Abstract
The generalized log-logistic distribution is especially useful for modelling survival data with variable hazard rate shapes because it extends the log-logistic distribution by adding an extra parameter to the classical distribution, resulting in greater flexibility in analyzing and modelling various data types. We derive the fundamental mathematical and statistical properties of the proposed distribution in this paper. Many well-known lifetime special submodels are included in the proposed distribution, including the Weibull, log-logistic, exponential, and Burr XII distributions. The maximum likelihood method was used to estimate the unknown parameters of the proposed distribution, and a Monte Carlo simulation study was run to assess the estimators' performance. This distribution is significant because it can model both monotone and nonmonotone hazard rate functions, which are quite common in survival and reliability data analysis. Furthermore, the proposed distribution's flexibility and usefulness are demonstrated in a real-world data set and compared to its submodels, the Weibull, log-logistic, and Burr XII distributions, as well as other three-parameter parametric survival distributions, such as the exponentiated Weibull distribution, the three-parameter log-normal distribution, the three-parameter (or the shifted) log-logistic distribution, the three-parameter gamma distribution, and an exponentiated Weibull distribution. The proposed distribution is plausible, according to the goodness-of-fit, log-likelihood, and information criterion values. Finally, for the data set, Bayesian inference and Gibb's sampling performance are used to compute the approximate Bayes estimates as well as the highest posterior density credible intervals, and the convergence diagnostic techniques based on Markov chain Monte Carlo techniques were used.
Highlights
Applied statisticians use many probability distributions for reliability and survival studies. e distributions could be applied in different fields such as medicine, engineering, economy, industrial and physical fields, and so many other fields
We focus on a modification of the loglogistic model because it resembles the log-normal distribution in shape but is better suited for the application in the analysis of survival data when dealing with incomplete data, such as censored observations which are common in such data [2]. e presence of incomplete observations causes difficulties when using log-normal or inverse Gaussian models, since the survival functions in these cases are complicated
For the applied cases, especially in the survival modelling, the GLL model could be applicable in the following cases: (1) modelling the “asymmetric monotonically right-skewed” heavy tail data sets; (2) modelling the “bathtub-shaped hazard rate” data sets like data set I; (3) in “survival analysis,” the GLL distribution could be chosen for modelling proportional hazard frameworks; (4) in the medical field, the GLL distribution could be considered in modelling the “bladder cancer data sets” which have “reversed bathtub-shaped HRF” as illustrated in data set I; and (5) in the reliability and survival analysis, the proposed distribution can be an alternative to the Weibull distribution since it can be closed under both accelerated failure time (AFT) and PH models since the Weibull distribution fails to model unimodal data
Summary
Applied statisticians use many probability distributions for reliability and survival studies. e distributions could be applied in different fields such as medicine, engineering, economy, industrial and physical fields, and so many other fields. For the applied cases, especially in the survival modelling, the GLL model could be applicable in the following cases: (1) modelling the “asymmetric monotonically right-skewed” heavy tail data sets; (2) modelling the “bathtub-shaped hazard rate” data sets like data set I; (3) in “survival analysis,” the GLL distribution could be chosen for modelling proportional hazard frameworks; (4) in the medical field, the GLL distribution could be considered in modelling the “bladder cancer data sets” which have “reversed bathtub-shaped HRF” as illustrated in data set I; and (5) in the reliability and survival analysis, the proposed distribution can be an alternative to the Weibull distribution since it can be closed under both accelerated failure time (AFT) and PH models since the Weibull distribution fails to model unimodal data For these based on ground reasons, we are motivated to study and introduce the GLL distribution. Submodels. e proposed distribution consists of a number of important submodels that are widely used in parametric survival modelling. ese include the log-logistic distribution, the standard log-logistic distribution, the Burr XII distribution, the Weibull distribution, and the exponential distribution. e propositions below relate the GLL to the log-logistic, standard log-logistic, Burr XII, Weibull, and exponential distributions
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