Page - 339 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 TherelationM2= f(2)+ f(1)establishedabovegives ∂2M/∂t2j =N(2)a 2 j fN−2(0)+NajfN−1(0). (A3) Repeateduseof (A3)nowgives M2,2=N(4)a 2 i a 2 j fN−4(0)+N(3)aiaj(ai+aj)fN−3(0)+N(2)aiaj fN−2(0) (A4) so that E(n2in 2 j)=N(4)π 2 iπ 2 j+N(3)πiπj(πi+πj)+N(2)πiπj. We further lookat themixedmomentsneeded for the thirdmomentofW. For the skewness ofW,weneedE(n2in 4 j) for i = jand,whenk>1,E(n2in2jn2l) for i, j,ldistinct.Wefindthesesimilarly, as follows,assumingk>1andN≥6. Equation(A4)abovegives ∂2M/∂t2j∂t 2 l =N(4)a 2 ja 2 l fN−4(0)+N(3)ajal(aj+al)fN−3(0)+N(2)ajal fN−2(0) fromwhich,using(A3)repeatedly,wehave M2,2,2= a2ja 2 l{N(6)a2i fN−6(0)+N(5)ai fN−5(0)}+ajal(aj+al){N(5)a2i fN−5(0)+N(4)ai fN−4(0)}+ ajal{N(4)a2i fN−4(0)+N(3)ai fN−3(0)} =N(6)a 2 i a 2 ja 2 l fN−6(0)+N(5)aiajal{aiaj+ajal+alai}fN−5(0)+N(4)aiajal{ai+aj+al}fN−4(0)+ N(3)aiajal fN−3(0) so thatE(n2in 2 jn 2 l)equals N(6)π 2 iπ 2 jπ 2 l +N(5)πiπjπl{πiπj+πjπl+πlπi}+N(4)πiπjπl{πi+πj+πl}+N(3)πiπjπl. Finally, therelationM4= f(4)+6f(3)+7f(2)+ f(1)establishedabovegives ∂4M/∂t4j = N(4)a 4 j fN−4(0)+6N(3)a 3 j fN−3(0)+7N(2)a 2 j fN−2(0)+ NajfN−1(0) so that, againusing(A3)repeatedly,yields E(n2in 4 j)=N(6)π 2 iπ 4 j+N(5)πiπ 3 j(6πi+πj)+N(4)πiπ 2 j(7πi+6πj)+N(3)πiπj(πi+7πj)+N(2)πiπj. Combiningaboveresults,weobtainhere thefirst threemomentsofW.Highermomentsmaybe foundsimilarly. Wefirst lookatE(W).WehaveW= 1N2 k ∑ i=0 n2i πi −1andE(n2i)=N(2)π2i +Nπi, so that E(W)= N(2) N2 + (k+1) N −1= k N . Thevariance iscomputedbyrecalling thatN2(W+1)=∑i n2i πi ,whileE(W)= kN, Var(W)=Var(W+1)= A(2) N4 − ( k N +1 )2 , where A(2) :=N4E{(W+1)2}=∑i E(n4i) π2i +∑∑i =j E(n2in 2 j) πiπj . 339