[Paper Review] Bivariate Discrete Inverse Weibull Distribution

Mohamed S. Eliwa, Mahmoud El-Morshedy|arXiv (Cornell University)|Aug 22, 2018

Statistical Distribution Estimation and Applications28 references3 citations

TL;DR

This paper proposes a new bivariate discrete inverse Weibull (BDsIW) distribution with discrete inverse Weibull marginals, enabling flexible modeling of bivariate discrete lifetime data. Using maximum likelihood estimation, the model demonstrates superior fit to two real datasets—football injury data and nasal drainage severity scores—outperforming competing distributions in AIC, BIC, and likelihood ratio tests.

ABSTRACT

In this paper, we propose a new class of bivariate distributions, called the bivariate discrete inverse Weibull (BDsIW) distribution, whose marginals are discrete inverse Weibull (DsIW) distributions. Some statistical and mathematical properties are presented. The maximum likelihood method is used for estimating the model parameters. Simulations are presented to verify the performance of the direct maximum likelihood estimation. Finally, two real data sets are analyzed for illustrative purposes.

Motivation & Objective

• To develop a flexible bivariate discrete distribution suitable for modeling paired discrete lifetime data in reliability and survival analysis.
• To extend the discrete inverse Weibull (DsIW) distribution to a bivariate framework while preserving its marginal properties.
• To evaluate the performance of maximum likelihood estimation (MLE) for the new model through simulation studies.
• To compare the BDsIW model with existing bivariate discrete distributions using real-world datasets.
• To validate the model’s superiority through goodness-of-fit criteria and likelihood ratio tests on real data.

Proposed method

• Proposes a bivariate discrete inverse Weibull (BDsIW) distribution with joint cumulative distribution function (CDF) and probability mass function (PMF) derived from the survival copula structure.
• Defines the joint PMF using the survival function of the bivariate model, ensuring that the marginals are discrete inverse Weibull (DsIW) distributions.
• Employs maximum likelihood estimation (MLE) to estimate the parameters: θ₁, θ₂, θ₃, and ζ, with iterative numerical optimization.
• Conducts simulation studies to assess the accuracy and consistency of MLE estimates under varying sample sizes and parameter settings.
• Applies information criteria (AIC, BIC, CAIC, HQIC) and likelihood ratio tests (LRT) to compare model fit across competing models.
• Validates the model on two real datasets: football injury data and nasal drainage severity scores, using marginal and joint distribution fits.

Experimental results

Research questions

• RQ1 Can a bivariate discrete distribution be constructed such that its marginals are discrete inverse Weibull (DsIW) distributions?
• RQ2 How well does the proposed BDsIW model fit real bivariate discrete lifetime data compared to existing models?
• RQ3 Does the maximum likelihood estimation method provide reliable and consistent parameter estimates for the BDsIW model?
• RQ4 Are the special cases of BDsIW (BDsIE and BDsIR) statistically inadequate for the data, as indicated by likelihood ratio tests?
• RQ5 What is the relative performance of BDsIW in terms of information criteria and goodness-of-fit across different datasets?

Key findings

• The BDsIW distribution provides the best fit among competing models for the football injury data, with the lowest -L (61.96), AIC (131.82), CAIC (133.82), BIC (136.95), and HQIC (133.37) values.
• The likelihood ratio test strongly rejects the null hypotheses H₀₁: ζ=1 (BDsIE) and H₀₂: ζ=2 (BDsIR), with p-values < 0.01 and 0.0384 respectively, indicating BDsIW is significantly better.
• For the nasal drainage severity score data, BDsIW again outperforms competitors, with the lowest -L (56.72), AIC (121.44), and other information criteria values.
• The likelihood ratio test for this dataset also rejects both BDsIE and BDsIR models (p-values < 0.01 and 0.0381), confirming BDsIW as the preferred model.
• Simulation results confirm that MLE provides consistent and accurate parameter estimates for the BDsIW model across various sample sizes and parameter settings.
• The joint PMF plots visually support the superior fit of BDsIW over BDsIE and BDsIR in both real data applications.