Estimation of a Likelihood Ratio Ordered Family of Distributions

TL;DR

This paper introduces an empirical likelihood-based method to estimate a family of distributions under likelihood ratio order constraints, improving estimation and prediction in bivariate data analysis.

Contribution

It develops a novel algorithm for estimating conditional distributions under likelihood ratio order, offering a stronger regularization than stochastic order.

Findings

Enhanced estimation accuracy demonstrated on simulated data

Improved predictive performance over traditional methods

Applicable to real-world bivariate datasets

Abstract

Consider bivariate observations $(X_1,Y_1), \ldots, (X_n,Y_n) \in \mathbb{R}\times \mathbb{R}$ with unknown conditional distributions $Q_x$ of $Y$, given that $X = x$. The goal is to estimate these distributions under the sole assumption that $Q_x$ is isotonic in $x$ with respect to likelihood ratio order. If the observations are identically distributed, a related goal is to estimate the joint distribution $\mathcal{L}(X,Y)$ under the sole assumption that it is totally positive of order two in a certain sense. An algorithm is developed which estimates the unknown family of distributions $(Q_x)_x$ via empirical likelihood. The benefit of the stronger regularization imposed by likelihood ratio order over the usual stochastic order is evaluated in terms of estimation and predictive performances on simulated as well as real data.