# Implementing Monte Carlo Tests with P-value Buckets

arXiv: https://arxiv.org/abs/1703.09305 · HTML: https://mlsanity.com/paper/1703.09305/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2017.

## Built on by (most cited first, 2 of 2)

- [seMCD: Sequentially implemented Monte Carlo depth computation with statistical guarantees](https://mlsanity.com/paper/2507.06227/) (2025)
- [A Family of Divergence Measures for Evaluating the Reconstruction Quality of Explainable Ensemble Trees](https://mlsanity.com/paper/2605.19618/) (2026)

## Abstract (arXiv)

Software packages usually report the results of statistical tests using p-values. Users often interpret these by comparing them to standard thresholds, e.g. 0.1%, 1% and 5%, which is sometimes reinforced by a star rating (***, **, *). We consider an arbitrary statistical test whose p-value p is not available explicitly, but can be approximated by Monte Carlo samples, e.g. by bootstrap or permutation tests. The standard implementation of such tests usually draws a fixed number of samples to approximate p. However, the probability that the exact and the approximated p-value lie on different sides of a threshold (the resampling risk) can be high, particularly for p-values close to a threshold. We present a method to overcome this. We consider a finite set of user-specified intervals which cover [0,1] and which can be overlapping. We call these p-value buckets. We present algorithms that, with arbitrarily high probability, return a p-value bucket containing p. We prove that for both a bounded resampling risk and a finite runtime, overlapping buckets need to be employed, and that our methods both bound the resampling risk and guarantee a finite runtime for such overlapping buckets. To interpret decisions with overlapping buckets, we propose an extension of the star rating system. We demonstrate that our methods are suitable for use in standard software, including for low p-value thresholds occurring in multiple testing settings, and that they can be computationally more efficient than standard implementations.