Page - 371 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 Proposition3states theconvergenceof themaximumlikelihoodestimate YˆN to the truevalueof theparameter Y¯. It isbasedonLemma1,whichstates that theparameter Y¯ is theRiemannianmedian of thedistributionL(Y¯,σ) in thesenseofdefinitionEquation(12). Proposition2 (MLEandmedian). Themaximumlikelihoodestimateof theparameter Y¯ is theRiemannian medianYˆN ofY1, . . . ,YN.Moreover, themaximumlikelihoodestimateof theparameterσ is the solution σˆN of: σ2× d dσ logζm(σ)= 1 N N ∑ n=1 d(Y¯,Yn) (20) Both YˆN and σˆN exist andareunique foranyrealizationof the samplesY1, . . . ,YN. ProofofProposition2. The log-likelihoodfunction,of theparameters Y¯andσ, canbewrittenas: N ∑ n=1 log p(Yn|Y¯,σ) = N ∑ n=1 log ( 1 ζm(σ) e− d(Y¯,Yn) σ ) = −N logζm(σ)− 1 σ N ∑ n=1 d(Y¯,Yn) Asthefirst terminthe lastexpressiondoesnotcontain Y¯, argmaxY¯ N ∑ n=1 log p(Yn|Y¯,σ)=argminY¯ N ∑ n=1 d(Y¯,Yn) Thequantityontheright isexactly YˆN byEquation(11). Thisprovesthefirstclaim.Now,consider the function: F(η)=−N log(ζm(−1 η ))+η N ∑ n=1 d(YˆN,Yn), η< −1 σm This function isstrictlyconcave, since it is the logarithmof themomentgeneratingfunctionofa positivemeasure.Note that limη→−1σm F(η)=−∞, andadmit foramoment that limη→−∞F(η)=−∞. By the strict concavityofF, there exists aunique ηˆN< −1σm (which is themaximumofF), such that F′(ηˆN)= 0. It follows that σˆN = −1ηˆN lies in (0,σm)andsatisfiesEquation (20). Theuniquenessof σˆN isaconsequenceof theuniquenessof ηˆN. Thus, theproof iscomplete.Now, it remains tocheckthat limη→−∞F(η)=−∞or just limσ→+∞ 1σ log(ζm(1σ))=0.Clearly: ∏ i<j sinh (|ri−rj| 2 ) ≤AmeBm|r| whereAm andBm are twoconstantsonlydependingonm. Usingthis, it followsthat: 1 σ log(ζm( 1 σ ))≤ 1 σ log(cmAm)+ 1 σ log (∫ Rm exp((−σ+Bm)|r|)dr1 · · ·drm ) (21) However, forsomeconstantCmonlydependingonm, ∫ Rm exp((−σ+Bm)|r|)dr1 · · ·drm=Cm ∫ ∞ 0 exp((−σ+Bm)u)um−1du ≤ (m−1)!Cm ∫ ∞ 0 exp((−σ+Bm+1)u)du= (m−1)!Cm σ−Bm−1 CombiningthisboundandEquation(21)yields limσ→+∞ 1σ log(ζm( 1 σ))=0. 371