Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane’s Air Conditioning System and Precipitation Data

Tiago M Magalhães,Diego I Gallardo,Osvaldo Venegas,Yolanda M Gómez

doi:10.3390/sym12050851

Tiago M Magalhães, Diego I Gallardo + Show 2 more

Open Access

https://doi.org/10.3390/sym12050851

Copy DOI

Abstract

The Marshall-Olkin extended family of distributions is an alternative for modeling lifetimes, and considers more or less asymmetry than its parent model, achieved by incorporating just one extra parameter. We investigate the bias of maximum likelihood estimators and use it to develop an estimator with less bias than traditional estimators, by a modification of the score function. Unlike other proposals, in this paper, we consider a bias reduction methodology that can be applied to any member of the family and not necessarily to any particular distribution. We conduct a Monte Carlo simulation in order to study the performance of the corrected estimators in finite samples. This simulation shows that the maximum likelihood estimator is quite biased and the proposed estimator is much less biased; in small sample sizes, the bias is reduced by around 50 percent. Two applications, related to the air conditioning system of an airplane and precipitations, are presented to illustrate the results. In those applications, the bias reduction for the shape parameters is close to 25% and the bias reduction also reduces, among others things, the width of the 95% confidence intervals for quantiles lower than 0.594.

Highlights

IntroductionMarshall and Olkin [1] proposed a way of introducing a parameter in a family of distributions to compete with such commonly used distributions as the Weibull, gamma and lognormal distributions
Marshall and Olkin [1] proposed a way of introducing a parameter in a family of distributions to compete with such commonly used distributions as the Weibull, gamma and lognormal distributions.Let G and g be the cumulative distribution function and the probability density function respectively, indexed by the vector of parameter λ ∈ Λ
We present a simulation study to illustrate the performance of our methodology compared with the traditional maximum likelihood estimators (MLE) in the Marshall-Olkin extended exponential (MOEE) and Marshall-Olkin extended Rayleigh (MOER) models

Summary

Introduction

Marshall and Olkin [1] proposed a way of introducing a parameter in a family of distributions to compete with such commonly used distributions as the Weibull, gamma and lognormal distributions. Let G and g be the cumulative distribution function (cdf) and the probability density function (pdf) respectively, indexed by the vector of parameter λ ∈ Λ. The pdf of the Marshall-Olkin extended model is given by f ( x; α, λ) = αg( x; λ). Olkin [2] called α the “tilt” parameter because the hazard rate of MOEg (α, λ) is shifted below (α ≥ 1). Castellares and Lemonte [3] showed an interpretation of MOEg (α, λ) based on the distribution of order statistics.

Objectives

Results

Conclusion