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Distance between sampling with and without replacement
Author(s) -
Stam A. J.
Publication year - 1978
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1978.tb01387.x
Subject(s) - mathematics , combinatorics , sample (material) , epoch (astronomy) , sampling (signal processing) , set (abstract data type) , type (biology) , statistics , upper and lower bounds , mathematical analysis , physics , computer science , ecology , stars , astronomy , detector , biology , optics , thermodynamics , programming language
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 (L 1 ‐)distance and I the K ullback ‐L eibler 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 2 I. For fixed H we have d 0, I 0 as N if and only if n/N 0. Let W r 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 W 1 ., W r without and with replacement are given.