Product-form estimators: exploiting independence to scale up Monte Carlo

We introduce a class of Monte Carlo estimators that aim to overcome the rapid growth of variance with dimension often observed for standard estimators by exploiting the target’s independence structure. We identify the most basic incarnations of these estimators with a class of generalized U-statistics and thus establish their unbiasedness, consistency, and asymptotic normality. Moreover, we show that they obtain the minimum possible variance amongst a broad class of estimators, and we investigate their computational cost and delineate the settings in which they are most efficient. We exemplify the merger of these estimators with other well known Monte Carlo estimators so as to better adapt the latter to the target’s independence structure and improve their performance. We do this via three simple mergers: one with importance sampling, another with importance sampling squared, and a final one with pseudo-marginal Metropolis–Hastings. In all cases, we show that the resulting estimators are well founded and achieve lower variances than their standard counterparts. Lastly, we illustrate the various variance reductions through several examples.

Non-Parametric Tests for Testing of Scale Parameters

One of the fundamental problems in testing of equality of populations is of testing the equality of scale parameters. The subsequent usages for scale are dispersion, spread and variability. In this paper, we proposed non-parametric tests based on U-Statistics for the testing of equality of scale parameters. The null distribution of proposed tests is developed and its Pitman efficiency is worked out to compare proposed tests with respect to some existing tests. Simulation study is carried out to compute the asymptotic power of proposed tests. An illustrative example is also provided.