Page - 338 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 When there is no risk of confusion, we may abbreviate M(t;N) to M and fN,i(t;r) to fN(r), or even to f(r)—so that (A1) becomes M = f(0). Again, we may write ∂rM(t;N)/∂tri as Mr, ∂r+sM(t;N)/∂tri∂t s j asMr,s and∂ r+s+uM(t;N)/∂tri∂t s j∂t u l asMr,s,u,withsimilarconventions forhigher ordermixedderivatives. We can nowuse this to explicitly calculate lowordermoments of the count vectors. Using E(nri)= ∂ rM(t;N)/∂tri|t=0, thefirstNmomentsofninowfollowfrom(A1)andrepeateduseof (A2), notingthatm(0)=1and ai(0)=πi. Inparticular, thefirst6momentsofeachni canbeobtainedas follows,whereN≥6 isassumed. Using(A1)and(A2),wehave M1= f(1) M2= f(2)+ f(1) M3= f(3)+2f(2)+ f(2)+ f(1)= f(3)+3f(2)+ f(1) M4= f(4)+ 6f(3)+ 7f(2)+ f(1) M5= f(5)+10f(4)+25f(3)+15f(2)+ f(1) M6= f(6)+15f(5)+65f(4)+90f(3)+31f(2)+ f(1). Substituting in,wehave E(ni)= Nπi E(n2i)=N(2)π 2 i + Nπi E(n3i)=N(3)π 3 i + 3N(2)π 2 i + Nπi E(n4i)=N(4)π 4 i + 6N(3)π 3 i + 7N(2)π 2 i + Nπi E(n5i)=N(5)π 5 i +10N(4)π 4 i +25N(3)π 3 i +15N(2)π 2 i + Nπi E(n6i)=N(6)π 6 i +15N(5)π 5 i +65N(4)π 4 i +90N(3)π 3 i +31N(2)π 2 i +Nπi. Thiscanbeformalised in the followingLemma LemmaA1. The integer coefficients inanyexpansion Mr= r ∑ s=1 cr(s)f(s) (1≤ r≤N) canbecomputedusingcr(1)= cr(r)=1 together, for r≥3,with theupdate: cr(s)= cr−1(s−1)+scr−1(s) (1< s< r). Wenote that ifMr is requiredfor r>N,wemayrepeatedlydifferentiate MN= N ∑ s=1 cN(s)f(s) w.r.t. ti,notingthat f(N)=N!aNi nolongerdependsonm(t)sothat, forallh>0,∂ hf(N)/∂thi =N hf(N). Mixedmomentsofanyordercanbederivedfromthoseof lowerorder,exploitingthefact that ai dependson tonlyvia ti. We illustrate this byderiving those required for the secondand third momentsofW. First consider themixedmoments requiredfor thesecondmomentofW. Ofcourse,Var(W)=0 ifk=0.Otherwise,k>0,andcomputingVar(W) requiresE(n2in 2 j) for i = j.Wefindthisas follows, assumingN≥4. 338