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# On the Conditional Moments of the k-Statistics for the Poisson Distribution

John J. Gart and Hugh M. Pettigrew
Biometrika
Vol. 57, No. 3 (Dec., 1970), pp. 661-664
DOI: 10.2307/2334785
Stable URL: http://www.jstor.org/stable/2334785
Page Count: 4
Consider a random sample, x1,..., xn, from a Poisson distribution. The Rao-Blackwell theorem and the completeness property of this distribution are exploited to show that $E(k_j\mid \Sigmax_i = X) = \bar{x} (j = 1, 2,\ldots)$ 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.