Distance between sampling with and without replacement

A. J. Stam

Statistica Neerlandica, 1978, vol. 32, issue 2, 81-91

Abstract: Summary Two random samples of size n are taken from a set containing N objects of H types, first with and then without replacement. Let d be the absolute (L1‐)distance and I the Kullback‐Leibler information distance between the distributions of the sample compositions without and with replacement. Sample composition is meant with respect to types; it does not matter whether order of sampling is included or not. A bound on I and d is derived, that depends only on n, N, H. The bound on I is not higher than 2I. For fixed H we have d0, I0 as N if and only if n/N0. Let Wr be the epoch at which for the r‐th time an object of type I appears. Bounds on the distances between the joint distributions of W1., Wr without and with replacement are given.

Date: 1978
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