Two-stage importance sampling with mixture proposals

Wentao Li, Zhiqiang Tan, Rong Chen

Research output: Contribution to journalArticlepeer-review


For importance sampling (IS), multiple proposals can be combined to address different aspects of a target distribution. There are various methods for IS with multiple proposals, including Hesterberg's stratified IS estimator, Owen and Zhou's regression estimator, and Tan's maximum likelihood estimator. For the problem of efficiently allocating samples to different proposals, it is natural to use a pilot sample to select the mixture proportions before the actual sampling and estimation. However, most current discussions are in an empirical sense for such a two-stage procedure. In this article, we establish a theoretical framework of applying the two-stage procedure for various methods, including the asymptotic properties and the choice of the pilot sample size. By our simulation studies, these two-stage estimators can outperform estimators with naive choices of mixture proportions. Furthermore, while Owen and Zhou's and Tan's estimators are designed for estimating normalizing constants, we extend their usage and the two-stage procedure to estimating expectations and show that the improvement is still preserved in this extension.
Original languageEnglish
Pages (from-to)1350-1365
Number of pages16
JournalJournal of the American Statistical Association
Issue number504
Publication statusPublished - 19 Dec 2013


  • control variates
  • normalizing constant
  • pilot samples


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