A generalised matching distribution for the problem of coincidences

TL;DR

This paper introduces a generalized matching distribution that accounts for a preliminary allocation phase, combining classical matching with random sampling, and explores its properties and applications in matching tests.

Contribution

It extends the classical matching distribution by incorporating a preliminary allocation step, providing a new distribution for improved matching analysis.

Findings

The generalized distribution is a convolution of classical matching and binomial distributions.

Methods for computing the probability functions of the new distribution are provided.

Applications include matching tests and inference of matching ability.

Abstract

This paper examines the classical matching distribution arising in the "problem of coincidences". We generalise the classical matching distribution with a preliminary round of allocation where items are correctly matched with some fixed probability, and remaining non-matched items are allocated using simple random sampling without replacement. Our generalised matching distribution is a convolution of the classical matching distribution and the binomial distribution. We examine the properties of this latter distribution and show how its probability functions can be computes. We also show how to use the distribution for matching tests and inferences of matching ability.