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Stochastic orderings between two finite mixture models with inverted-Kumaraswamy distributed components

Raju Bhakta, Pradip Kundu, Suchandan Kayal, Morad Alizadeh

Abstract

In this paper, we consider two finite mixture models (FMMs), with inverted-Kumaraswamy distributed components' lifetimes. Several stochastic ordering results between the FMMs have been obtained. Mainly, we focus on three different cases in terms of the heterogeneity of parameters. The usual stochastic order between the FMMs have been established when heterogeneity presents in one parameter as well as two parameters. In addition, we have also studied ageing faster order in terms of the reversed hazard rate between two FMMs when heterogeneity is in two parameters. For the case of heterogeneity in three parameters, we obtain the comparison results based on reversed hazard rate and likelihood ratio orders. The theoretical developments have been illustrated using several examples and counterexamples.

Stochastic orderings between two finite mixture models with inverted-Kumaraswamy distributed components

Abstract

In this paper, we consider two finite mixture models (FMMs), with inverted-Kumaraswamy distributed components' lifetimes. Several stochastic ordering results between the FMMs have been obtained. Mainly, we focus on three different cases in terms of the heterogeneity of parameters. The usual stochastic order between the FMMs have been established when heterogeneity presents in one parameter as well as two parameters. In addition, we have also studied ageing faster order in terms of the reversed hazard rate between two FMMs when heterogeneity is in two parameters. For the case of heterogeneity in three parameters, we obtain the comparison results based on reversed hazard rate and likelihood ratio orders. The theoretical developments have been illustrated using several examples and counterexamples.
Paper Structure (7 sections, 19 theorems, 53 equations, 7 figures)

This paper contains 7 sections, 19 theorems, 53 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The basic definitions and some prerequisites
  3. Model description and results
  4. Ordering results for MMs when heterogeneity presents in one parameter
  5. Ordering results for MMs when heterogeneity presents in two parameters
  6. Ordering results for MMs when heterogeneity presents in three parameters
  7. Concluding remarks

Key Result

Lemma 2.1

A differentiable function $\Upsilon:\mathbb{R}_4^+\rightarrow\mathbb{R}^+$ satisfies $\Upsilon(P)\geq\Upsilon(Q)$ for all $P$, $Q$ such that $P\in\mathcal{L}_2$, and $P\gg Q$ if and only if

Figures (7)

  • Figure 1: (a) Plots of the sfs of $R_3(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p^*})$ (blue curve) and $R_3(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p})$ (red curve) in Example \ref{['example3.1']}. (b) Plot of the difference between the sfs of $R_3(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p^*})$ and $R_3(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p})$ in Counterexample \ref{['counterexample3.1']}.
  • Figure 2: (a) Plots of the sfs of $R_3(\boldsymbol{X}_{\alpha,\boldsymbol{\beta}};\boldsymbol{p^*})$ (purple curve) and $R_3(\boldsymbol{X}_{\alpha,\boldsymbol{\beta}};\boldsymbol{p})$ (orange curve) in Example \ref{['example3.2']}. (b) Plots of the sfs of $R_3(\boldsymbol{X}_{\boldsymbol{\alpha^*},\beta};\boldsymbol{p})$ (green curve) and $R_3(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p})$ (pink curve) in Example \ref{['example3.3']}.
  • Figure 3: (a) Plot of the difference between the sfs of $R_3(\boldsymbol{X}_{\boldsymbol{\alpha^*},\beta};\boldsymbol{p})$ and $R_3(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p})$ in Counterexample \ref{['counterexample3.3']}. (b) Plots of the sfs of $R_2(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p})$ (brown curve) and $R_2(\boldsymbol{X}_{\boldsymbol{\alpha^*},\beta};\boldsymbol{p^*})$ (cyan curve) in Example \ref{['example3.4']}.
  • Figure 4: (a) Plot of the difference between the sfs of $R_2(\boldsymbol{X}_{\boldsymbol{\alpha},\beta};\boldsymbol{p})$ and $R_2(\boldsymbol{X}_{\boldsymbol{\alpha^*},\beta};\boldsymbol{p^*})$ in Counterexample \ref{['counterexample3.4']}. (b) Plots of the sfs of $R_2(\boldsymbol{X}_{\alpha,\boldsymbol{\beta}};\boldsymbol{p})$ (purple curve) and $R_2(\boldsymbol{X}_{\alpha,\boldsymbol{\beta^*}};\boldsymbol{p^*})$ (black curve) in Example \ref{['example3.5']}.
  • Figure 5: (a) Plot of the difference between the sfs of $R_2(\boldsymbol{X}_{\alpha,\boldsymbol{\beta}};\boldsymbol{p})$ and $R_2(\boldsymbol{X}_{\alpha,\boldsymbol{\beta^*}};\boldsymbol{p^*})$ in Counterexample \ref{['counterexample3.5']}. (b) Plot of the ratio of the rhs of $R_3(\boldsymbol{X}_{\alpha,\boldsymbol{\beta}};\boldsymbol{p})$ and $R_3(\boldsymbol{X}_{\alpha,\boldsymbol{\beta^*}};\boldsymbol{p^*})$ in Example \ref{['example3.6']}.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • ...and 43 more