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Estimating ordered variance of two scale mixture of normal distributions

Shrajal Bajpai, Lakshmi Kanta Patra

TL;DR

The paper addresses estimating ordered variances in scale mixtures of normal distributions under $L_1$ (squared error) and $L_2$ (entropy) losses. It derives the BAEE for each variance and constructs improved, shrinkage-type estimators based on ratio statistics such as $Z_1=S_2/S_1$ and higher-order ratios, proving domination over BAEE in many settings with explicit piecewise forms. The authors extend the approach to a broader class of estimators, provide risk-dominance theorems, and apply the results to the multivariate $t$ distribution, including a detailed simulation study that confirms theoretical risk improvements. These results offer more accurate, order-constrained variance estimates in scale-mixture models, with practical implications for MV-normal and MV$t$ applications.

Abstract

This study investigates component wise estimation of ordered variances of scale mixture of two normal distributions. For this study two special loss functions are considered namely squared error loss function and entropy loss function. We have derived the general improvement results and based on these results the estimators that outperform BAEE are obtained. Moreover under certain sufficient conditions a class of improved estimators is proposed for both loss functions. As a special case of scale mixture of normal distribution the results are applied to the multivariate t-distribution and obtained the improvement results. For this case a detailed numerical comparison is carried out which validates our theoretical findings.

Estimating ordered variance of two scale mixture of normal distributions

TL;DR

The paper addresses estimating ordered variances in scale mixtures of normal distributions under (squared error) and (entropy) losses. It derives the BAEE for each variance and constructs improved, shrinkage-type estimators based on ratio statistics such as and higher-order ratios, proving domination over BAEE in many settings with explicit piecewise forms. The authors extend the approach to a broader class of estimators, provide risk-dominance theorems, and apply the results to the multivariate distribution, including a detailed simulation study that confirms theoretical risk improvements. These results offer more accurate, order-constrained variance estimates in scale-mixture models, with practical implications for MV-normal and MV applications.

Abstract

This study investigates component wise estimation of ordered variances of scale mixture of two normal distributions. For this study two special loss functions are considered namely squared error loss function and entropy loss function. We have derived the general improvement results and based on these results the estimators that outperform BAEE are obtained. Moreover under certain sufficient conditions a class of improved estimators is proposed for both loss functions. As a special case of scale mixture of normal distribution the results are applied to the multivariate t-distribution and obtained the improvement results. For this case a detailed numerical comparison is carried out which validates our theoretical findings.
Paper Structure (8 sections, 25 theorems, 50 equations, 4 figures, 2 tables)

This paper contains 8 sections, 25 theorems, 50 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Estimation of $\sigma_1^2$
  3. A class of improved estimators
  4. Estimation of $\sigma_2^2$
  5. A class of improved estimator
  6. Application to the multivariate $t$ distribution
  7. Simulation study
  8. Conclusions

Key Result

Lemma 1.1

Figures (4)

  • Figure 1: Percentage risk improvement of $d_{12}^Q$ under squared error loss $L_1(t)$ for $\sigma_{1}^2$
  • Figure 2: Percentage risk improvement of $d_{12}^Q$ under squared error loss $L_1(t)$ for $\sigma_{1}^2$
  • Figure 3: Percentage risk improvement of $d_{12}^E$ under entropy loss $L_2(t)$ for $\sigma_{1}^2$
  • Figure 4: Percentage risk improvement of $d_{12}^E$ under entropy loss $L_2(t)$ for $\sigma_{1}^2$

Theorems & Definitions (25)

  • Lemma 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 3.1
  • ...and 15 more