The Distribution of the Domination Number of Class Cover Catch Digraphs for Non-uniform One-dimensional Data

TL;DR

This paper studies the distribution of the domination number in class cover catch digraphs for non-uniform one-dimensional data, extending previous uniform case results to more general distributions and multiple class points.

Contribution

It derives the distribution of the domination number for CCCDs with non-uniform data and multiple class points, broadening the understanding beyond the uniform case.

Findings

Derived the distribution for non-uniform data on an interval.

Extended calculations to multiple class points.

Provided finite sample distribution formulas.

Abstract

For two or more classes of points in $\R^d$ with $d \ge 1$, the class cover catch digraphs (CCCDs) can be constructed using the relative positions of the points from one class with respect to the points from the other class. The CCCDs were introduced by (Priebe, DeVinney, and Mar-chette, (2001). On the distribution of the domination number of random class catch cover di-graphs. Statistics and Probability Letters, 55:239-246) who investigated the case of two classes, $\X$ and $\Y$. They calculated the exact (finite sample) distribution of the domination number of the CCCDs based on $\X$ points relative to $\Y$ points both of which were uniformly distri-buted on a bounded interval. We investigate the distribution of the domination number of the CCCDs based on data from non-uniform $\X$ points on an interval with end points from $\Y$. Then we extend these calculations for multiple $\Y$ points on bounded intervals.