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Withers, C. Expressions for Moments and Cumulants in Terms of Each Other Using Bell Polynomials, P

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Withers, C. Expressions for Moments and Cumulants in Terms of Each Other Using Bell Polynomials, P Expressions for moments and cumulants in terms of each other using Bell polynomials C S Withers Applied Mathematics Group The New Zealand Institute for Industrial Research and Development P O Box 31-310 Lower Hutt New Zealand Abstract Moments and cumulants are expressed in terms of each other using Bell poly­ nomials. The recurrence formulas for the latter make these expressions amenable to use by algebraic manipulation programs. One of the four formulas given is an explicit version of Kendall’s use of Faa di Bruno’s chain rule to express cumulants in terms of moments. 1 Introduction and summary . The classic expressions for moments and cumulants in terms of each other involve the partition functions: see for example (3.33), (3.40) of Stuart and Ord (1987). Recurrence relations between them have been given by Smith (1992). In §2 we express cumulants in terms of moments. In §3 we do the reverse. The method uses the exponential Bell polynomials, defined by oo oo ( H Sf,■*•/*■!)*/*! = for k = 0 ,1 ,... (1.1) t=l i=k These are tabled on page 307, 308 of Comtet (1974): Bik — Bik{y) is a polynomial in yi,... and satisfies the recurrence relation For example Bio = <5«o, (1 or 0 for i = 0 or i ± 0), Bn - yt, (1.3) AMS 1990 Subject Classifications: Primary 62E30, Secondary 60E05. Keywords and phrases: moments, cumulants, Bell polynomials, recurrence formulas, Faa di Bruno’s chain rule. 226 Bn = y\ and Biti- 1 = j y\ 2 y2. (1.4) ^From (1.2), (1.3) Bik is easily computed by an algebraic manipulation program. Given a random variable X on R set mr = E X r , fir = E(X — m\)r and «r = rth cumulant of X In §4 we give an alternative derivation of 2 of our 4 formulas using Faa di Bruno’s chain rule for differentiating a function of a function. One of these gives an explicit form for a method advocated by Kendall. 2 Cumulants in terms of moments Set 5 = Eetx - 1 = x trmr/r\. Then Sk/k\ = RHS (1.1) at y = m where m = (mi,m2,...). Taking the coefficient of tJ/j\ in oo log (Eetx) = log(l + S) = £(-1 )*-*s7fc k= 1 gives = (2.1) k=l as given by (2) pl60 of Comtet (1974). Set oo T = Ee‘<x-"“> - 1 = £ t y r/r! = t £ t%/j\ r= 2 j=l where y3 = pj+1 /( j + 1). (2.2) So oo Tk/k\ = £ tk+rBTk(y)/r\. r=k Taking the coefficient of V /j! in oo log(Eetx) - rn.it = log(l + T ) = £ ( - 1 ) * ' ' Tk/k k= 1 gives = mi and i/2 Kj = ^ ( - l ) fc-1(^ - # ,-* ,* (2/) for i > 2 (2.3) *=i where (j)k = j\/{j ~ A:)! = ;'(;' - 1)... (j - k + 1), 227 and sums over a < k < b. For example *5 = - 1)!(5)fc Bs-k,k k= 1 — 5^41 — 20^32 = 5y4 - 60yly2 by (1.3), (1.4) = /x5 - 10/i2/i3. 3 Moments in terms of cumulants . Set oo K = I<x(t) = log (Eetx) = E trKr/r\. r = 1 Then oo oo oo £ e ‘* = £ A'Vfc! = £ E Bik(K)e/H (3.1) A:=0 fc=0 :=Jfc by (1.1). Taking the coefficient of tJ/j\ gives m, = E BA*), (3-2) k=0 as given by (2) pl60 of Comtet (1974). If mi = 0, K = t £ £ i t'yi/i! where Vi = K.+i/(* + 1) (3-3) so K k/k'. = tkY ,B ik(y)ti/i\ t=k and the coefficient of V/j\ in (3.1) gives i/2 = 5Z 0‘)ik Bj-kAv) for J > 1- (3-4) fc=i For example fiQ = 6 £ 5i + 30^42 + 120^33 = 6^5 + 30(4y iy 3 + 3j/2 ) + 12 0 j/j using (1.2) or Comtet’s table, so fiQ — AC6 -h 15/c2 ac4 -}- 10/Cg -f- 15/c|. 228 4 The connection with Faa di Bruno’s chain rule . Faa di Bruno’s chain rule for the rth derivative of a function of a function may be written f(9{t))0) = Bjk{g)h for j > 1, where gt = g(l){t) and fk = f (k](g{t)). (4.1) k=1 See for example Comtet (1974). (For a more traditional approach with the coef­ ficient of fk expressed in terms of the partition function, see for example Withers (1984)). If f(g) = log(</), then /<*>(<?) = ( - 1 )*-■(* - 1)!<T*. So taking g(t) = EetX and setting t = 0 gives gt = m,-, so (4.1) implies (2.1). This is essentially the method advocated by Kendall in (3.16) of Stuart and Ord (1987), although neither the connection with Bell polynomials nor an explicit formula like (2.1) is given. If f(g) = exp{g), then f (k){g) = exp(g), so taking g(t) = log(EetX) and setting t = 0 gives gi = so (4.1) implies (3.2). REFERENCES Comtet, L. (1974) Advanced combinatorics. Reidel, Dordrecht. Smith, P. (1992) A recursive formulation of the old problem of obtaining moments from cumulants and vice-versa. Submitted. Stuart, A. and Ord, J.K. (1987) Kendall’s advanced theory of statistics, Griffin, London. Withers, C.S. (1984) A chain rule for differentiation with applications to multi­ variate Hermite polynomials. Bulletin, Austral Math Soc. 30, 247-250. 229.
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