  $\textbf{Motivation:}$ 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.
