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For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation.
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leads to good results in most practical settingsâmaking MVUE a natural starting point for a broad range of analysesâa targeted specification may perform better for a given problem; thus, MVUE is not always the best stopping point.
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2009:{\displaystyle \eta (X)=\operatorname {E} (\delta (X)\mid T)=\operatorname {E} \left(\left.{\frac {T^{2}}{2}}\,\right|\,T\right)={\frac {T^{2}}{2}}={\frac {\log(1+e^{-X})^{2}}{2}}}
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600:{\displaystyle \operatorname {var} (\delta (X_{1},X_{2},\ldots ,X_{n}))\leq \operatorname {var} ({\tilde {\delta }}(X_{1},X_{2},\ldots ,X_{n}))}
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2152:. This is a scaled and shifted (so unbiased) transform of the sample maximum, which is a sufficient and complete statistic. See
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1161:. In some cases biased estimators have lower MSE because they have a smaller variance than does any unbiased estimator; see
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1650:{\displaystyle \operatorname {E} (T)={\frac {1}{\theta }},\quad \operatorname {var} (T)={\frac {1}{\theta ^{2}}}}
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991:{\displaystyle \eta (X_{1},X_{2},\ldots ,X_{n})=\operatorname {E} (\delta (X_{1},X_{2},\ldots ,X_{n})\mid T)\,}
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This example illustrates that an unbiased function of the complete sufficient statistic will be UMVU, as
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734:, an unbiased estimator that is a function of a complete, sufficient statistic is the UMVUE estimator.
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Further, for other distributions the sample mean and sample variance are not in general MVUEs â for a
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that has lower variance than any other unbiased estimator for all possible values of the parameter.
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1479:{\displaystyle {\frac {e^{-x}}{1+e^{-x}}}\exp(-\theta \log(1+e^{-x})+\log(\theta ))}
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need not exist, but if it does and if it is unbiased, it is the MVUE. Since the
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1296:{\displaystyle p_{\theta }(x)={\frac {\theta e^{-x}}{(1+e^{-x})^{\theta +1}}}}
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one can also prove that determining the MVUE is simply a matter of finding a
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exists, then one can prove there is an essentially unique MVUE. Using the
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is a complete sufficient statistic for the family of densities. Then
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Unbiased estimators and their applications, Vol.1: Univariate case
1715:{\displaystyle \operatorname {E} (T^{2})={\frac {2}{\theta ^{2}}}}
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Statistical Theory: Notes for a Course in
Theoretical Statistics
1546:. In fact this is a full rank exponential family, and therefore
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are the MVUEs for the population mean and population variance.
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For a normal distribution with unknown mean and variance, the
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is not unbiased for the population standard deviation â see
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First we recognize that the density can be written as
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Consider the data to be a single observation from an
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uniformly minimum-variance unbiased estimator (UMVUE)
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Autoregressive conditional heteroskedasticity (ARCH)
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Here we use
LehmannâScheffĂ© theorem to get the MVUE
1831:is complete sufficient, thus the UMVU estimator is
1351:{\displaystyle g(\theta )={\frac {1}{\theta ^{2}}}}
60:. Unsourced material may be challenged and removed.
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2071:exemplars are chosen (without replacement) from a
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798:{\displaystyle \delta (X_{1},X_{2},\ldots ,X_{n})}
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27:Unbiased statistical estimator minimizing variance
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260:i.i.d. from some member of a family of densities
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2285:Theoretical statistics: Topics for a core course
3763:Multivariate adaptive regression splines (MARS)
716:{\displaystyle p_{\theta },\theta \in \Omega }
319:is the parameter space. An unbiased estimator
292:{\displaystyle p_{\theta },\theta \in \Omega }
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1770:{\displaystyle \delta (X)={\frac {T^{2}}{2}}}
2298:. Kluwer Academic Publishers. pp. 521p.
2293:
2257:: CS1 maint: multiple names: authors list (
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2261:) CS1 maint: numeric names: authors list (
1306:and we wish to find the UMVU estimator of
442:{\displaystyle \forall \theta \in \Omega }
138:minimum-variance unbiased estimator (MVUE)
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2058:with unknown upper and lower bounds, the
2049:unbiased estimation of standard deviation
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253:{\displaystyle X_{1},X_{2},\ldots ,X_{n}}
120:Learn how and when to remove this message
2075:over the set {1, 2, ...,
2227:U-statistics : theory and practice
14:
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4289:KaplanâMeier estimator (product limit)
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1050:
159:with the desirability metric of least
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69:"Minimum-variance unbiased estimator"
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4299:Accelerated failure time (AFT) model
2133:{\displaystyle {\frac {k+1}{k}}m-1,}
2062:is the MVUE for the population mean.
1489:Which is an exponential family with
58:adding citations to reliable sources
29:
4611:
3894:Analysis of variance (ANOVA, anova)
2746:
2294:Voinov V. G., Nikulin M.S. (1993).
2193:
635:{\displaystyle {\tilde {\delta }}.}
24:
18:Minimum variance unbiased estimator
3989:CochranâMantelâHaenszel statistics
2615:Pearson product-moment correlation
2279:. Springer. pp. 47â48, 57â58.
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1175:absolutely continuous distribution
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155:While combining the constraint of
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610:for any other unbiased estimator
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4363:
1824:{\displaystyle T=\log(1+e^{-x})}
1539:{\displaystyle T=\log(1+e^{-x})}
34:
4248:Least-squares spectral analysis
1611:
45:needs additional citations for
3229:Mean-unbiased minimum-variance
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2172:Best linear unbiased estimator
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4542:Geographic information system
3758:Simultaneous equations models
2289:DOI 10.1007/978-0-387-93839-4
2211:
2073:discrete uniform distribution
1570:for a derivation which shows
167:
3725:Coefficient of determination
3336:Uniformly most powerful test
1566:is complete sufficient. See
1192:{\displaystyle \mathbb {R} }
644:If an unbiased estimator of
7:
4294:Proportional hazards models
4238:Spectral density estimation
4220:Vector autoregression (VAR)
3654:Maximum posterior estimator
2886:Randomized controlled trial
2160:
2079:} with unknown upper bound
1026:{\displaystyle g(\theta ).}
10:
4655:
4054:Multivariate distributions
2474:Average absolute deviation
2283:Keener, Robert W. (2010).
2275:Keener, Robert W. (2006).
2225:Lee, A. J., 1946- (1990).
1168:
827:{\displaystyle g(\theta )}
727:unbiased estimator on it.
666:{\displaystyle g(\theta )}
409:{\displaystyle g(\theta )}
194:{\displaystyle g(\theta )}
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4042:Structural equation model
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2793:Sample size determination
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2205:Minimum mean square error
2045:sample standard deviation
1159:among unbiased estimators
1045:minimum mean square error
684:statistic for the family
4537:Environmental statistics
4059:Elliptical distributions
3852:Generalized linear model
3781:Simple linear regression
3551:HodgesâLehmann estimator
3008:Probability distribution
2917:Stochastic approximation
2479:Coefficient of variation
4197:Cross-correlation (XCF)
3805:Non-standard predictors
3239:LehmannâScheffĂ© theorem
2912:Adaptive clinical trial
2229:. New York: M. Dekker.
2183:LehmannâScheffĂ© theorem
2021:LehmannâScheffĂ© theorem
1157:the MVUE minimizes MSE
732:LehmannâScheffĂ© theorem
312:{\displaystyle \Omega }
172:Consider estimation of
4593:Mathematics portal
4414:Engineering statistics
4322:NelsonâAalen estimator
3899:Analysis of covariance
3786:Ordinary least squares
3710:Pearson product-moment
3114:Statistical functional
3025:Empirical distribution
2858:Controlled experiments
2587:Frequency distribution
2365:Descriptive statistics
2287:. New York: Springer.
2178:Biasâvariance tradeoff
2134:
2010:
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1063:(MSE) of an estimator
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737:Put formally, suppose
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4509:Population statistics
4451:System identification
4185:Autocorrelation (ACF)
4113:Exponential smoothing
4027:Discriminant analysis
4022:Canonical correlation
3886:Partition of variance
3748:Regression validation
3592:(JonckheereâTerpstra)
3491:Likelihood-ratio test
3180:Frequentist inference
3092:Locationâscale family
3013:Sampling distribution
2978:Statistical inference
2945:Cross-sectional study
2932:Observational studies
2891:Randomized experiment
2720:Stem-and-leaf display
2522:Central limit theorem
2135:
2011:
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675:RaoâBlackwell theorem
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4432:Probabilistic design
4017:Principal components
3860:Exponential families
3812:Nonlinear regression
3791:General linear model
3753:Mixed effects models
3743:Errors and residuals
3720:Confounding variable
3622:Bayesian probability
3600:Van der Waerden test
3590:Ordered alternative
3355:Multiple comparisons
3234:RaoâBlackwellization
3197:Estimating equations
3153:Statistical distance
2871:Factorial experiment
2404:Arithmetic-Geometric
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2056:uniform distribution
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1491:sufficient statistic
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54:improve this article
4504:Official statistics
4427:Methods engineering
4108:Seasonal adjustment
3876:Poisson regressions
3796:Bayesian regression
3735:Regression analysis
3715:Partial correlation
3687:Regression analysis
3286:Prediction interval
3281:Likelihood interval
3271:Confidence interval
3263:Interval estimation
3224:Unbiased estimators
3042:Model specification
2922:Up-and-down designs
2610:Partial correlation
2566:Index of dispersion
2484:Interquartile range
2154:German tank problem
1057:efficient estimator
1051:Estimator selection
4524:Spatial statistics
4404:Medical statistics
4304:First hitting time
4258:Whittle likelihood
3909:Degrees of freedom
3904:Multivariate ANOVA
3837:Heteroscedasticity
3649:Bayesian estimator
3614:Bayesian inference
3463:KolmogorovâSmirnov
3348:Randomization test
3318:Testing hypotheses
3291:Tolerance interval
3202:Maximum likelihood
3097:Exponential family
3030:Density estimation
2990:Statistical theory
2950:Natural experiment
2896:Scientific control
2813:Survey methodology
2499:Standard deviation
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2006:
1821:
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1568:exponential family
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1061:mean squared error
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146:unbiased estimator
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4499:National accounts
4469:Actuarial science
4461:Social statistics
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4281:Survival function
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4128:Granger causality
3969:Contingency table
3944:Survival analysis
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3773:Linear regression
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3639:Credible interval
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3207:Method of moments
3076:Parametric family
3037:Statistical model
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2881:Random assignment
2803:Statistical power
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2582:Contingency table
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2419:Generalized/power
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2004:
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1559:{\displaystyle T}
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847:{\displaystyle T}
723:and conditioning
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16:(Redirected from
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4230:Frequency domain
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4123:Structural break
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4032:Cluster analysis
3979:Log-linear model
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3842:Homoscedasticity
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3576:(KruskalâWallis)
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3501:Cross validation
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3468:AndersonâDarling
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3402:
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3373:Likelihood-ratio
3365:Parametric tests
3343:Permutation test
3326:1- & 2-tails
3217:Minimum distance
3189:Point estimation
3185:
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3136:Optimal decision
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2955:Quasi-experiment
2905:Adaptive designs
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2194:Bayesian analogs
2167:CramĂ©râRao bound
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1643:
1631:
1607:
1599:
1565:
1563:
1562:
1557:
1545:
1543:
1542:
1537:
1532:
1531:
1485:
1483:
1482:
1477:
1451:
1450:
1408:
1406:
1405:
1404:
1385:
1384:
1372:
1357:
1355:
1354:
1349:
1347:
1345:
1344:
1332:
1302:
1300:
1299:
1294:
1292:
1290:
1289:
1288:
1273:
1272:
1250:
1249:
1248:
1232:
1218:
1217:
1198:
1196:
1195:
1190:
1188:
1153:
1151:
1150:
1145:
1141:
1140:
1139:
1032:
1030:
1029:
1024:
1001:is the MVUE for
997:
995:
994:
989:
974:
973:
955:
954:
942:
941:
911:
910:
892:
891:
879:
878:
853:
851:
850:
845:
833:
831:
830:
825:
805:is unbiased for
804:
802:
801:
796:
791:
790:
772:
771:
759:
758:
730:Further, by the
722:
720:
719:
714:
700:
699:
672:
670:
669:
664:
641:
639:
638:
633:
628:
627:
619:
606:
604:
603:
598:
590:
589:
571:
570:
558:
557:
545:
544:
536:
515:
514:
496:
495:
483:
482:
448:
446:
445:
440:
415:
413:
412:
407:
386:
384:
383:
378:
373:
372:
354:
353:
341:
340:
318:
316:
315:
310:
298:
296:
295:
290:
276:
275:
259:
257:
256:
251:
249:
248:
230:
229:
217:
216:
200:
198:
197:
192:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
4654:
4653:
4649:
4648:
4647:
4645:
4644:
4643:
4629:
4628:
4627:
4622:
4585:
4556:
4518:
4455:
4441:quality control
4408:
4390:Clinical trials
4367:
4342:
4326:
4314:Hazard function
4308:
4262:
4224:
4208:
4171:
4167:BreuschâGodfrey
4155:
4132:
4072:
4047:Factor analysis
3993:
3974:Graphical model
3946:
3913:
3880:
3866:
3846:
3800:
3767:
3729:
3692:
3691:
3660:
3604:
3591:
3583:
3575:
3559:
3544:
3523:Rank statistics
3517:
3496:Model selection
3484:
3442:Goodness of fit
3436:
3413:
3387:
3359:
3312:
3257:
3246:Median unbiased
3174:
3085:
3018:Order statistic
2980:
2959:
2926:
2900:
2852:
2807:
2750:
2748:Data collection
2729:
2641:
2596:
2570:
2548:
2508:
2460:
2377:Continuous data
2367:
2354:
2336:
2331:
2271:
2270:
2250:
2249:
2237:
2223:
2219:
2214:
2200:Bayes estimator
2196:
2163:
2102:
2100:
2098:
2095:
2094:
2083:, the MVUE for
2038:sample variance
2036:and (unbiased)
2029:
1994:
1990:
1981:
1977:
1961:
1959:
1945:
1941:
1939:
1910:
1906:
1904:
1903:
1900:
1899:
1895:
1839:
1836:
1835:
1809:
1805:
1782:
1779:
1778:
1756:
1752:
1750:
1733:
1730:
1729:
1704:
1700:
1695:
1683:
1679:
1668:
1665:
1664:
1639:
1635:
1630:
1598:
1578:
1575:
1574:
1551:
1548:
1547:
1524:
1520:
1497:
1494:
1493:
1443:
1439:
1397:
1393:
1386:
1377:
1373:
1371:
1369:
1366:
1365:
1340:
1336:
1331:
1314:
1311:
1310:
1278:
1274:
1265:
1261:
1251:
1241:
1237:
1233:
1231:
1213:
1209:
1207:
1204:
1203:
1184:
1182:
1179:
1178:
1171:
1135:
1131:
1075:
1072:
1071:
1053:
1041:Bayes estimator
1006:
1003:
1002:
969:
965:
950:
946:
937:
933:
906:
902:
887:
883:
874:
870:
862:
859:
858:
839:
836:
835:
810:
807:
806:
786:
782:
767:
763:
754:
750:
742:
739:
738:
695:
691:
689:
686:
685:
649:
646:
645:
618:
617:
615:
612:
611:
585:
581:
566:
562:
553:
549:
535:
534:
510:
506:
491:
487:
478:
474:
457:
454:
453:
425:
422:
421:
392:
389:
388:
368:
364:
349:
345:
336:
332:
324:
321:
320:
304:
301:
300:
271:
267:
265:
262:
261:
244:
240:
225:
221:
212:
208:
206:
203:
202:
177:
174:
173:
170:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
4652:
4642:
4641:
4624:
4623:
4621:
4620:
4608:
4596:
4582:
4569:
4566:
4565:
4562:
4561:
4558:
4557:
4555:
4554:
4549:
4544:
4539:
4534:
4528:
4526:
4520:
4519:
4517:
4516:
4511:
4506:
4501:
4496:
4491:
4486:
4481:
4476:
4471:
4465:
4463:
4457:
4456:
4454:
4453:
4448:
4443:
4434:
4429:
4424:
4418:
4416:
4410:
4409:
4407:
4406:
4401:
4396:
4387:
4385:Bioinformatics
4381:
4379:
4369:
4368:
4356:
4355:
4352:
4351:
4348:
4347:
4344:
4343:
4341:
4340:
4334:
4332:
4328:
4327:
4325:
4324:
4318:
4316:
4310:
4309:
4307:
4306:
4301:
4296:
4291:
4285:
4283:
4274:
4268:
4267:
4264:
4263:
4261:
4260:
4255:
4250:
4245:
4240:
4234:
4232:
4226:
4225:
4223:
4222:
4217:
4212:
4204:
4199:
4194:
4193:
4192:
4190:partial (PACF)
4181:
4179:
4173:
4172:
4170:
4169:
4164:
4159:
4151:
4146:
4140:
4138:
4137:Specific tests
4134:
4133:
4131:
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4095:
4089:
4087:
4080:
4074:
4073:
4071:
4070:
4069:
4068:
4067:
4066:
4051:
4050:
4049:
4039:
4037:Classification
4034:
4029:
4024:
4019:
4014:
4009:
4003:
4001:
3995:
3994:
3992:
3991:
3986:
3984:McNemar's test
3981:
3976:
3971:
3966:
3960:
3958:
3948:
3947:
3923:
3922:
3919:
3918:
3915:
3914:
3912:
3911:
3906:
3901:
3896:
3890:
3888:
3882:
3881:
3879:
3878:
3862:
3856:
3854:
3848:
3847:
3845:
3844:
3839:
3834:
3829:
3824:
3822:Semiparametric
3819:
3814:
3808:
3806:
3802:
3801:
3799:
3798:
3793:
3788:
3783:
3777:
3775:
3769:
3768:
3766:
3765:
3760:
3755:
3750:
3745:
3739:
3737:
3731:
3730:
3728:
3727:
3722:
3717:
3712:
3706:
3704:
3694:
3693:
3690:
3689:
3684:
3678:
3670:
3669:
3666:
3665:
3662:
3661:
3659:
3658:
3657:
3656:
3646:
3641:
3636:
3635:
3634:
3629:
3618:
3616:
3610:
3609:
3606:
3605:
3603:
3602:
3597:
3596:
3595:
3587:
3579:
3563:
3560:(MannâWhitney)
3555:
3554:
3553:
3540:
3539:
3538:
3527:
3525:
3519:
3518:
3516:
3515:
3514:
3513:
3508:
3503:
3493:
3488:
3485:(ShapiroâWilk)
3480:
3475:
3470:
3465:
3460:
3452:
3446:
3444:
3438:
3437:
3435:
3434:
3426:
3417:
3405:
3399:
3397:Specific tests
3393:
3392:
3389:
3388:
3386:
3385:
3380:
3375:
3369:
3367:
3361:
3360:
3358:
3357:
3352:
3351:
3350:
3340:
3339:
3338:
3328:
3322:
3320:
3314:
3313:
3311:
3310:
3309:
3308:
3303:
3293:
3288:
3283:
3278:
3273:
3267:
3265:
3259:
3258:
3256:
3255:
3250:
3249:
3248:
3243:
3242:
3241:
3236:
3221:
3220:
3219:
3214:
3209:
3204:
3193:
3191:
3182:
3176:
3175:
3173:
3172:
3167:
3162:
3161:
3160:
3150:
3145:
3144:
3143:
3133:
3132:
3131:
3126:
3121:
3111:
3106:
3101:
3100:
3099:
3094:
3089:
3073:
3072:
3071:
3066:
3061:
3051:
3050:
3049:
3044:
3034:
3033:
3032:
3022:
3021:
3020:
3010:
3005:
3000:
2994:
2992:
2982:
2981:
2969:
2968:
2965:
2964:
2961:
2960:
2958:
2957:
2952:
2947:
2942:
2936:
2934:
2928:
2927:
2925:
2924:
2919:
2914:
2908:
2906:
2902:
2901:
2899:
2898:
2893:
2888:
2883:
2878:
2873:
2868:
2862:
2860:
2854:
2853:
2851:
2850:
2848:Standard error
2845:
2840:
2835:
2834:
2833:
2828:
2817:
2815:
2809:
2808:
2806:
2805:
2800:
2795:
2790:
2785:
2780:
2778:Optimal design
2775:
2770:
2764:
2762:
2752:
2751:
2739:
2738:
2735:
2734:
2731:
2730:
2728:
2727:
2722:
2717:
2712:
2707:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2651:
2649:
2643:
2642:
2640:
2639:
2634:
2633:
2632:
2627:
2617:
2612:
2606:
2604:
2598:
2597:
2595:
2594:
2589:
2584:
2578:
2576:
2575:Summary tables
2572:
2571:
2569:
2568:
2562:
2560:
2554:
2553:
2550:
2549:
2547:
2546:
2545:
2544:
2539:
2534:
2524:
2518:
2516:
2510:
2509:
2507:
2506:
2501:
2496:
2491:
2486:
2481:
2476:
2470:
2468:
2462:
2461:
2459:
2458:
2453:
2448:
2447:
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2409:Contraharmonic
2406:
2401:
2390:
2388:
2379:
2369:
2368:
2356:
2355:
2353:
2352:
2347:
2341:
2338:
2337:
2330:
2329:
2322:
2315:
2307:
2301:
2300:
2291:
2281:
2269:
2268:
2235:
2216:
2215:
2213:
2210:
2209:
2208:
2202:
2195:
2192:
2191:
2190:
2185:
2180:
2175:
2169:
2162:
2159:
2158:
2157:
2150:sample maximum
2142:
2141:
2140:
2129:
2126:
2123:
2120:
2115:
2111:
2108:
2105:
2089:
2088:
2065:
2064:
2063:
2052:
2028:
2027:Other examples
2025:
2017:
2016:
2003:
1997:
1993:
1987:
1984:
1980:
1976:
1973:
1970:
1967:
1964:
1958:
1953:
1948:
1944:
1938:
1934:
1930:
1925:
1918:
1913:
1909:
1902:
1898:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1820:
1815:
1812:
1808:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1764:
1759:
1755:
1749:
1746:
1743:
1740:
1737:
1723:
1722:
1707:
1703:
1699:
1694:
1691:
1686:
1682:
1678:
1675:
1672:
1658:
1657:
1642:
1638:
1634:
1629:
1626:
1623:
1620:
1617:
1614:
1610:
1605:
1602:
1597:
1594:
1591:
1588:
1585:
1582:
1555:
1535:
1530:
1527:
1523:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1487:
1486:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1449:
1446:
1442:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1403:
1400:
1396:
1392:
1389:
1383:
1380:
1376:
1359:
1358:
1343:
1339:
1335:
1330:
1327:
1324:
1321:
1318:
1304:
1303:
1287:
1284:
1281:
1277:
1271:
1268:
1264:
1260:
1257:
1254:
1247:
1244:
1240:
1236:
1230:
1227:
1224:
1221:
1216:
1212:
1187:
1170:
1167:
1163:estimator bias
1155:
1154:
1138:
1134:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1052:
1049:
1022:
1019:
1016:
1013:
1010:
999:
998:
986:
983:
980:
977:
972:
968:
964:
961:
958:
953:
949:
945:
940:
936:
932:
929:
926:
923:
920:
917:
914:
909:
905:
901:
898:
895:
890:
886:
882:
877:
873:
869:
866:
843:
823:
820:
817:
814:
794:
789:
785:
781:
778:
775:
770:
766:
762:
757:
753:
749:
746:
712:
709:
706:
703:
698:
694:
662:
659:
656:
653:
631:
625:
622:
608:
607:
596:
593:
588:
584:
580:
577:
574:
569:
565:
561:
556:
552:
548:
542:
539:
533:
530:
527:
524:
521:
518:
513:
509:
505:
502:
499:
494:
490:
486:
481:
477:
473:
470:
467:
464:
461:
438:
435:
432:
429:
405:
402:
399:
396:
376:
371:
367:
363:
360:
357:
352:
348:
344:
339:
335:
331:
328:
308:
288:
285:
282:
279:
274:
270:
247:
243:
239:
236:
233:
228:
224:
220:
215:
211:
201:based on data
190:
187:
184:
181:
169:
166:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
4651:
4640:
4637:
4636:
4634:
4619:
4618:
4609:
4607:
4606:
4597:
4595:
4594:
4589:
4583:
4581:
4580:
4571:
4570:
4567:
4553:
4550:
4548:
4547:Geostatistics
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4529:
4527:
4525:
4521:
4515:
4514:Psychometrics
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4466:
4464:
4462:
4458:
4452:
4449:
4447:
4444:
4442:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4419:
4417:
4415:
4411:
4405:
4402:
4400:
4397:
4395:
4391:
4388:
4386:
4383:
4382:
4380:
4378:
4377:Biostatistics
4374:
4370:
4366:
4361:
4357:
4339:
4338:Log-rank test
4336:
4335:
4333:
4329:
4323:
4320:
4319:
4317:
4315:
4311:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4286:
4284:
4282:
4278:
4275:
4273:
4269:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4235:
4233:
4231:
4227:
4221:
4218:
4216:
4213:
4211:
4209:(BoxâJenkins)
4205:
4203:
4200:
4198:
4195:
4191:
4188:
4187:
4186:
4183:
4182:
4180:
4178:
4174:
4168:
4165:
4163:
4162:DurbinâWatson
4160:
4158:
4152:
4150:
4147:
4145:
4144:DickeyâFuller
4142:
4141:
4139:
4135:
4129:
4126:
4124:
4121:
4119:
4118:Cointegration
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4093:Decomposition
4091:
4090:
4088:
4084:
4081:
4079:
4075:
4065:
4062:
4061:
4060:
4057:
4056:
4055:
4052:
4048:
4045:
4044:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4004:
4002:
4000:
3996:
3990:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3970:
3967:
3965:
3964:Cohen's kappa
3962:
3961:
3959:
3957:
3953:
3949:
3945:
3941:
3937:
3933:
3928:
3924:
3910:
3907:
3905:
3902:
3900:
3897:
3895:
3892:
3891:
3889:
3887:
3883:
3877:
3873:
3869:
3863:
3861:
3858:
3857:
3855:
3853:
3849:
3843:
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3817:Nonparametric
3815:
3813:
3810:
3809:
3807:
3803:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3778:
3776:
3774:
3770:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3740:
3738:
3736:
3732:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3707:
3705:
3703:
3699:
3695:
3688:
3685:
3683:
3680:
3679:
3675:
3671:
3655:
3652:
3651:
3650:
3647:
3645:
3642:
3640:
3637:
3633:
3630:
3628:
3625:
3624:
3623:
3620:
3619:
3617:
3615:
3611:
3601:
3598:
3594:
3588:
3586:
3580:
3578:
3572:
3571:
3570:
3567:
3566:Nonparametric
3564:
3562:
3556:
3552:
3549:
3548:
3547:
3541:
3537:
3536:Sample median
3534:
3533:
3532:
3529:
3528:
3526:
3524:
3520:
3512:
3509:
3507:
3504:
3502:
3499:
3498:
3497:
3494:
3492:
3489:
3487:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3457:
3453:
3451:
3448:
3447:
3445:
3443:
3439:
3433:
3431:
3427:
3425:
3423:
3418:
3416:
3411:
3407:
3406:
3403:
3400:
3398:
3394:
3384:
3381:
3379:
3376:
3374:
3371:
3370:
3368:
3366:
3362:
3356:
3353:
3349:
3346:
3345:
3344:
3341:
3337:
3334:
3333:
3332:
3329:
3327:
3324:
3323:
3321:
3319:
3315:
3307:
3304:
3302:
3299:
3298:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3268:
3266:
3264:
3260:
3254:
3251:
3247:
3244:
3240:
3237:
3235:
3232:
3231:
3230:
3227:
3226:
3225:
3222:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3199:
3198:
3195:
3194:
3192:
3190:
3186:
3183:
3181:
3177:
3171:
3168:
3166:
3163:
3159:
3156:
3155:
3154:
3151:
3149:
3146:
3142:
3141:loss function
3139:
3138:
3137:
3134:
3130:
3127:
3125:
3122:
3120:
3117:
3116:
3115:
3112:
3110:
3107:
3105:
3102:
3098:
3095:
3093:
3090:
3088:
3082:
3079:
3078:
3077:
3074:
3070:
3067:
3065:
3062:
3060:
3057:
3056:
3055:
3052:
3048:
3045:
3043:
3040:
3039:
3038:
3035:
3031:
3028:
3027:
3026:
3023:
3019:
3016:
3015:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2995:
2993:
2991:
2987:
2983:
2979:
2974:
2970:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2937:
2935:
2933:
2929:
2923:
2920:
2918:
2915:
2913:
2910:
2909:
2907:
2903:
2897:
2894:
2892:
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2863:
2861:
2859:
2855:
2849:
2846:
2844:
2843:Questionnaire
2841:
2839:
2836:
2832:
2829:
2827:
2824:
2823:
2822:
2819:
2818:
2816:
2814:
2810:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2763:
2761:
2757:
2753:
2749:
2744:
2740:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2670:Control chart
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2652:
2650:
2648:
2644:
2638:
2635:
2631:
2628:
2626:
2623:
2622:
2621:
2618:
2616:
2613:
2611:
2608:
2607:
2605:
2603:
2599:
2593:
2590:
2588:
2585:
2583:
2580:
2579:
2577:
2573:
2567:
2564:
2563:
2561:
2559:
2555:
2543:
2540:
2538:
2535:
2533:
2530:
2529:
2528:
2525:
2523:
2520:
2519:
2517:
2515:
2511:
2505:
2502:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2471:
2469:
2467:
2463:
2457:
2454:
2452:
2449:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2400:
2397:
2396:
2395:
2392:
2391:
2389:
2387:
2383:
2380:
2378:
2374:
2370:
2366:
2361:
2357:
2351:
2348:
2346:
2343:
2342:
2339:
2335:
2328:
2323:
2321:
2316:
2314:
2309:
2308:
2305:
2297:
2292:
2290:
2286:
2282:
2278:
2273:
2272:
2264:
2260:
2254:
2246:
2242:
2238:
2232:
2228:
2221:
2217:
2206:
2203:
2201:
2198:
2197:
2189:
2186:
2184:
2181:
2179:
2176:
2173:
2170:
2168:
2165:
2164:
2155:
2151:
2147:
2143:
2127:
2124:
2121:
2118:
2113:
2109:
2106:
2103:
2093:
2092:
2091:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2061:
2057:
2053:
2050:
2046:
2043:However, the
2042:
2041:
2039:
2035:
2031:
2030:
2024:
2022:
2001:
1995:
1985:
1982:
1978:
1974:
1971:
1965:
1962:
1956:
1951:
1946:
1942:
1936:
1932:
1928:
1923:
1916:
1911:
1907:
1896:
1892:
1886:
1880:
1877:
1871:
1865:
1859:
1853:
1847:
1841:
1834:
1833:
1832:
1813:
1810:
1806:
1802:
1799:
1793:
1790:
1787:
1784:
1762:
1757:
1753:
1747:
1741:
1735:
1726:
1705:
1701:
1697:
1692:
1684:
1680:
1673:
1663:
1662:
1661:
1640:
1636:
1632:
1627:
1621:
1615:
1612:
1608:
1603:
1600:
1595:
1589:
1583:
1573:
1572:
1571:
1569:
1553:
1528:
1525:
1521:
1517:
1514:
1508:
1505:
1502:
1499:
1492:
1467:
1461:
1458:
1455:
1447:
1444:
1440:
1436:
1433:
1427:
1424:
1421:
1418:
1412:
1409:
1401:
1398:
1394:
1390:
1387:
1381:
1378:
1374:
1364:
1363:
1362:
1341:
1337:
1333:
1328:
1322:
1316:
1309:
1308:
1307:
1285:
1282:
1279:
1269:
1266:
1262:
1258:
1255:
1245:
1242:
1238:
1234:
1228:
1222:
1214:
1210:
1202:
1201:
1200:
1199:with density
1176:
1166:
1164:
1160:
1136:
1125:
1119:
1116:
1110:
1104:
1098:
1095:
1092:
1086:
1080:
1077:
1070:
1069:
1068:
1066:
1062:
1058:
1048:
1046:
1042:
1038:
1033:
1020:
1014:
1008:
981:
978:
970:
966:
962:
959:
956:
951:
947:
943:
938:
934:
927:
921:
915:
907:
903:
899:
896:
893:
888:
884:
880:
875:
871:
864:
857:
856:
855:
841:
818:
812:
787:
783:
779:
776:
773:
768:
764:
760:
755:
751:
744:
735:
733:
728:
726:
707:
704:
701:
696:
692:
683:
680:
676:
657:
651:
642:
629:
620:
586:
582:
578:
575:
572:
567:
563:
559:
554:
550:
537:
528:
525:
522:
511:
507:
503:
500:
497:
492:
488:
484:
479:
475:
468:
462:
459:
452:
451:
450:
433:
430:
419:
400:
394:
369:
365:
361:
358:
355:
350:
346:
342:
337:
333:
326:
283:
280:
277:
272:
268:
245:
241:
237:
234:
231:
226:
222:
218:
213:
209:
185:
179:
165:
162:
158:
153:
149:
147:
143:
139:
135:
124:
121:
113:
110:November 2009
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: â
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
4615:
4603:
4584:
4577:
4489:Econometrics
4439: /
4422:Chemometrics
4399:Epidemiology
4392: /
4365:Applications
4207:ARIMA model
4154:Q-statistic
4103:Stationarity
3999:Multivariate
3942: /
3938: /
3936:Multivariate
3934: /
3874: /
3870: /
3644:Bayes factor
3543:Signed rank
3455:
3429:
3421:
3409:
3228:
3104:Completeness
2940:Cohort study
2838:Opinion poll
2773:Missing data
2760:Study design
2715:Scatter plot
2637:Scatter plot
2630:Spearman's Ï
2592:Grouped data
2295:
2284:
2276:
2226:
2220:
2156:for details.
2145:
2084:
2080:
2076:
2068:
2018:
1727:
1724:
1659:
1488:
1360:
1305:
1172:
1158:
1156:
1064:
1054:
1039:analog is a
1034:
1000:
736:
729:
724:
643:
609:
417:
171:
157:unbiasedness
154:
150:
141:
137:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
4617:WikiProject
4532:Cartography
4494:Jurimetrics
4446:Reliability
4177:Time domain
4156:(LjungâBox)
4078:Time-series
3956:Categorical
3940:Time-series
3932:Categorical
3867:(Bernoulli)
3702:Correlation
3682:Correlation
3478:JarqueâBera
3450:Chi-squared
3212:M-estimator
3165:Asymptotics
3109:Sufficiency
2876:Interaction
2788:Replication
2768:Effect size
2725:Violin plot
2705:Radar chart
2685:Forest plot
2675:Correlogram
2625:Kendall's Ï
2188:U-statistic
2034:sample mean
1660:Therefore,
834:, and that
4484:Demography
4202:ARMA model
4007:Regression
3584:(Friedman)
3545:(Wilcoxon)
3483:Normality
3473:Lilliefors
3420:Student's
3296:Resampling
3170:Robustness
3158:divergence
3148:Efficiency
3086:(monotone)
3081:Likelihood
2998:Population
2831:Stratified
2783:Population
2602:Dependence
2558:Count data
2489:Percentile
2466:Dispersion
2399:Arithmetic
2334:Statistics
2236:0824782534
2212:References
682:sufficient
168:Definition
134:statistics
80:newspapers
4639:Estimator
3865:Logistic
3632:posterior
3558:Rank sum
3306:Jackknife
3301:Bootstrap
3119:Bootstrap
3054:Parameter
3003:Statistic
2798:Statistic
2710:Run chart
2695:Pie chart
2690:Histogram
2680:Fan chart
2655:Bar chart
2537:L-moments
2424:Geometric
2253:cite book
2122:−
2060:mid-range
1983:−
1966:
1893:
1878:∣
1866:δ
1860:
1842:η
1811:−
1794:
1736:δ
1702:θ
1674:
1637:θ
1616:
1604:θ
1584:
1526:−
1509:
1468:θ
1462:
1445:−
1428:
1422:θ
1419:−
1413:
1399:−
1379:−
1338:θ
1323:θ
1280:θ
1267:−
1243:−
1235:θ
1215:θ
1126:δ
1120:
1105:δ
1099:
1087:δ
1081:
1015:θ
979:∣
960:…
928:δ
922:
897:…
865:η
819:θ
777:…
745:δ
711:Ω
708:∈
705:θ
697:θ
658:θ
624:~
621:δ
576:…
541:~
538:δ
529:
523:≤
501:…
469:δ
463:
437:Ω
434:∈
431:θ
428:∀
401:θ
359:…
327:δ
307:Ω
287:Ω
284:∈
281:θ
273:θ
235:…
186:θ
4633:Category
4579:Category
4272:Survival
4149:Johansen
3872:Binomial
3827:Isotonic
3414:(normal)
3059:location
2866:Blocking
2821:Sampling
2700:QâQ plot
2665:Box plot
2647:Graphics
2542:Skewness
2532:Kurtosis
2504:Variance
2434:Heronian
2429:Harmonic
2245:21523971
2161:See also
2023:states.
1728:Clearly
1047:(MMSE).
1037:Bayesian
679:complete
299:, where
161:variance
4605:Commons
4552:Kriging
4437:Process
4394:studies
4253:Wavelet
4086:General
3253:Plug-in
3047:L space
2826:Cluster
2527:Moments
2345:Outline
2148:is the
1169:Example
94:scholar
4474:Census
4064:Normal
4012:Manova
3832:Robust
3582:2-way
3574:1-way
3412:-test
3083:
2660:Biplot
2451:Median
2444:Lehmer
2386:Center
2243:
2233:
2207:(MMSE)
2174:(BLUE)
2144:where
1142:
144:is an
96:
89:
82:
75:
67:
4098:Trend
3627:prior
3569:anova
3458:-test
3432:-test
3424:-test
3331:Power
3276:Pivot
3069:shape
3064:scale
2514:Shape
2494:Range
2439:Heinz
2414:Cubic
2350:Index
418:UMVUE
101:JSTOR
87:books
4331:Test
3531:Sign
3383:Wald
2456:Mode
2394:Mean
2263:link
2259:link
2241:OCLC
2231:ISBN
1117:bias
73:news
3511:BIC
3506:AIC
2067:If
1963:log
1791:log
1613:var
1506:log
1459:log
1425:log
1410:exp
1177:on
1096:var
1078:MSE
1067:is
1055:An
725:any
526:var
460:var
420:if
416:is
387:of
140:or
132:In
56:by
4635::
2255:}}
2251:{{
2239:.
2087:is
1165:.
1035:A
449:,
136:a
3456:G
3430:F
3422:t
3410:Z
3129:V
3124:U
2326:e
2319:t
2312:v
2265:)
2247:.
2146:m
2128:,
2125:1
2119:m
2114:k
2110:1
2107:+
2104:k
2085:N
2081:N
2077:N
2069:k
2051:.
2002:2
1996:2
1992:)
1986:X
1979:e
1975:+
1972:1
1969:(
1957:=
1952:2
1947:2
1943:T
1937:=
1933:)
1929:T
1924:|
1917:2
1912:2
1908:T
1897:(
1890:E
1887:=
1884:)
1881:T
1875:)
1872:X
1869:(
1863:(
1857:E
1854:=
1851:)
1848:X
1845:(
1819:)
1814:x
1807:e
1803:+
1800:1
1797:(
1788:=
1785:T
1763:2
1758:2
1754:T
1748:=
1745:)
1742:X
1739:(
1706:2
1698:2
1693:=
1690:)
1685:2
1681:T
1677:(
1671:E
1641:2
1633:1
1628:=
1625:)
1622:T
1619:(
1609:,
1601:1
1596:=
1593:)
1590:T
1587:(
1581:E
1554:T
1534:)
1529:x
1522:e
1518:+
1515:1
1512:(
1503:=
1500:T
1474:)
1471:)
1465:(
1456:+
1453:)
1448:x
1441:e
1437:+
1434:1
1431:(
1416:(
1402:x
1395:e
1391:+
1388:1
1382:x
1375:e
1342:2
1334:1
1329:=
1326:)
1320:(
1317:g
1286:1
1283:+
1276:)
1270:x
1263:e
1259:+
1256:1
1253:(
1246:x
1239:e
1229:=
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