Printed in Great Britain On the conditional moments of the k-statistics for the Poisson distribution

Consider a random sample, xx,...,xn, from a Poisson distribution. The Rao-Blaokwell theorem and the completeness property of this distribution are exploited to show that E(ki\Lxi = X) — x {j = 1,2,...) and to find higher conditional moments of the sample cumulants. Some possible applications to the testing of the fit of the Poisson distribution are suggested. 1. THE CONDITIONAL EXPECTATION OF kt Consider a random sample, x1,...,xn, from a Poisson distribution with mean, A. Fisher (1950) suggested that statistics testing the fit of the Poisson distribution be conditioned on the observed value of the sufficient statistic X = Ea;4. Rao & Chakravarti (1956) found a ample device for deriving the conditional moments of several such statistics, in particular, k2 = S(x< — x)*l(n — 1), where x = Xjn. These and other results may be found in an alterna-tive way by exploiting the Rao-Blackwell theorem and the concept of completeness (Hogg & Craig, 1965, pp. 216-23). The sample cumulants, kt (Kendall & Stuart, 1969, Chapter 12), are unbiased estimators of the respective population cumulants, Kt (j = 1,2,...). Since all the cumulants of the Poisson equal its mean, it follows that E(kj) = A for all,?. By the Rao-Blackwell theorem, E{k]\X) = fy(X), where Qj(X) is an unbiased miniim-im variance estimator of A. By the completeness property, Q>j{X) = x. Thus E(kj\X) = x for all j. Lukacs (see Lukacs & Laha, 1964, p. 109) found the following related result: among nonnegative disorete distributions possessing finite moments up to order r, for some r> s> 1, E{kT—ks\X) = A+Bx, where A and B are constants, if and only if the population sampled is Poisson. Of course, from what we have just seen, A = B = 0. Bartko, Greenhous