758:
6755:
6753:
6750:
6749:
6748:
6745:
6744:
6742:
6740:
6736:
6730:
6727:
6725:
6722:
6720:
6717:
6715:
6712:
6710:
6707:
6705:
6702:
6700:
6697:
6695:
6692:
6691:
6689:
6687:
6683:
6679:
6675:
6670:
6666:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6599:
6597:
6596:Control chart
6594:
6592:
6589:
6587:
6584:
6582:
6579:
6578:
6576:
6574:
6570:
6564:
6561:
6557:
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6548:
6547:
6544:
6542:
6539:
6537:
6534:
6533:
6531:
6529:
6525:
6519:
6516:
6514:
6511:
6509:
6506:
6505:
6503:
6499:
6493:
6490:
6489:
6487:
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6481:
6469:
6466:
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6459:
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6437:
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6411:
6408:
6406:
6403:
6401:
6398:
6397:
6395:
6393:
6389:
6383:
6380:
6378:
6375:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6351:
6348:
6346:
6343:
6341:
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6336:
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6331:
6328:
6326:
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6321:
6318:
6317:
6315:
6313:
6309:
6306:
6304:
6300:
6296:
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6283:
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6225:
6221:
6219:
6215:
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6210:
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6201:
6197:
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6184:
6180:
6176:
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6164:
6160:
6159:
6153:
6149:
6145:
6140:
6135:
6131:
6127:
6123:
6118:
6114:
6112:0-333-30110-2
6108:
6104:
6099:
6098:
6094:
6087:
6081:
6078:
6074:
6070:
6066:
6065:
6058:
6055:
6051:
6047:
6043:
6037:
6034:
6030:
6024:
6021:
6017:
6011:
6008:
6004:
6003:
5996:
5993:
5990:
5988:, CRC Press.
5987:
5981:
5978:
5974:
5968:
5965:
5961:
5960:
5953:
5950:
5946:
5945:
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5856:
5848:
5845:
5841:
5838:
5834:
5830:
5826:
5822:
5818:
5814:
5813:Trevor Hastie
5809:
5806:
5801:
5799:0-387-95361-2
5795:
5791:
5784:
5781:
5775:
5772:
5767:
5763:
5759:
5755:
5751:
5747:
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5473:
5470:
5467:
5455:
5453:
5449:
5445:
5441:
5433:
5431:
5429:
5425:
5424:error ellipse
5421:
5416:
5412:
5408:
5404:
5400:
5396:
5391:
5387:
5382:
5380:
5376:
5372:
5368:
5364:
5360:
5356:
5351:
5349:
5341:
5339:
5319:
5311:
5308:
5299:
5290:
5264:
5250:
5236:
5232:
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5224:
5205:
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5112:
5090:
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5071:
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5026:
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4997:
4994:
4991:
4988:
4982:
4979:
4976:
4973:
4966:
4952:
4940:
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4925:
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4766:
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4757:
4754:
4745:
4734:
4731: â
4730:
4726:
4723: â
4722:
4718:
4715: â
4714:
4710:
4702:
4700:
4684:
4680:
4659:
4651:
4650:Gaussian blur
4646:
4644:
4625:
4617:
4613:
4602:
4592:
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4573:
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4208:
4204:
4201:data, a t or
4200:
4196:
4191:
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3936:
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3899:
3895:
3894:-distribution
3893:
3886:
3884:
3863:
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2106:
2083:
2079:
2075:
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2069:
2064:
2060:
2050:
2048:
2044:
2040:
2039:linear models
2036:
2028:
2026:
2024:
2020:
2016:
2012:
2008:
2003:
2001:
1997:
1992:
1990:
1971:
1968:
1963:
1956:
1953:
1944:
1940:
1936:
1933:
1930:
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1533:
1532:in the model
1531:
1527:
1523:
1522:least squares
1518:
1516:
1512:
1508:
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1501: â
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766:
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755:
751:
747:
744:
740:
735:
733:
729:
725:
721:
705:
702:
690:
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679:
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661:
658:
655:
651:
641:
621:
595:
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578:
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563:
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536:
531:
527:
520:
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510:
504:
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473:
467:
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450:
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429:
422:
413:
412:
411:
409:
387:
375:
373:
369:
350:
345:
337:
333:
325:
316:
312:
305:
296:
295:
294:
292:
291:random vector
289:-dimensional
288:
268:
263:
259:
255:
252:
249:
244:
240:
232:
231:
230:
223:
221:
219:
215:
211:
207:
203:
199:
191:
189:
187:
186:Ronald Fisher
183:
178:
177:
172:
168:
160:
158:
154:
152:
148:
144:
140:
136:
132:
131:linear models
127:
125:
124:random vector
121:
116:
102:
99:
96:
88:
72:
64:
60:
55:
52:Estimates of
50:
48:
44:
40:
33:
19:
8541:
8529:
8510:
8503:
8415:Econometrics
8365: /
8348:Chemometrics
8325:Epidemiology
8318: /
8291:Applications
8133:ARIMA model
8080:Q-statistic
8029:Stationarity
7925:Multivariate
7868: /
7864: /
7862:Multivariate
7860: /
7834:
7800: /
7796: /
7570:Bayes factor
7469:Signed rank
7381:
7355:
7347:
7335:
7030:Completeness
6866:Cohort study
6764:Opinion poll
6699:Missing data
6686:Study design
6641:Scatter plot
6563:Scatter plot
6556:Spearman's Ď
6518:Grouped data
6191:
6187:
6162:
6156:
6132:(3): 75â78.
6129:
6125:
6102:
6080:
6062:
6057:
6041:
6036:
6028:
6023:
6015:
6010:
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5985:
5980:
5972:
5967:
5958:
5952:
5943:
5909:
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5885:
5880:
5869:. Retrieved
5854:
5847:
5820:
5808:
5789:
5783:
5774:
5749:
5745:
5735:
5713:(1): 87â94.
5710:
5706:
5696:
5671:
5667:
5657:
5640:
5636:
5623:
5612:. Retrieved
5607:
5597:
5586:. Retrieved
5581:
5572:
5456:
5454:in geodesy.
5451:
5447:
5439:
5437:
5426:for a given
5420:a posteriori
5419:
5417:
5410:
5406:
5402:
5398:
5394:
5389:
5385:
5383:
5378:
5374:
5370:
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5234:
5230:
5226:
5222:
5220:
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4732:
4728:
4724:
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4716:
4712:
4711:replaced by
4708:
4706:
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4511:
4507:
4503:
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4493:
4489:
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4439:, and other
4428:
4426:
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4340:
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4216:
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4199:heavy-tailed
4187:
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4130:
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3938:
3933:
3923:
3919:
3909:
3902:
3897:
3896:with 2 and 3
3891:
3887:
3882:
3880:
3666:
3496:
3492:
3482:
3478:
3346:
3344:
2677:
2528:
2192:
2051:
2041:, including
2034:
2032:
2022:
2018:
2014:
2010:
2006:
2004:
1999:
1995:
1993:
1988:
1986:
1816:
1713:
1709:
1646:
1642:
1637:
1631:
1627:
1625:
1529:
1525:
1519:
1514:
1510:
1506:
1502:
1497:
1493:
1484:
1431:
1348:
1339:
1287:
1277:
1273:
1270:
1266:
1257:Of residuals
1251:
1246:
1243:
1238:
1234:
1230:
1228:
1191:
1185:
1033:
1030:
1025:
938:
762:
757:
749:
736:
731:
727:
723:
719:
718:. The first
642:
610:
376:
371:
367:
365:
286:
284:
227:
213:
209:
206:R. A. Fisher
197:
195:
174:
173:in his 1908
164:
155:
128:
117:
86:
51:
42:
36:
8543:WikiProject
8458:Cartography
8420:Jurimetrics
8372:Reliability
8103:Time domain
8082:(LjungâBox)
8004:Time-series
7882:Categorical
7866:Time-series
7858:Categorical
7793:(Bernoulli)
7628:Correlation
7608:Correlation
7404:JarqueâBera
7376:Chi-squared
7138:M-estimator
7091:Asymptotics
7035:Sufficiency
6802:Interaction
6714:Replication
6694:Effect size
6651:Violin plot
6631:Radar chart
6611:Forest plot
6601:Correlogram
6551:Kendall's Ď
5842:(eq.(5.16))
5674:(1): 1â25.
5549:Sample size
4247:generalized
4243:regularized
3929:chi-squared
3910:approximate
1188:Student's t
1038:statistic,
408:sample mean
200:(lowercase
151:chi-squared
8410:Demography
8128:ARMA model
7933:Regression
7510:(Friedman)
7471:(Wilcoxon)
7409:Normality
7399:Lilliefors
7346:Student's
7222:Resampling
7096:Robustness
7084:divergence
7074:Efficiency
7012:(monotone)
7007:Likelihood
6924:Population
6757:Stratified
6709:Population
6528:Dependence
6484:Count data
6415:Percentile
6392:Dispersion
6325:Arithmetic
6260:Statistics
6086:Biometrika
6042:J. Geodesy
5871:2020-08-28
5668:Biometrika
5614:2008-08-21
5588:2008-08-21
5565:References
5405: ' ÎŁ
5229:) to only
4482:), or the
4336:hat matrix
3922:Student's
3905:split-plot
2021:, leaving
1641:and hence
1351:, and let
1344:each with
1186:follows a
176:Biometrika
120:dimensions
39:statistics
7791:Logistic
7558:posterior
7484:Rank sum
7232:Jackknife
7227:Bootstrap
7045:Bootstrap
6980:Parameter
6929:Statistic
6724:Statistic
6636:Run chart
6621:Pie chart
6616:Histogram
6606:Fan chart
6581:Bar chart
6463:L-moments
6350:Geometric
6148:121982952
5766:2299-5684
5481:‖
5471:−
5465:‖
5323:^
5309:−
5305:Σ
5294:^
5261:‖
5254:^
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5176:−
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5140:≈
5128:^
5125:σ
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5060:
5051:−
5037:‖
5030:^
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4977:−
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4956:^
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4929:^
4926:σ
4879:−
4859:−
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4824:^
4815:‖
4797:^
4794:σ
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4681:χ
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4267:^
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4079:−
4046:∑
4010:σ
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1487:estimates
1462:¯
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1312:…
1249:will be.
1203:μ
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1078:μ
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1006:σ
975:σ
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377:Now, let
326:⋮
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100:−
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8559:Category
8505:Category
8198:Survival
8075:Johansen
7798:Binomial
7753:Isotonic
7340:(normal)
6985:location
6792:Blocking
6747:Sampling
6626:QâQ plot
6591:Box plot
6573:Graphics
6468:Skewness
6458:Kurtosis
6430:Variance
6360:Heronian
6355:Harmonic
5898:(eq.(7))
5559:Variance
5507:See also
5411:X '
5300:′
5279:becomes
5091:′
5005:′
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1024:), with
913:‖
843:‖
743:subspace
192:Notation
147:subspace
63:variance
8531:Commons
8478:Kriging
8363:Process
8320:studies
8179:Wavelet
8012:General
7179:Plug-in
6973:L space
6752:Cluster
6453:Moments
6271:Outline
6179:3087407
6073:2283275
5896:2669609
5727:2340521
5688:2331554
5446:, the
5342:General
4510: '
4334:is the
4221:(e.g.,
4190:integer
3881:with 3(
1489:of the
1284:Example
1237:of the
746:spanned
406:be the
161:History
8400:Census
7990:Normal
7938:Manova
7758:Robust
7508:2-way
7500:1-way
7338:-test
7009:
6586:Biplot
6377:Median
6370:Lehmer
6312:Center
6177:
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4297:where
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2529:where
1626:where
1491:errors
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8024:Trend
7553:prior
7495:anova
7384:-test
7358:-test
7350:-test
7257:Power
7202:Pivot
6995:shape
6990:scale
6440:Shape
6420:Range
6365:Heinz
6340:Cubic
6276:Index
6175:JSTOR
6144:S2CID
6069:JSTOR
5892:JSTOR
5723:JSTOR
5684:JSTOR
5633:(PDF)
4492:)/tr(
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2023:n - p
1036:-test
208:used
8257:Test
7457:Sign
7309:Wald
6382:Mode
6320:Mean
6107:ISBN
5915:ISBN
5860:ISBN
5825:ISBN
5794:ISBN
5762:ISSN
5179:1.25
4911:or:
4447:and
4427:The
3437:and
2144:and
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7437:BIC
7432:AIC
6196:doi
6167:doi
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2011:p
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1661:a
1647:i
1643:Y
1638:i
1632:i
1628:x
1611:n
1608:,
1602:,
1599:1
1596:=
1593:i
1583:i
1579:e
1575:+
1570:i
1566:x
1562:b
1559:+
1556:a
1553:=
1548:i
1544:Y
1530:b
1526:a
1515:n
1511:n
1507:n
1503:Îź
1498:i
1494:X
1468:n
1459:X
1448:i
1444:X
1415:n
1409:n
1405:X
1401:+
1395:+
1390:1
1386:X
1379:=
1374:n
1365:X
1349:Îź
1323:n
1319:X
1315:,
1309:,
1304:1
1300:X
1247:Ď
1239:Ď
1231:Ď
1207:0
1192:n
1168:)
1165:1
1159:n
1156:(
1152:/
1146:2
1142:)
1132:X
1121:i
1117:X
1113:(
1108:n
1103:1
1100:=
1097:i
1087:)
1082:0
1065:X
1059:(
1054:n
1034:t
1026:n
1010:2
979:2
952:i
948:X
924:.
919:2
901:X
890:n
886:X
865:X
854:1
850:X
837:=
832:2
828:)
818:X
807:i
803:X
799:(
794:n
789:1
786:=
783:i
758:n
750:n
732:n
728:n
724:n
720:n
706:0
703:=
700:)
691:X
680:i
676:X
672:(
667:n
662:1
659:=
656:i
622:X
596:.
591:)
579:X
568:n
564:X
543:X
532:1
528:X
521:(
516:+
511:)
505:1
491:1
485:(
474:X
468:=
463:)
455:n
451:X
434:1
430:X
423:(
388:X
372:n
368:n
351:.
346:)
338:n
334:X
317:1
313:X
306:(
287:n
269:.
264:n
260:X
256:,
250:,
245:1
241:X
214:n
210:n
198:ν
133:(
103:1
97:N
87:N
73:N
34:.
20:)
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