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

TL;DR

This paper introduces a novel method combining the cross-entropy approach and MCMC sampling to accurately estimate extremely small p-values in genomic data analysis, overcoming limitations of existing methods.

Contribution

It presents a general, efficient algorithm for estimating tiny p-values for complex test statistics where analytical solutions are unavailable.

Findings

Accurately estimates p-values as small as 10^{-100}

Demonstrates effectiveness through simulations and real genomic data

Improves existing statistical testing procedures in genomics

Abstract

$\textbf{Motivation:}$ 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 analytically 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 estimating 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. $\textbf{Results:}$ We evaluate the performance of the proposed algorithm through simulations and demonstrate its application to three real-world 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 for the improvement of some existing test procedures and the development of new test procedures in genomic studies.