Testing equality in distribution of random convex compact sets via theory of N-distances and its application to assessing similarity of general random sets

TL;DR

This paper introduces a new statistical test based on N-distances and characteristic functions to determine if two random convex compact sets or general random sets share the same distribution, supported by simulation results.

Contribution

It develops a novel metric-based testing method for distribution equality of random convex and general sets, including finite-dimensional approximations and practical application procedures.

Findings

The proposed test effectively distinguishes between different distributions.

Finite-dimensional approximations enable practical implementation.

Simulation studies validate the test's accuracy and robustness.

Abstract

This paper concerns a method of testing equality of distribution of random convex compact sets and the way how to use the test to distinguish between two realisations of general random sets. The family of metrics on the space of distributions of random convex compact sets is constructed using the theory of N-distances and characteristic functions of random convex compact sets. Further, the approximation of the metrics through its finite dimensional counterparts is proposed, which lead to a new statistical test for testing equality in distribution of two random convex compact sets. Then, it is described how to approximate a realisation of a general random set by a union of convex compact sets, and it is shown how to determine whether two realisations of general random sets come from the same process using the constructed test. The procedure is justified by an extensive simulation study.