Characterization of Order Statistics in Two Runs Using Conditional Expectation

TL;DR

This paper analyzes the order statistics of two runs-based test statistics, deriving their probability distributions, means, and variances using conditional expectation, to enhance understanding of runs tests in statistical independence testing.

Contribution

It introduces a detailed characterization of order statistics in two runs tests, providing explicit probability mass functions and moments, which were not previously available.

Findings

Derived probability mass functions for order statistics

Calculated means and variances using conditional expectation

Enhanced understanding of runs test statistics

Abstract

The runs test is a well-known test that is used for checking independence between elements of a sample data sequence. Some of runs tests are based on the longest run and others based on the total runs. In this paper, we consider order statistics of two runs statistics, and obtain their probability mass functions. In addition, the means and variances of the order statistics are derived using traditional conditional expectation.