Page - 425 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 375 Observingthat theEuclideansamplemeanis theminimiserof thesumofsquareddistances, this canbeput inthemoregeneral frameworkofFréchetmeans [6]: definethesetof circularsamplemeans tobe μˆn=argmin ζ∈S1 n ∑ k=1 |Zk−ζ|2 , (1) andanaloguouslydefinethe set of circularpopulationmeansof therandomvariableZ tobe μ=argmin ζ∈S1 E |Z−ζ|2 . (2) Then, as usual, the circular samplemeans are the circular populationmeanswith respect to the empiricaldistributionofZ1, . . . ,Zn. Thecircularpopulationmeancanberelated to theEuclideanpopulationmeanEZbynoting that E |Z−ζ|2=E |Z−EZ|2+ |EZ−ζ|2 (instatistics, this iscalledthebias-variancedecomposition), so that μ=argmin ζ∈S1 |EZ−ζ|2 (3) is the set ofpoints on the circle closest toEZ. It follows thatμ is unique if andonly ifEZ = 0 in whichcase it isgivenbyμ=EZ/|EZ|, theorthogonalprojectionofEZonto thecircle;otherwise, i.e., ifEZ=0, thesetof circularpopulationmeans isallofS1.Weconsider the informationofwhether the circularpopulationmeanisnotunique,e.g.,butnotexclusivelybecauseZ isuniformlydistributed over thecircle, toberelevant; it thusshouldbe inferredfromthedataaswell.Analogously, μˆn iseither allofS1 oruniquelygivenby Z¯n/|Z¯n|accordingtowhether Z¯n is0ornot.Note that Z¯n =0a.s. ifZ is continuouslydistributedonthecircle, evenifEZ=0. Z¯n iswhat isknownasthevector resultant, while Z¯n/|Z¯n| is sometimesreferredtoas themeandirection. Theexpectedsquareddistancesminimised inEquation(2)aregivenbythemetric inherited from theambientspaceC; therefore,μ isalsocalledthesetof extrinsicpopulationmeans. Ifwemeasured distances intrinsically along the circle, i.e., usingarc-length insteadof chordaldistance,wewould obtainwhat is called the set of intrinsicpopulationmeans. Wewill not consider the latter in the following, seee.g., [7] foracomparisonand[8,9] forgeneralizationsof theseconcepts. Ouraimis toconstruct confidence sets for thecircularpopulationmeanμ that formasupersetof μwithacertain (so-called) coverageprobability that is requiredtobenot less thansomepre-specified significance level1−α forα∈ (0,1). The classical approach is to construct an asymptotic confidence interval where the coverage probability converges to 1−α when n tends to infinity. This canbedone as follows: sinceZ is a boundedrandomvariable, √ n(Z¯n−EZ)convergestoabivariatenormaldistributionwhenidentifying CwithR2.Now,assumeEZ =0soμ isunique. Then, theorthogonalprojection isdifferentiable ina neighbourhoodofEZ, so theδ-method(seee.g., [1] (p.111)or [4] (Lemma3.1)) canbeappliedand oneeasilyobtains √ n Arg(μ−1μˆn) D→N ( 0, E(Im(μ−1Z))2 |EZ|2 ) , (4) where Arg : C\{0} → (−π,π] ⊂ R denotes the argument of a complex number (it is defined arbitrarilyat0∈C),whilemultiplyingwithμ−1 rotatessuchthatEZ=μ ismappedto0∈ (−π,π], see e.g., [4] (Proposition3.1) or [7] (Theorem5). Estimating theasymptoticvarianceandapplying Slutsky’s lemma,onearrivesat theasymptotic confidencesetCA = {ζ∈S1 : |Arg(ζ−1μˆn)|< δA} provided μˆn isunique,where theangledeterminingthe interval isgivenby δA= q1−α2 n|Z¯n| √ n ∑ k=1 ( Im(μˆ−1n Zk) )2, (5) 425