[Paper Review] Generalized inverse xgamma distribution: A non-monotone hazard rate model

Harsh Tripathi, Abhimanyu Singh Yadav+2 more|arXiv (Cornell University)|Nov 3, 2018

Statistical Distribution Estimation and Applications10 references4 citations

TL;DR

This paper proposes the generalized inverse xgamma distribution (GIXGD), a flexible two-parameter lifetime model that exhibits non-monotone hazard rates, derived via power transformation of the inverse xgamma distribution. The model is applied to guinea pig survival data, demonstrating superior fit over competing distributions using AIC, BIC, and Kolmogorov-Smirnov statistics, with maximum likelihood estimation used for parameter and reliability function inference.

ABSTRACT

In this article, a generalized inverse xgamma distribution (GIXGD) has been introduced as the generalized version of the inverse xgamma distribution. The proposed model exhibits the pattern of non-monotone hazard rate and belongs to family of positively skewed models. The explicit expressions of some distributional properties, such as, moments, inverse moments, conditional moments, mean deviation, quantile function have been derived. The maximum likelihood estimation procedure has been used to estimate the unknown model parameters as well as survival characteristics of GIXGD. The practical applicability of the proposed model has been illustrated through a survival data of guinea pigs.

Motivation & Objective

• To develop a more flexible lifetime distribution by generalizing the inverse xgamma distribution using a power transformation.
• To model non-monotone hazard rates commonly observed in reliability and clinical studies.
• To derive explicit expressions for key statistical properties such as moments, quantile functions, and reliability characteristics.
• To evaluate the model's performance through real data fitting and comparison with existing lifetime distributions.
• To provide a robust alternative to one- and two-parameter inverse family distributions in survival analysis.

Proposed method

• Propose the generalized inverse xgamma distribution (GIXGD) via power transformation $ Y = X^{1/α} $, where $ X \sim \text{IXGD}(\theta) $, introducing shape parameter $ \alpha $.
• Derive the probability density function (PDF) and cumulative distribution function (CDF) of GIXGD, showing that it reduces to IXGD when $ \alpha = 1 $.
• Define the survival function $ S(y;\alpha,\theta) = 1 - F(y;\alpha,\theta) $ and hazard rate function $ H(y;\alpha,\theta) = f(y;\alpha,\theta)/S(y;\alpha,\theta) $.
• Derive analytical expressions for moments, inverse moments, conditional moments, mean deviation, and quantile function.
• Use maximum likelihood estimation (MLE) to estimate parameters $ \alpha $ and $ \theta $, and apply invariance property to estimate survival and hazard functions.
• Validate model performance using Kolmogorov-Smirnov (K-S) test and information criteria (AIC, BIC, CAIC, HQIC) on a guinea pig survival data set.

Experimental results

Research questions

• RQ1 Can a generalized inverse xgamma distribution (GIXGD) be constructed to model non-monotone hazard rates more flexibly than existing inverse family distributions?
• RQ2 How do the statistical properties of GIXGD—such as moments, quantile function, and reliability measures—compare to those of the baseline inverse xgamma distribution?
• RQ3 Does GIXGD provide a better fit to real survival data than competing distributions like inverse Weibull, inverse Lindley, and generalized exponential?
• RQ4 What are the performance characteristics of MLE in estimating the parameters and reliability functions of GIXGD under different censoring and sample conditions?
• RQ5 Can the GIXGD serve as a viable alternative model in reliability and survival analysis where non-monotone hazard patterns are observed?

Key findings

• The GIXGD exhibits a non-monotone hazard rate, with an initial increase followed by a decline, making it suitable for modeling complex failure patterns in reliability and clinical studies.
• The model provides a superior fit to the guinea pig survival data compared to 6 competing distributions, as evidenced by the lowest AIC (787.98), BIC (792.54), CAIC (793.54), HQIC (789.79), and K-S statistic (0.1432) among all models.
• The maximum likelihood estimates for the shape and scale parameters were $ \hat{\alpha} = 1.624 $ and $ \hat{\theta} = 641.75 $, respectively, based on the data set of 72 observations.
• Estimated survival probabilities at $ y = 54, 70, 99, 112 $ days were 0.626, 0.475, 0.308, and 0.260, respectively, with corresponding hazard rates of 0.0177, 0.0165, 0.0136, and 0.0124.
• The generalized inverse xgamma distribution outperformed the generalized exponential (GED), inverse Weibull (IWD), and gamma (GD) distributions, with the latter showing a very high K-S statistic (0.998), indicating poor fit.
• The study confirms that GIXGD is a strong candidate for modeling positively skewed, non-monotone hazard rate data in survival analysis, especially when the baseline IXGD is insufficiently flexible.