Page - 341 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 ∑∑i =j E(n2in 4 j) πiπ 2 j =N(6)πiπ 2 j+N(5)πj(6πi+πj)+N(4)(7πi+6πj)+N(3)(πi/πj+7)+N(2)π −1 j =N(6){π(2)−π(3)}+N(5){6+(k−6)π(2)}+ 13N(4)k+N(3){π(−1)+(7k−1)(k+1)}+N(2)kπ(−1) and ∑∑∑i,j,ldistinct E(n2in 2 jn 2 l) πiπjπl =N(6){1+2π(3)−3π(2)}+3N(5)(k−1){1−π(2)}+ 3N(4)k(k−1)+N(3)k(k2−1) so that, after somesimplification, A(3)=N(6)+3N(5)(k+5)+N(4){3k(k+12)+65}+ N(3){k3+21k2+107k+87}+3N(3)π(−1)+N(2)(31+3k)π(−1)+Nπ(−2). Substituting inandsimplifying,wefindE[{W−E(W)}3] tobe: { π(−2)−(k+1)3 } −(3k+25−22N) { π(−1)−(k+1)2 } +g(k,N) N5 , where g(k,N)=4(N−1)k(k+2N−5)>0. Note that E[{W−E(W)}3] depends on (πi) only via π(−1) and the larger quantity π(−2). Inparticular, forgivenkandN, theskewnessofW tends to+∞asoneormoreπi→0+. AppendixB. TruncateandBoundApproximations In thenotationofLemma1, it suffices tofind truncateandboundapproximations for eachof E(Xμ),E(X.Xμ), andE(X2μ). For all r,s inN , definehr,s(μ) := E{(log(X+r))s}. Appropriate choicesofm∈N0 and a∈A in (9), togetherwith (10),give: E(Xμ)=μh1,1(μ)−μlogμ, E(X.Xμ)={μ2h2,1(μ)+μh1,1(μ)}−(μ2+μ)logμ, and: E(X2μ)=μ 2h2,2(μ)+μh1,2(μ)+(μ2+μ)(logμ)2−2logμ{μ2h2,1(μ)+μh1,1(μ)}, so that it suffices to truncateandboundhr,s(μ) for r,s∈{1,2}. Forall r,s inN , andforallm∈N0,wewrite: hr,s(μ)=h [m] r,s (μ)+ε [m] r,s (μ) inwhich: h[m]r,s (μ) :=∑mx=0{(log(x+r))s}p(x)and ε[m]r,s (μ) :=∑∞x=m+1{(log(x+r))s}p(x). Usingagain (7), the“error term” ε[m]r,s (μ)has lowerandupperbounds: 0< ε[m]r,s (μ)< ε [m] r,s (μ) :=∑∞x=m+1(x+(r−1))sp(x). 341