Seite - 426 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 375 withq1−α2 denotingthe (1− α 2)-quantileof thestandardnormaldistributionN(0,1). Thereare twomajordrawbacks to theuseofasymptoticconfidence intervals: firstly,bydefinition, theydonotguaranteeacoverageprobabilityofat least1−α forfiniten, so thecoverageprobability for a fixeddistribution and sample sizemaybemuch smaller. Indeed, Simulation 2 in Section 4 demonstrates that, evenforn=100, thecoverageprobabilitymaybeas lowas64%whenconstructing theasymptotic confidenceset for1−α= 90%. Secondly, they assume thatEZ = 0, so theyarenot applicable toalldistributionsonthecircle. Since inpractice it isunknownwhether thisassumption hold, onewould have to test the hypothesis EZ = 0, possibly again by an asymptotic test, and construct the confidence set conditionedon thishypothesis havingbeen rejected, settingCA = S1 otherwise.However, this sequentialprocedurewouldrequiresomeadaptation takingthepre-test into account (cf. e.g., [10])—wecomebackto thispoint inSection5—andit isnotcommonly implemented inpractice. Wethereforeaimtoconstructnon-asymptoticconfidencesets forμ, guaranteeingcoveragewithat least thedesiredprobability foranysamplesizen,which inadditionareuniversal in thesensethat they donotmakeanydistributionalassumptionsabout thecirculardatabesides thembeing independent and identicallydistributed. It hasbeen shown in [7] that this ispossible; however, the confidence sets thatwereconstructedtherewere far too large tobeofuse inpractice.Nonetheless,westartby varyingthatconstruction inSection2butusingHoeffding’s inequality insteadofChebyshev’sas in [7]. Considerable improvementsarepossible ifonetakes thevarianceE(Im(μ−1Z))2 “perpendicular to EZ”intoaccount; this isachievedbyasecondconstructioninSection3.Ofcourse, thelatterconfidence setswill stillbeconservativebutProposition2(iv) showsthat theyare (for1−α=95%)onlya factor ∼ 32 longerthantheasymptoticoneswhenthesamplesizen is large.Wefurther illustrateandcompare thoseconfidencesets in simulationsand foranapplication to realdata inSection4,discussing the resultsobtainedinSection5. 2.ConstructionUsingHoeffding’s Inequality We will construct a confidence set as the acceptance region of a series of tests. This idea has beenused before for the construction of confidence sets for the circular populationmean [7] (Section6);however,wewillmodify thatconstructionbyreplacingChebyshev’s inequality—which is tooconservativehere—bythreeapplicationsofHoeffding’s inequality [11] (Theorem1): ifU1, . . . ,Un are independent randomvariables takingvalues in theboundedinterval [a,b]with−∞< a< b<∞. Then, U¯n= 1n∑ n k=1UkwithEU¯n= ν fulfills P ( U¯n−ν≥ t )≤[( ν−a ν−a+ t )ν−a+t( b−ν b−ν− t )b−ν−t] nb−a (6) for any t ∈ (0,b−ν). The bound on the right-hand side—denoted β(t)—is continuous and strictlydecreasing in t (as expected; seeAppendixA)with β(0) = 1 and limt→b−ν β(t) = ( ν−a b−a )n whence aunique solution t= t(γ,ν,a,b) to the equation β(t) = γ exists for anyγ∈ (( ν−a b−a )n,1). Equivalently, t(γ,ν,a,b) is strictlydecreasing inγ. Furthermore,ν+ t(γ,ν,a,b) is strictly increasing in ν (seeAppendixAagain),which isalso tobeexpected.While there isnoclosedformexpressionfor t(γ,ν,a,b), it canwithoutdifficultybedeterminednumerically. Note that theestimate β(t)≤ exp(−2nt2/(b−a)2) (7) isoftenusedandcalledHoeffding’s inequality [11].While thiswouldallowtosolveexplicitly for t, weprefer toworkwithβas it is sharper, especially forνclose tobaswellas for large t.Nonetheless, it showsthat the tailboundβ(t) tends tozeroas fastas ifusing thecentral limit theoremwhich iswhyit iswidelyappliedforboundedvariables, seee.g., [12]. 426