Page - 428 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 375 Then,weobtain the followingresult: sp s0 0 λζ ζ P(Imζ−1Z¯n≥ sp)≤ 38α P(Reζ−1Z¯n≤−s0)≤ α4 P(Imζ−1Z¯n≤−sp)≤ 38α Figure2.Theconstructionfor the testof thehypothesisEZ=λζwithλ>0. sp Z¯n δH 0 λζ ζ Figure 3. The critical Z¯n regarding the rejection of ζ. δH bounds the angle between μˆn and any acceptedζ. Proposition1. LetZ1, . . . ,Zn be randomvariables takingvalueson theunit circleS1,α∈ (0,1), and letCH bedefinedas inEquation (8). (i) CH is a (1−α)-confidence set for the circular populationmean set. In particular, ifEZ = 0, i.e., the circular populationmean set equalsS1, then |Z¯n|> √ 2s0 with probability atmost α, so indeed CH=S1withprobabilityat least1−α. (ii) s0 andsp areof ordern −12 . (iii) IfEZ =0, then√nδH→0 inprobabilityand theprobabilityof obtaining the trivial confidence set, i.e., P(CH=S1)=P(|Z¯n|≤ √ 2s0), goes to0exponentially fast. Proof. (i)holdsbyconstruction. For (ii), recall Equation (7), from which we obtain the estimates α4 ≤ exp(−ns20/2) resp. 3 8α≤ exp(−ns2p/2), implying that s0 and sp are of order n− 1 2; the same holds stochastically for δH since Z¯n → EZ a.s. Regarding the second statement of (iii), if μ is unique, consider ζ =−μ; then, τ = EX¯n < 0 and − √ 2s0 is eventually less than τ2 and also α > 2 −n+2 eventually. Hence, the probability of obtaining the trivial confidence setCH = S1 is eventually bounded by P(ζ∈CH)≤P(X¯n>−s0)≤P(X¯n> τ2)=P(X¯n−EX¯n>−τ2)≤ exp(−nτ2/8), andthuswillgo to zeroexponentially fastasn tends to infinity. 3. EstimatingtheVariance From the central limit theorem for μˆn in case of unique μ, cf. Equation (4), we see that the aymptoticvarianceof μˆngetssmall if |EZ| isclose to1(thenEZ isclose totheboundaryS1 of theunit disc,which ispossibleonly if thedistribution isveryconcentrated)or if thevarianceE(Im(μ−1Z))2 in thedirectionperpendicular toμ is small (if thedistributionwereconcentratedon±μ, thisvariance wouldbezeroand μˆnwouldequalμwith largeprobability).WhileδH (|Z¯n|beingthedenominator 428