Estimation of a Likelihood Ratio Ordered Family of Distributions

TL;DR

This work advances distribution estimation by imposing likelihood ratio order across a covariate in a regression setting and showing its equivalence to estimating a TP2 joint distribution under empirical likelihood. The authors develop a convex, likelihood-based framework with a dimension-reduced parameterization and a quasi-Newton–style algorithm built from row- and column-wise isotonic updates, ensuring descent toward a unique LR-ordered estimator. Empirical results on Gamma-model simulations and real growth data demonstrate that LR-order regularization yields smoother, more accurate conditional distributions and modest yet notable improvements in predictive performance compared to traditional stochastic-order isotonic regression. The approach provides a scalable, regularized alternative for distributional regression with potential applications in finance and risk assessment where LR-order constraints are natural. The accompanying R package LRDistReg implements these methods and enables practical application and evaluation.

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.

Figures (7)

• Figure 1: In this specific example, $n \ge 8$ raw observations yielded $\ell = 6$ different values $x_j$ and $m = 7$ different values $y_k$. The green dots represent those $(j,k)$ with $w_{jk} > 0$. The green dots and black circles represent the set ${\cal P}$.
• Figure 2: The true conditional Gamma distribution function $G_x$, the estimate under likelihood ratio (LR) order constraint $\widehat{G}_x$ and the estimated under usual stochastic (ST) order constraint $\widecheck{G}_x$ are displayed from left to right for $x\in\{1.5,2,2.5,3,3.5\}$.
• Figure 3: Selection of $\beta$-quantile curves. Specifically, a taut-string Duembgen_Kovac_2009 is computed between the lower $\mathfrak{X}\ni x\mapsto \min\{y\in\mathbb{R}: \tilde{G}_x(y)\ge \beta\}$ and upper $\mathfrak{X}\ni x\mapsto \inf\{y\in\mathbb{R}: \tilde{G}_x(y)> \beta\}$ quantile curves for each $\tilde{G}\in\{G,\widehat{G},\widecheck{G}\}$ (corresponding respectively to 'Truth', 'LR' and 'ST') and $\beta\in\{0.1,0.25,0.5,0.75,0.9\}$.
• Figure 4: Monte Carlo simulations to evaluate estimation performances with a simple score. First row: Simple scores with $\tilde{G}$ being either $\widehat{G}$ (solid line), $\widecheck{G}$ (dashed line) or $\widehat{\mathbb{G}}$ (dotted line). Second row: Relative change of score when enforcing a likelihood ratio order constraint over the usual stochastic order constraint. The thicker line is the median variation, whereas the thin lines are the first and third quartiles. Negative values represent an improvement in score.
• Figure 5: Monte Carlo simulations to evaluate prediction performances using a CRPS-type score. First row: CRPS scores with $\tilde{G}$ being either $\widehat{G}$ (solid line), $\widecheck{G}$ (dashed line) or $\widehat{\mathbb{G}}$ (doted line). Second row: Relative change of score when enforcing a likelihood ratio order constraint over the usual stochastic order constraint.

Theorems & Definitions (12)

• Lemma 3.1
• Theorem 3.2
• Lemma 3.3
• Theorem 3.4
• Lemma A.1
• proof : Proof
• proof : Proof of Lemma \ref{['Lem:Dim.reduction']}
• proof : Proof of Theorem \ref{['Thm:Existence.uniqueness']}
• proof : Proof of Lemma \ref{['Lem:Properties.Psi']}
• proof : Proof of Theorem \ref{['Thm:Convergence']}