Seite - 340 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 Usingexpressions forE(n4i)andE(n 2 in 2 j)establishedabove,andputting π(α) :=∑iπαi , wehave ∑i E(n4i) π2i =∑i{N(4)π2i +6N(3)πi+7N(2)+Nπ−1i } =N(4)π (2)+6N(3)+7N(2)(k+1)+Nπ (−1) and ∑∑i =j E(n2in 2 j) πiπj =∑i =j{N(4)πiπj+N(3)(πi+πj)+N(2)} =N(4)(1−π(2))+2N(3)k+N(2)k(k+1), so that A(2)=N(4)+2N(3)(k+3)+N(2)(k+1)(k+7)+Nπ (−1), whence Var(W)= N(4)+2N(3)(k+3)+N(2)(k+1)(k+7)+Nπ(−1) N4 − ( 1+ k N )2 = { π(−1)−(k+1)2 } +2k(N−1) N3 , after somesimplification. Note thatVar(W)dependson (πi)onlyviaπ(−1)while,bystrict convexityofx→1/x (x>0), π(−1)≥ (k+1)2, equalityholding iffπi i≡1/(k+1). Thus, forgivenkandN,Var(W) is strictly increasingas (πi)departs fromuniformity, tendingto∞as oneormoreπi→0+. Finally, for thesecalculations,welookatE[{W−E(W)}3]. RecallingagainthatN2(W+1)=∑i n 2 i πi , E[{W−E(W)}3]=E[{(W+1)−E(W+1)}3] =N−6A(3)−3Var(W)(E(W)+1)−(E(W)+1)3, whereA(3) :=N6E{(W+1)3} isgivenby A(3)=∑i E(n6i) π3i +3∑∑i =j E(n2in 4 j) πiπ 2 j +∑∑∑i,j,ldistinct E(n2in 2 jn 2 l) πiπjπl . Giventhat E(W)= k/NandVar(W)= { π(−1)−(k+1)2 } +2k(N−1) N3 , it suffices tofindA(3). Usingexpressions forE(n6i),E(n 2 in 2 jn 2 l), andE(n 2 in 4 j)establishedabove,wehave ∑i E(n6i) π3i =N(6)π (3)+15N(5)π (2)+65N(4)+90N(3)(k+1)+31N(2)π (−1)+Nπ(−2) 340