Seite - 372 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 Remark1. ReplacingF in thepreviousproofwithF(η)=− log(ζm(−1η ))+ηcwhere c>0 shows that the equation: σ2× d dσ logζm(σ)= c has aunique solutionσ∈ (0,σm). This shows inparticular thatσ →σ2× ddσ logζm(σ) is a bijection from (0,σm) to (0,∞). Considernowthenumerical computationof themaximumlikelihoodestimates YˆN and σˆN given byProposition2.Computationof YˆN consists infindingtheRiemannianmedianofY1, . . . ,YN,defined byEquation(11). ThiscanbedoneusingtheRiemanniansub-gradientdescentalgorithmof [8]. The k-th iterationof thisalgorithmproducesanapproximation YˆkN of YˆN in the followingway. Fork=1,2, . . ., letΔk bethesymmetricmatrix: Δk= 1 N N ∑ n=1 Log Yˆ k−1 N (Yn) ||Log Yˆ k−1 N (Yn)|| (22) Here, Log is theRiemannian logarithmmapping inverse to the theRiemannian exponential mapping: ExpY (Δ)=Y 1/2 exp ( Y−1/2ΔY−1/2 ) Y1/2 (23) and ||Loga(b)||= √ ga(b,b). Then, Yˆ k N isdefinedtobe: Yˆ k N=ExpYˆk−1N (τkΔk) (24) whereτk>0 isastepsize,whichcanbedeterminedusingabacktrackingprocedure. Computationof σˆN requiressolvinganon-linearequation inonevariable. This is readilydone usingNewton’smethod. It is shownnowthat theempiricalRiemannianmedian YˆN convergesalmost surely to the true median Y¯. Thismeans that YˆN is a consistent estimator of Y¯. The proof of this fact requires few notationsandapreparatory lemma. For Y¯∈Pm andσ∈ (0,σm), let: E(Y|Y¯,σ)= ∫ Pm d(Y,Z)p(Z|Y¯,σ)dv(Z) Thefollowing lemmashowshowtofind Y¯andσ fromthefunctionY →E(Y|Y¯,σ). Lemma1. ForanyY¯∈Pm andσ∈ (0,σm), the followingpropertieshold (i) Y¯ isgivenby: Y¯=argminY E(Y|Y¯,σ) (25a) That is, Y¯ is theRiemannianmedianofL(Y¯,σ). (ii) σ isgivenby: σ=Φ(E(Y¯|Y¯,σ)) (25b) where the functionΦ is the inverse functionof σ →σ2×d logζm(σ)/dσ. ProofofLemma1. (i)LetE(Y)= E(Y|Y¯,σ). According toTheorem2.1 in [28], this functionhasa uniqueglobalminimum,whichisalsoauniquestationarypoint. Thus, toprovethat Y¯ is theminimum 372