Page - 429 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 375 of its sine) takes the former intoaccount, the latterhasnotbeenexploitedyet. Todoso,weneedto estimateE(Im(μ−1Z))2. Consider Vn = 1n∑ n k=1Y 2 k that is under the hypothesis that the corresponding ζ is the unique circular population mean has expectation σ2 = Var(Yk) = E(Im(ζ−1Z))2. Now, 1−Vn= 1n∑nk=1(1−Y2k) is themeanofn independent randomvariables takingvalues in [0,1]and having expectation 1−σ2. By another application of Equation (6), we obtain P(σ2 ≥ Vn+ t) = P(1−Vn ≥ 1−σ2+ t) ≤ α4 for t = t(α4,1−σ2,0,1), the latter existing if α4 > (1−σ2)n. Since 1−σ2+ t(α4,1−σ2,0,1) increases with 1−σ2, there is a minimal σ2 for which 1−Vn ≥ 1−σ2+ t(α4,1−σ2,0,1)holdsandbecomesanequality;wedenote itby σ̂2=Vn+ t(α4,1− σ̂2,0,1). Inserting intoEquation(6), itbyconstructionfulfills α 4 = [( 1− σ̂2 1−Vn )1−Vn( σ̂2 Vn )Vn]n . (9) It is easy to see that the right-handsidedependscontinuouslyonand is strictlydecreasing in σ̂2 ∈ [Vn,1] (seeAppendixA), thereby traversing the interval [0,1] so thatone canagain solve the equationnumerically.Wethenmay,withanerrorprobabilityofatmost α4,use σ̂ 2 asanupperbound forσ2.Note that σ̂2>Vn exists if α4 > (1− σ̂2)n.The latter is fulfilledforanyVn<1sinceEquation(9) isequivalent to α 4 = ( 1− σ̂2)n[( 1 1−Vn ) ︸ ︷︷ ︸ >1 ( 1− σ̂2 1−Vn )−Vn ︸ ︷︷ ︸ >1 ( σ̂2 Vn )Vn ︸ ︷︷ ︸ >1 ]n . ForVn=1, let σ̂2=1be the trivialbound. With such anupper bound on its variance, we now can get a better estimate forP(Y¯n > t). Indeed,onemayuseanother inequalitybyHoeffding[11] (Theorem3): themeanW¯n= 1n∑ n k=1Wkof asequenceW1, . . . ,Wnof independentrandomvariables takingvalues in (−∞,1], eachhavingzero expectationaswellasvarianceρ2 fulfills P ( W¯n≥w )≤[(1+ w ρ2 )−ρ2−w( 1−w )w−1] n1+ρ2 , (10) ≤ exp(−nt[(1+ ρ2t ) ln(1+ tρ2)−1]). (11) for any w ∈ (0,1). Again, an elementary calculation (analogous to Lemma A1) shows that the right-hand side of Equation (10) is strictly decreasing in w, continuously ranging between 1 and( ρ2 1+ρ2 )n aswvaries in (0,1), so that thereexistsauniquew=w(γ,ρ2) forwhichtheright-handside equalsγ,providedγ∈ (( ρ2 1+ρ2 )n,1).Moreover, theright-handside increaseswithρ2 (asexpected), so thatw(γ,ρ2) is increasing inρ2, too (cf.AppendixA). Therefore,under thehypothesis that thecorrespondingζ is theuniquecircularpopulationmean, P (|Y¯n| ≥ w(α4,σ2)) ≤ 2α4 = α2. Now, sinceP(w(α4,σ2) ≥ w(α4, σ̂2)) = P(σ2 ≥ σ̂2) ≤ α4, setting sV=w(α4, σ̂ 2)wegetP (|Y¯n|≥ sV)≤ 34α.Note that ρ21+ρ2 increaseswithρ2, so incase s0 exists, σ̂2≤1 implies α4 >2 −n≥ ( σ̂2 1+σ̂2 )n , i.e., theexistenceof sV. FollowingtheconstructionforCH fromSection2,wecanagainobtainaconfidenceset forμwith coverageprobability at least 1−α as shown inourprevious article [13]. Inpracticehowever, this confidenceset ishard tocalculatesince σ̂2= σ̂2(ζ)has tobecalculatedforeveryζ∈S1.Thoughthese confidencesetscanbeapproximatedbyusingagridas in [13],wesuggestusingasimultaneousupper boundfor thevarianceof Imζ−1Zk. 429