Permutation tests from biased samples for the equality of two distributions

Abstract

Suppose that independent samples are taken from two distributions according to two biased (nonrandom) sampling plans. This study constructs a class of biased-adjusted permutation tests for the equality of the two distributions. A resampling algorithm that generates permutations with unequal probabilities is proposed to estimate the tests’ exact P-values with a manageable Monte Carlo simulation error. This algorithm leads to the derivation of large-sample, normal theory tests. Conditions under which these tests are consistent are given. The tests’ power at finite sample sizes is examined via a simulation study.

Keywords:

• weighted distribution
• rank sum test
• rejective sampling
• importance sampling
• asymptotic normality

AMS Subject Classification :

• Primary: 62G10
• Secondary: 62D05
• 62G20

Acknowledgements

We would like to thank the editor, the associate editor, and the anonymous referees for their helpful comments on this paper.