Further Results on the Bivariate Semi-parametric Singular Family of Distributions

TL;DR

This paper explores a semi-parametric family of bivariate distributions, providing a characterization via functional equations, and investigates conditions for these solutions to form valid distributions, extending previous work on modeling bivariate data with ties.

Contribution

It introduces a new characterization of the semi-parametric bivariate distribution family through functional equations and analyzes conditions for valid distribution solutions.

Findings

Characterization of the distribution family via functional equations

Conditions for solutions to be valid bivariate distributions

Extension of previous models for data with ties

Abstract

General classes of bivariate distributions are well studied in literature. Most of these classes are proposed via a copula formulation or extensions of some characterisation properties in the univariate case. In Kundu(2022) we see one such semi-parametric family useful to model bivariate data with ties. This model is a general semi-parametric model with a baseline. In this paper we present a characterisation property of this class of distributions in terms of a functional equation. The general solution to this equation is explored. Necessary and sufficient conditions under which the solution becomes a bivariate distribution is investigated.