Page - 337 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 0.0 0.2 0.4 0.6 0.8 1.0 N = 50, 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 = 50, nominal level = 0.05 πc 0.00 0.02 0.04 0.06 0.08 0.10 Figure8.Heatmapof theactual levelof the test forN=50atnominal levels0.1and0.05; thestandard rule-of-thumb(whereexpectedcountsaregreater than5)applies inside theclosedblackcurvedregion. 5.Discussion Thispaperhas illustrated thekey importanceofworkingwith theboundaryof theclosureof exponential familieswhenstudyinggoodness-of-fit testing in thehighdimensional, lowsamplesize context. Someof thiswork isnew(Section2),whilesomeuses thestructureofextendedexponential families toaddinsight tostandardresults in the literature (Section3). The last section,Section4,uses simulationstudies tostart toexploreopenquestions in thisarea. Oneopenquestion—relatedto theresultsofTheorems1and2—is tosee ifaunifiedtheory, forall valuesofα, andover largeclassesofextendedexponential families, canbedeveloped. Acknowledgments:Theauthorswould like to thanktheEPSRCfor thesupportofgrantnumberEP/L010429/1. GermainVanBeverwouldalso like to thankFRS-FNRSfor its support throughthegrantFC84444.Wewouldalso like to thankthereferees forveryhelpfulcomments. AuthorContributions: All fourauthorsmadecritical contributions to thepaper. R.S.madekeycontribution to,especially,Section4. P.M.andF.C.providedtheoverall structureandkeycontentdetailsof thepaper.G.V.B. providedinvaluablesuggestions throughout. Conflictsof Interest:Theauthorsdeclarenoconflictsof interest. AppendixA. ProofofTheorem1 Westartbynotinganimportant recurrencerelationwhichwillbeexploited in thecomputations below.Bydefinition, forany t :=(ti)∈Rk+1,n=(ni)hasmomentgeneratingfunction M(t;N) :=E{exp(tTn)}=[m(t)]N withm(t)=∑ki=0ai and ai= ai(ti)=πie ti. Putting fN,i(t;r) :=N(r) [m(t)] N−r ari (0≤ r≤N), where N(r) := NPr= { 1 if r=0 N(N−1)...(N−(r−1)) if r∈{1,...,N} , wehave M(t;N)= fN,i(t;0) (0≤ i≤ k) (A1) andtherecurrencerelation: ∂fN,i(t;r) ∂ti = fN,i(t;r+1)+rfN,i(t;r) (0≤ i≤ k; 0≤ r<N) . (A2) 337