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Stochastic comparisons of finite mixtures with general exponentiated location-scale distributed components

Raju Bhakta, Kaushik Gupta, Ghobad Saadat Kia, Suchandan Kayal

TL;DR

The paper studies stochastic-order relationships between finite mixtures whose subpopulations follow the Exponentiated Location-Scale (ELS) family, distinguishing single-outlier and multiple-outlier configurations. It develops a set of sufficient conditions for usual stochastic order $≤_{st}$, reversed hazard rate order $≤_{rh}$, and likelihood ratio order $≤_{lr}$ (and AFO for the reversed hazard rate) using majorization and Schur-convexity arguments, with proofs tailored to both common and non-common mixing weights. The main contributions include Theorems 3.1–3.4 for single-outlier FMMs and Theorems 4.1–4.3 for multi-outlier FMMs, complemented by numerical illustrations and counterexamples that delineate the necessity of assumptions. These results extend stochastic-order theory to flexible ELS-based mixtures and have practical implications for reliability, survival analysis, and risk modeling where heterogeneous subpopulations are present. Overall, the work provides rigorous tools for comparing heterogeneous systems modeled by FMMs with ELS components, facilitating model selection and inference in applications with skewed, heavy-tailed, or multimodal lifetimes and risks.

Abstract

In this paper, we study stochastic ordering results between two finite mixtures with single and multiple outliers, assuming subpopulations follow general exponentiated location-scale distributions. For single-outlier mixtures, several sufficient conditions are derived under which the mixture variables are ordered in the usual stochastic, reversed hazard rate, and likelihood ratio orders, using majorization concepts. For multiple-outlier mixtures, results are obtained for the reversed hazard rate, likelihood ratio, and ageing faster orders in reversed hazard rate. Numerical examples and counterexamples are presented to illustrate and support the established theoretical findings.

Stochastic comparisons of finite mixtures with general exponentiated location-scale distributed components

TL;DR

The paper studies stochastic-order relationships between finite mixtures whose subpopulations follow the Exponentiated Location-Scale (ELS) family, distinguishing single-outlier and multiple-outlier configurations. It develops a set of sufficient conditions for usual stochastic order , reversed hazard rate order , and likelihood ratio order (and AFO for the reversed hazard rate) using majorization and Schur-convexity arguments, with proofs tailored to both common and non-common mixing weights. The main contributions include Theorems 3.1–3.4 for single-outlier FMMs and Theorems 4.1–4.3 for multi-outlier FMMs, complemented by numerical illustrations and counterexamples that delineate the necessity of assumptions. These results extend stochastic-order theory to flexible ELS-based mixtures and have practical implications for reliability, survival analysis, and risk modeling where heterogeneous subpopulations are present. Overall, the work provides rigorous tools for comparing heterogeneous systems modeled by FMMs with ELS components, facilitating model selection and inference in applications with skewed, heavy-tailed, or multimodal lifetimes and risks.

Abstract

In this paper, we study stochastic ordering results between two finite mixtures with single and multiple outliers, assuming subpopulations follow general exponentiated location-scale distributions. For single-outlier mixtures, several sufficient conditions are derived under which the mixture variables are ordered in the usual stochastic, reversed hazard rate, and likelihood ratio orders, using majorization concepts. For multiple-outlier mixtures, results are obtained for the reversed hazard rate, likelihood ratio, and ageing faster orders in reversed hazard rate. Numerical examples and counterexamples are presented to illustrate and support the established theoretical findings.

Paper Structure

This paper contains 6 sections, 10 theorems, 48 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic comparisons between two single-outlier FMMs
  4. Stochastic comparisons between two multiple-outlier FMMs
  5. Numerical examples and counterexamples
  6. Concluding remarks

Key Result

Lemma 2.1

(marshall2011inequalities, Theorem $A.3$, p. $83$) Consider the function $\zeta:\mathcal{D}_n\rightarrow\mathbb{R}$ is continuous on $\mathcal{D}_n$ and continuously differentiable on the interior of $\mathcal{D}_n$. Then, $\zeta$ is Schur-convex (Schur-concave) on $\mathcal{D}_n$, if and only if $\

Figures (8)

  • Figure 1: $(a)$ Graph of the SFs of $U_3(\boldsymbol{r};\boldsymbol{\alpha},\boldsymbol{\sigma},\boldsymbol{\lambda})$ (magenta curve) and $U_3(\boldsymbol{r};\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\theta})$ (cyan curve) in Example \ref{['example4.1']}. $(b)$ Representation of the SFs of $U_3(\boldsymbol{r};\boldsymbol{\alpha},\boldsymbol{\sigma},\boldsymbol{\lambda})$ (green curve) and $U_3(\boldsymbol{r};\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\theta})$ (orange curve) in Counterexample \ref{['Coun4.1']}.
  • Figure 2: $(a)$ Graph of the ratio between the CDFs of $U_3(\boldsymbol{r};\boldsymbol{\alpha},\boldsymbol{\sigma},\boldsymbol{\lambda})$ and $U_3(\boldsymbol{r};\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\theta})$ in Counterexample \ref{['Coun4.2']}. $(b)$ Graph of the ratio between the PDFs of $U_3(\boldsymbol{r};\boldsymbol{\alpha},\boldsymbol{\sigma},\boldsymbol{\lambda})$ and $U_3(\boldsymbol{r};\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\theta})$ in Counterexample \ref{['Coun4.22']}.
  • Figure 3: $(a)$ Plot of the ratio between the CDFs of $U_3(\boldsymbol{s};\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\theta})$ and $U_3(\boldsymbol{r};\boldsymbol{\alpha},\boldsymbol{\sigma},\boldsymbol{\lambda})$ in Example \ref{['example4.2']}. $(b)$ Plot of the ratio between the CDFs of $U_3(\boldsymbol{s};\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\theta})$ and $U_3(\boldsymbol{r};\boldsymbol{\alpha},\boldsymbol{\sigma},\boldsymbol{\lambda})$ in Counterexample \ref{['Coun4.3']}.
  • Figure 4: $(a)$ Graph of the ratio between the PDFs of $U_3(\boldsymbol{s};\boldsymbol{\beta},\sigma,\lambda)$ and $U_3(\boldsymbol{r};\boldsymbol{\alpha},\sigma,\lambda)$ in Example \ref{['example4.3']}. $(b)$ Graph of the ratio between the PDFs of $U_3(\boldsymbol{s};\boldsymbol{\beta},\sigma,\lambda)$ and $U_3(\boldsymbol{r};\boldsymbol{\alpha},\sigma,\lambda)$ in Counterexample \ref{['Coun4.4']}.
  • Figure 5: $(a)$ Plots of the CDFs of $U_3(\boldsymbol{r};\boldsymbol{\alpha},\sigma,\lambda)$ (red curve) and $U_3(\boldsymbol{s};\boldsymbol{\beta},\mu,\theta)$ (green curve) in Example \ref{['example4.4']}. $(b)$ Plots of the CDFs of $U_3(\boldsymbol{r};\boldsymbol{\alpha},\sigma,\lambda)$ (pink curve) and $U_3(\boldsymbol{s};\boldsymbol{\beta},\mu,\theta)$ (cyan curve) in Counterexample \ref{['Coun5.6']}.
  • ...and 3 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • ...and 22 more