A Test Statistic for Weighted Runs

TL;DR

The paper introduces a new test statistic based on weighted runs to detect local deviations in data, complementing the chi-squared test by considering observation order, with detailed distribution derivation and an efficient p-value computation algorithm.

Contribution

It presents a novel weighted runs test statistic, derives its exact distribution, and develops an efficient algorithm for p-value calculation, enhancing local deviation detection.

Findings

The new test improves sensitivity to local deviations.

Exact distribution of the statistic is derived for non-parametric cases.

An efficient algorithm for p-value computation is proposed.

Abstract

A new test statistic based on success runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic $\chi^{2}$ test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute $p$-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.