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The generalized half-Cauchy distribution: Mathematical properties and regression models with censored data
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We study some mathematical properties of a new generalized half-Cauchy distribution, which extends the classical half-Cauchy model. We derive a power series expansion for the quantile function, explicit expressions for the ordinary and incomplete moments and generating function, order statistics and their moments. We also present characterizations of the generalized half-Cauchy distribution. The estimation of the model parameters is performed by maximum likelihood. We introduce thelog -generalized half-Cauchy regression model based on the half-Cauchy distribution. The new regression model represents a parametric family of models that includes as special cases several widely known regression models that can be applied to censored survival data. In addition, the sensitivity analysis is proposed and the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. We demonstrate the potentiality of the new models by means of two applications to two real data.
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... Several alternative hyperpriors can be considered as substitutes, such as the Half-Cauchy [10], Penalized Complexity [11], Uniform, generalized logGamma [12], generalized Half-Cauchy [13], and generalized Uniform [14]. However, practitioners may encounter challenges when trying to implement the generalized distribution, as it is not currently supported in common Bayesian software such as R-INLA 2023 (https://www.r-inla.org/) ...
... To ensure comprehensive future investigations, it is imperative to examine the utilization of generalized logGamma [12], generalized Half-Cauchy [13], and generalized Uniform [14] distributions. These alternatives have the potential to be strong replacements for hyperpriors, potentially improving the accuracy of predictions and accounting for additional variability in the data. ...
Spatiotemporal disease mapping modeling with count data is gaining increasing prominence. This approach serves as a benchmark in developing early warning systems for diverse disease types. Spatiotemporal modeling, characterized by its inherent complexity, integrates spatial and temporal dependency structures, as well as interactions between space and time. A Bayesian approach employing a hierarchical structure serves as a solution for spatial model inference, addressing the identifiability problem often encountered when utilizing classical approaches like the maximum likelihood method. However, the hierarchical Bayesian approach faces a significant challenge in determining the hyperprior distribution for the variance components of hierarchical Bayesian spatiotemporal models. Commonly used distributions include logGamma for log inverse variance, Half-Cauchy, Penalized Complexity, and Uniform distribution for hyperparameter standard deviation. While the logGamma approach is relatively straightforward with faster computing times, it is highly sensitive to changes in hyperparameter values, specifically scale and shape. This research aims to identify the most optimal hyperprior distribution and its parameters under various conditions of spatial and temporal autocorrelation, as well as observation units, through a Monte Carlo study. Real data on dengue cases in West Java are utilized alongside simulation results. The findings indicate that, across different conditions, the Uniform hyperprior distribution proves to be the optimal choice.
We present various characterizations of a recently introduced distribution (Ghosh 2014), called KumaraswamyHalf-
Cauchy distribution based on: (i) a simple relation between two truncated moments; (ii) truncated moment
of certain function of the 1st order statistic; (iii) truncated moment of certain function of the random variable; (iv)
hazard function; (v) distribution of the 1st order statistic; (vi) via record values. We also provide some remarks on
bivariate Gumbel copula distribution whose marginal distributions are Kumaraswamy- Half-Cauchy distributions.
We present characterizations of certain families of generalized gamma convolution distributions of L. Bondesson [Ann. Probab. 7, 965–979 (1979; Zbl 0421.60014)] based on a simple relationship between two truncated moments. We also present a list of well-known random variables whose distributions or the distributions of certain functions of them belong to the class of generalized gamma convolutions.
On the basis of the half-Cauchy distribution, we propose the called beta-half-Cauchy
distribution for modeling lifetime data. Various explicit expressions for its moments, generating
and quantile functions, mean deviations, and density function of the order statistics and their moments
are provided. The parameters of the new model are estimated by maximum likelihood, and the observed
information matrix is derived. An application to lifetime real data shows that it can yield a
better fit than three- and two-parameter Birnbaum-Saunders, gamma, and Weibull models.
Half Title Title Copyright Dedication Preface Contents
We introduce the log-odd Weibull regression model based on the odd Weibull distribution (Cooray, 2006). We derive some mathematical properties of the log-transformed distribution. The new regression model represents a parametric family of models that includes as sub-models some widely known regression models that can be applied to censored survival data. We employ a frequentist analysis and a parametric bootstrap for the parameters of the proposed model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to assess global influence. Further, for different parameter settings, sample sizes and censoring percentages, some simulations are performed. In addition, the empirical distribution of some modified residuals are given and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We define martingale and deviance residuals to check the model assumptions. The extended regression model is very useful for the analysis of real data.
In this article, for the first time, we propose the negative binomial–beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.
Graphical methods based on the analysis of residuals are considered for the setting of the highly-used D. R. Cox [J. R. Stat. Soc., Ser. B 34, 187-220 (1972; Zbl 0243.62041)] regression model and for the P. K. Andersen and R. D. Gill [Ann. Stat. 10, 1100-1120 (1982; Zbl 0526.62026)] generalization of that model. We start with a class of martingale-based residuals as proposed by W. E. Barlow and R. L. Prentice [Biometrika 75, 65-74 (1988; Zbl 0632.62102)]. These residuals and/or their transforms are useful for investigating the functional form of covariate, the proportional hazards assumption, the leverage of each subject upon the estimates of β, and the lack of model fit to a given subject.