Page - 334 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 Foreachtopic, theresultspresented invite further investigation. 4.1. TransitionBetweenDiscrete andContinuousFeaturesofSamplingDistributions Earlierwork[2]usedthedecomposition: D∗/2= ∑ {0≤i≤k:n∗i>0} n∗i log(n ∗ i/μi)=Γ ∗+Δ∗, Γ∗ := k ∑ i=0 αin∗i andΔ ∗ := ∑ {0≤i≤k:n∗i>1} n∗i logn ∗ i ≥0, whereαi :=− logμi, toshowthataparticularlybadcasefor theadequacyofanycontinuousapproximationtothesampling distributionof thedevianceD :=D∗|(N∗=N) is theuniformdiscretedistribution:πi=1/(k+1). In thiscase, theΓ∗ termcontributesaconstant to thedeviance,while theΔ∗ termhasnocontributions fromcellswith0or1observations—thesebeinginthevastmajorityintheN<< ksituationconsidered here. Inotherwords, allof thevariability inD comes fromthatbetween theni lognivalues for the (relativelyrare) cell countsabove1. Thisgivesrise toadiscretenessphenomenontermed“granularity” in [2],whosemeaningwas conveyedgraphically there in the caseN= 30 and k= 200. Workby Holst [19]predicts thatcontinuous (indeed,normal)approximationswill improvewith largervalues ofN/k, as is intuitive. Remarkably, simplydoubling thesamplesize toN=60wasshownin[2] tobe sufficient togiveagoodenoughapproximationformostgoodness-of-fit testingpurposes. Inother words,Nbeing30%ofk=200wasfoundtobegoodenoughforpracticalpurposes. Here,we illustrate the roleof k-asymptotics (Section2) in this transitionbetweendiscreteand continuous features by repeating the above analyses for different values of k. Figures 3 and 4 (wherek= 100whileN= 20and40, respectively)arequalitatively thesameas thosepresentedin[2]. Thedifferencehere is that thesmallervalueof kmeans thatahighervalueofN/k (40%) isneeded inFigure4 toadequately remove thegranularityevident inFigure3. For k= 400, thefigureswith N=50andN=100(omittedhere forbrevity)are,again,qualitatively thesameas in [2]—the larger valueofkneedingonlyasmallervalueofN/k (25%) forpracticalpurposes. Note theQQ-plotsused in these twofiguresarerelative tonormalquantiles. ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0 20 40 60 80 100 (a) Null distribution, N = 20 Rank of cell probability ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 200 400 600 800 1000 (b) Sample of Deviance Statistic Index ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ●●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −3 −2 −1 0 1 2 3 (c) QQplot Deviance Statistic Theoretical Quantiles Figure3. k=100,N=20. 334