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Statistica Sinica 17(2007), 895-907



Kim-Hung Li

The Chinese University of Hong Kong

Abstract: The sampling/importance resampling algorithm is an approximate noniterative sampling method. The algorithm has been used on many occasions to select an approximate random sample of size $m$ from a target distribution from $M$ input random variates. The selection mechanism is an unequal probability sampling with weights being the importance weights. As the weights are random, sampling without replacement is not always possible and some input variates may have more than one copy in the final sample.Duplication of values in the final output is undesirable as it means dependence among the output variates. In this paper a general and simple determination rule for $M$ is proposed. It keeps the duplication problem at a tolerably low level when a tight resampling method is used. We show that (a) $M=O(m)$ if and only if the importance weight is bounded above, (b) if the importance weight has a moment generating function, the suggested $M$ is of order $O(m
\ln(m))$, and (c) $M$ may need to be as large as $O(m^{c/(c-1)})$ if the importance weight has finite $c$-th moment for a $c>1$. A procedure is suggested to determine $M$ numerically. The method is tested on the Pareto, Gamma and Beta distributions, and gives satisfactory results.

Key words and phrases: Importance weight, Monte Carlo sampling, resampling method, sample size, tight resampling algorithm.

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