Abstract
In this paper, from the practical point of view, we focus on modeling traumatic brain injury data considering different stages of hospitalization, related to patients’ survival rates following traumatic brain injury caused by traffic accidents. From the statistical point of view, the primary objective is related to overcoming the limited number of traumatic brain injury patients available for studying by considering different estimation methods to obtain improved estimators of the model parameters, which can be recommended to be used in the presence of small samples. To have a general methodology, at least in principle, we consider the very flexible Generalized Gamma distribution. We compare various estimation methods using extensive numerical simulations. The results reveal that the penalized maximum likelihood estimators have the smallest mean square errors and biases, proving to be the most efficient method among the investigated ones, mainly to be used in the presence of small samples. The Simulated Annealing technique is used to avoid numerical problems during the optimization process, as well as the need for good initial values. Overall, we considered an amount of three real data sets related to traumatic brain injury caused by traffic accidents to demonstrate that the Generalized Gamma distribution is a simple alternative to be used in this type of applications for different occurrence rates and risks, and in the presence of small samples.
Highlights
Gamma distribution plays an important role in statistics as one of the most used generalizations of the Exponential distribution due to its various special cases
In order to overcome this problem, for small sample sizes, we considered the bootstrap approach presented by DiCiccio and Efron [25] to construct improved confidence intervals based on the penalized maximum likelihood (PML) estimates
We observed that the ordinary least squares (OLS), weighted least squares (WLS), and MM methods failed in finding the parameter estimates for a significant number of samples
Summary
Gamma distribution plays an important role in statistics as one of the most used generalizations of the Exponential distribution due to its various special cases (such as Exponential and Chi-square). This distribution has been used in different scenarios, such as reliability engineering, environmental modeling, and health research, to list a few (see Louzada and Ramos [1] and the references therein). Modeling traumatic brain injury lifetime data with improved estimators for the Generalized Gamma distribution distribution to unify other relevant distributions, e.g., Weibull and Lognormal. Huang and Hwang [7] used the method of moments to perform inference for the GG distribution. And Jones [9] discussed some different approaches to maximize the likelihood function; the proposed numerical technique returned smaller proportions of errors during the maximization process, but still failed in a significant number of samples, which is undesirable
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