Page 134 - Elements of Distribution Theory

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052184472Xc04 CUNY148/Severini May 24, 2005 2:39

120 Moments and Cumulants

where the sum is over all nonnegative integers x 1 , x 2 ,..., x m summing to n. Writing
n x j
m m m
x j exp(t j )θ j
exp(t j x j )θ = ,
j exp(t j )θ j m
j=1 j=1 j=1 j=1 exp(t j )θ j
it follows from the properties of the multinomial distribution that the moment-generating
function of X is
n n
m m

m
M X (t) = exp(t j )θ j = exp(t j )θ j , t = (t 1 ,..., t m ) ∈ R .
j=1 j=1
The cumulant-generating function is therefore given by

m

K X (t) = n log exp(t j )θ j .
j=1
It follows that, for j = 1,..., m,
E(X j ) = nθ j , Var(X j ) = nθ j (1 − θ j )

and, for j, k = 1,..., m,
Cov(X j , X k ) =−nθ j θ k .
Thus, the covariance matrix of X is the m × m matrix with ( j, k)th element given by
nθ j (1 − θ j )if j = k

σ jk = .
−nθ j θ k if j
= k

4.5 Moments and Cumulants of the Sample Mean
Let X 1 , X 2 ,..., X n denote independent, identically distributed, real-valued random vari-
ables. Let
n
1
¯
X n = X j
n
j=1
denote the sample mean based on X 1 , X 2 ,..., X n .
¯
In this section, we consider the moments and cumulants of X n . First consider the cumu-
¯
¯
lants. Let κ 1 ,κ 2 ,... denote the cumulants of X 1 and let κ 1 (X n ),κ 2 (X n ),... denote the
¯
cumulants of X n . Using Theorems 4.14 and 4.15, it follows that
1
¯
κ j (X n ) = j−1 κ j , j = 1, 2,.... (4.4)
n
For convenience, here we are assuming that all cumulants of X 1 exist; however, it is clear
that the results only require existence of the cumulants up to a given order. For instance,
¯
κ 2 (X n ) = κ 2 /n holds only provided that κ 2 exists.
¯
To obtain results for the moments of X n ,we can use the expressions relating moments
and cumulants; see Lemma 4.1. Then
2 n − 1 1
2
¯
E(X n ) = E(X 1 ) and E X ¯ = E(X 1 ) + E X 2 .
n 1
n n