Generalized Marshall–Olkin Lomax Distribution

• The GMOL distribution is an extension of the classical Lomax model using a two-parameter generator, enhancing flexibility in lifetime and reliability modeling.
• It provides closed-form density, quantile, and moment functions while supporting regression analysis for censored survival data.
• Empirical applications to AIDS and COVID-19 data demonstrate superior model fit compared to traditional Lomax-based approaches.

The generalized Marshall–Olkin Lomax (GMOL) distribution is a parametric family extending the classical Lomax (Pareto Type II) distribution through an explicit two-parameter generator, enabling enhanced flexibility for lifetime and reliability modeling. GMOL incorporates additional shape and scale modifications, admitting several familiar distributions as special instances and supporting regression extensions for censored survival data analysis. Recent work has established closed-form expressions for density, quantile, moment, and generating functions, and demonstrated the practical utility of GMOL for real datasets exhibiting skewness, leptokurtosis, and high censoring rates, notably in AIDS and COVID-19 applications (Ferreira et al., 15 Oct 2025).

1. Construction and Formal Definition

Let $G(x; \tau, \beta)$ and $g(x; \tau, \beta)$ denote, respectively, the cumulative distribution function (CDF) and probability density function (PDF) of the baseline Lomax distribution: $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$ where $\tau$ is the shape parameter and $\beta$ is the scale.

The GMOL generator introduces two additional parameters, $\alpha \in (0, 1]$ and $\lambda \in [0, 1]$, yielding the GMOL CDF as

$$F(x) = \frac{ \lambda\, G(x) + (1 - \lambda)\, [G(x)]^2 }{ \alpha + (1 - \alpha)\, G(x) }.$$

Replacing $G(x)$ by the Lomax distribution gives

$$F(x; \alpha, \lambda, \tau, \beta) = \frac{ \lambda\, \big[1 - (\frac{\beta}{\beta + x})^\tau\big] + (1-\lambda)\, \big[1 - (\frac{\beta}{\beta + x})^\tau\big]^2 }{ \alpha + (1-\alpha)\, \big[1 - (\frac{\beta}{\beta + x})^\tau\big] }.$$

The closed-form PDF and hazard rate function are derived by differentiation and algebraic manipulation; all standard cases and limits are directly obtainable.

2. Distributional Properties and Special Cases

2.1. Quantile Function

Given $g(x; \tau, \beta)$0, the GMOL quantile function $g(x; \tau, \beta)$1 has the explicit representation

$g(x; \tau, \beta)$2

where $g(x; \tau, \beta)$3 is a function of $g(x; \tau, \beta)$4 as prescribed by the generator structure.

2.2. Limits and Submodels

Specific parameterizations recover well-known distributions:

• $g(x; \tau, \beta)$5: GMOL reduces to the Marshall–Olkin Lomax (MOL).
• $g(x; \tau, \beta)$6: The distribution becomes the transmuted Lomax.
• $g(x; \tau, \beta)$7: Recovers the classical Lomax distribution.

GMOL admits infinite mixture representations with exponentiated Lomax densities, implying that moments, incomplete moments, Lorenz/Bonferroni curves, and entropic measures may be deduced directly from the properties of the Lomax family.

3. Regression for Censored Data

GMOL supports a regression model for censored survival analysis by allowing both scale and shape parameters to depend on explanatory covariates. Specifically, for observation $g(x; \tau, \beta)$8: $g(x; \tau, \beta)$9 where $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$0 is a covariate vector, and $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$1 parameter vectors. Generator parameters $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$2, $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$3 remain common across observations. For censored datasets, the likelihood contribution is partitioned according to event/failure indicators and the associated survival function, with log-likelihood formulas for both uncensored and censored components provided explicitly.

4. Estimation and Simulation

Parameters are estimated by the method of maximum likelihood. For the standard GMOL without covariates, the log-likelihood is

$$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$4

For regression with censoring, the likelihood incorporates covariate effects in $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$5 and $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$6 and appropriately combines event and censored likelihood terms. Optimization is performed numerically using standard packages (e.g., AdequacyModel for R, supporting algorithms such as BFGS, Nelder–Mead, and SANN). Monte Carlo simulations (1000 replicates per sample size) show empirical consistency; bias and mean squared error decrease with increasing $$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$7.

5. Applications and Model Comparison

Three real datasets illustrate the capacity and superiority of the GMOL model:

• AIDS data: 695 HIV-positive patients, time to opportunistic disease; GMOL exhibited lower values in Cramér–von Mises, Anderson–Darling, AIC, BIC, and generalized likelihood ratio tests than beta Lomax, Kumaraswamy Lomax, Weibull Lomax, and MOL.
• COVID-19 data (Paraíba): 109 deaths, symptom onset to death; the flexible GMOL achieved best fit for right-skewed, leptokurtic lifetimes.
• COVID-19 regression data (Distrito Federal): 485 patients, 78% right-censored, covariates include age and obesity; regression model indicated both covariates significantly affected survival, with GMOL providing superior fit evidenced by likelihood ratio tests and quantile residual diagnostics.

A tabular summary of model specializations:

Parameterization	GMOL Reduces to	Main Reference
$$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$8	Marshall–Olkin Lomax (MOL)	(Ferreira et al., 15 Oct 2025)
$$G(x; \tau, \beta) = 1 - \left( \frac{\beta}{\beta + x} \right)^\tau,\quad g(x; \tau, \beta) = \frac{\tau \beta^\tau}{(\beta + x)^{\tau+1}},\qquad x > 0,\; \tau > 0,\; \beta > 0,$$9	Transmuted Lomax	(Ferreira et al., 15 Oct 2025)
$\tau$0	Standard Lomax	(Ferreira et al., 15 Oct 2025)

6. Interpretive Notes and Implications

The GMOL distribution generalizes previous Lomax-based models by introducing additional generator parameters, thereby supporting a broad spectrum of hazard rate shapes, tail behaviors, and regression structures. Model selection via information criteria and likelihood ratio tests consistently favors GMOL in the analyzed real-world life data. This suggests that GMOL serves as a robust, fully parametric alternative for datasets characterized by censoring, heavy-tail behavior, and complex covariate dependencies. Given its closed-form quantile and mixture representations, GMOL also enables tractable computation of moments, entropic measures, and ordering indices, extending naturally to statistical inference and simulation for reliability, medical survival analysis, and risk modeling contexts.

The distribution's mathematical tractability and performance in empirical applications underscore its value as an extension of the Marshall–Olkin-Lomax paradigm, with extensive applicability in advanced survival analysis and reliability models (Ferreira et al., 15 Oct 2025).