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In this paper, we propose modeling for a single repairable system with a hierarchical structure under the assumption that the failures follow a nonhomogeneous Poisson process (which corresponds to minimal repair action) with a power-law intensity function. The properties of the new model are discussed in detail. The parameter estimators are obtained using the maximum likelihood method. A corrective approach is used to remove bias with order O(n −1), and the respective exact confidence intervals are proposed. A simulation study is conducted to show that our estimators are bias-free. The proposed modeling is illustrated via a toy example on a butterfly valve system, an example of an early-stage real project related to the traction system of an in-pipe robot, and also a real example on a blowout preventer system.

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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.

In this paper we propose to make Bayesian inferences for the parameters of the Lomax distribution using non-informative priors, namely the Jeffreys prior and the reference prior. We assess Bayesian estimation through a Monte Carlo study with 500 simulated data sets. To evaluate the possible impact of prior specification on estimation, two criteria were considered: the bias and square root of the mean square error. The developed procedures are illustrated on a real data set.

Multiple longitudinal outcomes are common in public health research and adequate methods are required when there is interest in the joint evolution of response variables over time. However, the main drawback of joint modeling procedures is the requirement to specify the joint density of all outcomes and their correlation structure, as well as numerical difficulties in statistical inference, when the dimension of these outcomes increases. To overcome such difficulty, we present two procedures to deal with multivariate longitudinal data. We first present an univariate approach, for which linear mixed-effects models are considered for each response variable separately. Then, a novel copula-based modeling is presented, in order to characterize relationships among the response variables. Both methodologies are applied to a real Brazilian data set on child growth.

The Poisson distribution is a discrete model widely used to analyze count data. Statistical control charts based on this distribution, such as the c$c$ and u$u$ charts, are relatively well‐established in the literature. Nevertheless, many studies suggest the need for alternative approaches that allow for modeling overdispersion, a phenomenon that can be observed in several fields, including biology, ecology, healthcare, marketing, economics, and industry. The one‐parameter Poisson mixture distributions, whose literature is extensive and essential, can model extra‐Poisson variability, accommodating different overdispersion levels. The distributions belonging to this class of models, including the Poisson‐Lindley (PL), Poisson‐Shanker (PSh), and Poisson‐Sujatha (PSu) models, can thus be used as interesting alternatives to the usual Poisson and COM‐Poisson distributions for analyzing count data in several areas. In this paper, we consider the class of probabilistic models mentioned above (as well as the cited three members of such a class) to develop novel and useful statistical control charts for counting processes, monitoring count data that exhibit overdispersion. The performance of the so‐called one‐parameter Poisson mixture charts, namely the PLc$\text{PL}_c$‐PLu$\text{PL}_u$, PShc$\text{PSh}_c$‐PShu$\text{PSh}_u$, and PSuc$\text{PSu}_c$‐PSuu$\text{PSu}_u$ charts, is measured by the average run length in exhaustive numerical simulations. Some data sets are used to illustrate the applicability of the proposed methodology.