Page - 336 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 0 50 100 150 200 Index Figure6.Exponentiallydecreasingvaluesofπi. 4.3.Variation inFiniteSampleAdequacyofAsymptoticDistributionsacross theParameterSpace Pearson’sχ2 statistic (α=3) iswidelyusedtotest independenceincontingencytables,astandard rule-of-thumbforitsvaliditybeingthateachexpectedcellfrequencyshouldbeatleast5. Forillustrative purposes,weconsider 2×2 contingency tables, the relevantN-asymptoticnulldistributionbeing χ21. We assess the adequacy of this asymptotic approximation by comparing nominal and actual significance levelsof this test,basedon10,000replications. Particular interest lies inhowtheseactual levelsvaryacrossdifferentdatagenerationprocesseswithinthesamenullhypothesisof independence. Figures7and8showtheactual levelof thePearsonχ2 test fornominal levels0.1and0.05 for samplesizesN=20andN=50,withπr andπcdenotingrowandcolumnprobabilities, respectively. Theabovegeneral ruleappliesonlyat thecentralblackdot inFigure7,and inside theclosedblack curvedregion inFigure8. Theactual levelwascomputedforallpairsofvaluesofπr andπc, averaged using thesymmetryof theparameterspace,andsmoothedusingthekernel smoother for irregular2D data (implemented in thepackagefields inR). Ineachcase, thewhite tonecontains thenominal level, while redtonescorrespondto liberalandblue tones toconservativeactual levels. Thefinitesampleadequacyof this standardasymptotic test clearlyvariesacross theparameter space. Inparticular, itsnominalandactual levelsagreewellatsomeparametervaluesoutsidethestandard rule-of-thumbregion;and,conversely,disagreesomewhatatotherparametervalues insideit. Intriguingly, theagreementbetweennominalandactual levelsdoesnot improveeverywherewithsamplesize.Overall, theclearpatternsevident inthis lackofuniformityinvitefurthertheoretical investigation. 0.0 0.2 0.4 0.6 0.8 1.0 N = 20, nominal level = 0.1 πc 0.00 0.05 0.10 0.15 0.20 ● 0.0 0.2 0.4 0.6 0.8 1.0 N = 20, nominal level = 0.05 πc 0.00 0.02 0.04 0.06 0.08 0.10 ● Figure7.Heatmapof theactual levelof the test forN=20atnominal levels0.1and0.05; thestandard rule-of-thumb(whereexpectedcountsaregreater than5)appliesonlyat theblackdot. 336