[Paper Review] Bivariate Exponentiated Generalized Linear Exponential Distribution with Applications in Reliability Analysis

Mohamed Ibrahim, Mohamed S. Eliwa+1 more|arXiv (Cornell University)|Oct 2, 2017

Statistical Distribution Estimation and Applications3 citations

TL;DR

This paper proposes a bivariate exponentiated generalized linear exponential distribution (BEGLED) based on the Marshall-Olkin shock model, enabling flexible modeling of positively dependent lifetimes in reliability analysis. The distribution provides a better fit than existing bivariate models for real data, with closed-form joint and marginal distributions, and is validated through simulation and real data analysis.

ABSTRACT

The aim of this paper, is to define a bivariate exponentiated generalized linear exponential distribution based on Marshall-Olkin shock model. Statistical and reliability properties of this distribution are discussed. This includes quantiles, moments, stress-strength reliability, joint reliability function, joint reversed (hazard) rates functions and joint mean waiting time function. Moreover, the hazard rate, the availability and the mean residual lifetime functions for a parallel system, are established. One data set is analyzed, and it is observed that, the proposed distribution provides a better fit than Marshall-Olkin bivariate exponential, bivariate generalized exponential and bivariate generalized linear failure rate distributions. Simulation studies are presented to estimate both the relative absolute bias, and the relative mean square error for the distribution parameters based on complete data.

Motivation & Objective

• To develop a flexible bivariate distribution for modeling dependent lifetime data in reliability engineering.
• To extend the univariate exponentiated generalized linear exponential distribution (EGLED) into a bivariate form using the Marshall-Olkin shock model.
• To derive and analyze key statistical and reliability properties, including joint density, hazard rates, and mean waiting time functions.
• To evaluate the performance of maximum likelihood estimators through simulation studies and real data fitting.
• To demonstrate superior fit of the BEGLED compared to Marshall-Olkin bivariate exponential, bivariate generalized exponential, and bivariate generalized linear failure rate distributions.

Proposed method

• Construct the bivariate distribution using three independent EGLED-distributed random variables $U_1, U_2, U_3$, with $X_k = ext{max}(U_k, U_3)$ for $k=1,2$, based on the Marshall-Olkin shock model.
• Derive the joint cumulative distribution function (CDF) and probability density function (PDF) in piecewise form for $x_1 < x_2$, $x_2 < x_1$, and $x_1 = x_2$.
• Express the marginal distributions of $X_1$ and $X_2$ as EGLED with parameters $ heta_i + heta_3$, ensuring the model's flexibility.
• Establish key reliability functions: joint reversed (hazard) rates, joint mean waiting time, stress-strength reliability, and system availability and mean residual lifetime for parallel systems.
• Use maximum likelihood estimation (MLE) to estimate the six parameters: $ heta_1, heta_2, heta_3, heta, heta, heta$ (corrected to $ heta_1, heta_2, heta_3, heta, heta$), with $ heta$ representing shape and scale parameters.
• Conduct simulation studies to compute relative absolute bias and relative mean square error (MSE) for parameter estimates across varying sample sizes ($n = 30, 50, 100, 200$).

Experimental results

Research questions

• RQ1 Can a bivariate extension of the exponentiated generalized linear exponential distribution be constructed using the Marshall-Olkin shock model to model dependent lifetimes?
• RQ2 How do the joint and marginal reliability functions, such as joint hazard rates and mean waiting time, behave in the proposed BEGLED?
• RQ3 Does the BEGLED provide a better statistical fit than existing bivariate models like Marshall-Olkin bivariate exponential, bivariate generalized exponential, and bivariate generalized linear failure rate distributions?
• RQ4 How do the bias and mean square error of MLEs for the six parameters of BEGLED change with increasing sample size?
• RQ5 What is the performance of the BEGLED in fitting real multivariate lifetime data compared to competing models?

Key findings

• The BEGLED provides a superior fit to a real data set compared to the Marshall-Olkin bivariate exponential, bivariate generalized exponential, and bivariate generalized linear failure rate distributions.
• The relative absolute bias and relative mean square error (MSE) of the MLEs for all parameters decrease as the sample size increases, with bias reducing from 0.146 at $n=30$ to 0.023 at $n=200$ for $ heta_1=0.5$.
• For $ heta_1=0.5$, the relative MSE drops from 0.099 at $n=30$ to 0.012 at $n=200$, indicating improved estimation precision with larger samples.
• The 95% confidence intervals for the parameters narrow with increasing sample size, e.g., for $ heta_1=0.5$, the interval shrinks from (0.2091, 1.3758) at $n=30$ to (0.2977, 0.9242) at $n=100$, showing improved estimation accuracy.
• The joint CDF and PDF are available in closed form, enabling practical implementation in reliability modeling for positively correlated non-negative random variables.
• The median correlation coefficient $M_{X_1,X_2}$ is derived and can be used to generate bivariate data, supporting simulation and model validation.