Accurate and Efficient Estimation of Small P-values with the Cross-Entropy Method: Applications in Genomic Data Analysis

Small $p$-values are often required to be accurately estimated in large scale genomic studies for the adjustment of multiple hypothesis tests and the ranking of genomic features based on their statistical significance. For those complicated test statistics whose cumulative distribution functions are analytical intractable, existing methods usually do not work well with small $p$-values due to lack of accuracy or computational restrictions. We propose a general approach for accurately and efficiently calculating small $p$-values for a broad range of complicated test statistics based on the principle of the cross-entropy method and Markov chain Monte Carlo sampling techniques.We evaluate the performance of the proposed algorithm through simulations and demonstrate its application to three real examples in genomic studies. The results show that our approach can accurately evaluate small to extremely small $p$-values (e.g. $10^{-6}$ to $10^{-100}$). The proposed algorithm is helpful to the improvement of existing test procedures and the development of new test procedures in genomic studies.