# On approximation of the distribution for Pearson statistic

## TL;DR

This paper proposes approximating the distribution of the Pearson goodness-of-fit statistic using a Gamma distribution with parameters estimated from the first two moments, simplifying quantile calculations especially for small samples.

## Contribution

It introduces a novel method to approximate the Pearson statistic distribution with a Gamma distribution based on moment estimation, improving small-sample quantile calculations.

## Key findings

- Gamma approximation aligns well with empirical distributions
- Simplifies quantile computation for small samples
- Validated through simulation experiments

## Abstract

The paper considers the classical Goodness of Fit test. It suggests to use the Gamma distribution for the approximation of the distribution of the Pearson statistics with unknown parameters estimated from raw data. The parameters of these Gamma distribution can be estimated from the first two moments of the statistic after averaging over a distribution of the unknown parameter over its range. This allows to simplify calculation of the quantiles for the Pearson statistic, as is shown in some simulation experiments with medium and small sample sizes.