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Inference of Half Logistic Geometric Distribution Based on Generalized Order Statistics

Neetu Gupta, S. K. Neogy, Qazi J. Azhad, Bhagwati Devi

TL;DR

This paper studies the half-logistic-geometric (HLG) distribution within the generalized order statistics (gos) framework, deriving marginal and joint moment generating functions and associated moments, including recurrence relations. It then develops Bayesian estimation for the shape parameter $\theta$ using Lindley approximation and Metropolis-Hastings MCMC under symmetric and asymmetric losses, with practical guidance on order statistics as a submodel. Simulation shows Lindley-based estimators outperform MCMC in many settings, and GE loss often yields the best performance for certain loss scales. The methods are validated on demography and reliability data, demonstrating applicability to lifetime modeling with flexible ordered data.

Abstract

As the unification of various models of ordered quantities, generalized order statistics act as a simplistic approach introduced in \cite{kamps1995concept}. In this present study, results pertaining to the expressions of marginal and joint moment generating functions from half logistic geometric distribution are presented based on generalized order statistics framework. We also consider the estimation problem of $θ$ and provides a Bayesian framework. The two widely and popular methods called Markov chain Monte Carlo and Lindley approximations are used for obtaining the Bayes estimators.The results are derived under symmetric and asymmetric loss functions. Analysis of the special cases of generalized order statistics, \textit{i.e.,} order statistics is also presented. To have an insight into the practical applicability of the proposed results, two real data sets, one from the field of Demography and, other from reliability have been taken for analysis.

Inference of Half Logistic Geometric Distribution Based on Generalized Order Statistics

TL;DR

This paper studies the half-logistic-geometric (HLG) distribution within the generalized order statistics (gos) framework, deriving marginal and joint moment generating functions and associated moments, including recurrence relations. It then develops Bayesian estimation for the shape parameter using Lindley approximation and Metropolis-Hastings MCMC under symmetric and asymmetric losses, with practical guidance on order statistics as a submodel. Simulation shows Lindley-based estimators outperform MCMC in many settings, and GE loss often yields the best performance for certain loss scales. The methods are validated on demography and reliability data, demonstrating applicability to lifetime modeling with flexible ordered data.

Abstract

As the unification of various models of ordered quantities, generalized order statistics act as a simplistic approach introduced in \cite{kamps1995concept}. In this present study, results pertaining to the expressions of marginal and joint moment generating functions from half logistic geometric distribution are presented based on generalized order statistics framework. We also consider the estimation problem of and provides a Bayesian framework. The two widely and popular methods called Markov chain Monte Carlo and Lindley approximations are used for obtaining the Bayes estimators.The results are derived under symmetric and asymmetric loss functions. Analysis of the special cases of generalized order statistics, \textit{i.e.,} order statistics is also presented. To have an insight into the practical applicability of the proposed results, two real data sets, one from the field of Demography and, other from reliability have been taken for analysis.

Paper Structure

This paper contains 9 sections, 4 theorems, 67 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Marginal moments
  3. Joint Moments
  4. Preliminary setup for Bayesian estimation
  5. Lindley's Approximation
  6. Markov Chain Monte Carlo
  7. Simulation Study
  8. Data Analysis
  9. Conclusion

Key Result

Theorem 2.1

For a continuous non-negative random variable X distributed as (1.8), with real $m,k$, $m\ge-1$, $k\ge1$

Figures (3)

  • Figure 1: Trace plots of $\theta$
  • Figure 2: Fitting of Mortality Data with HLG Distribution
  • Figure 3: Fitting of age for traction motors data with HLG Distribution

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof