Measures of relative variability, such as the Pearson's coefficient of variation (CV_p), give much insight into the spread of lifetime distributions, like the Weibull distribution. The estimation of the Weibull CV_p in modern statistics has traditionally been prioritized only when complete data is available. In this article, we estimate the Weibull CV_p and its second-order alternative, denoted as CV_k, under type-I progressively interval censoring, which is a typical scenario in survival analysis and reliability theory. Point estimates are obtained using the methods of maximum likelihood, least squares, and the Bayesian approach with MCMC simulation. A nonlinear least squares method is proposed for estimating the CV_p and CV_k. We also perform interval estimation of the CV_p and CV_k using the asymptotic confidence intervals, bootstrap intervals through the least squares estimates, and the highest posterior density intervals. A comprehensive Monte Carlo simulation study is carried out to understand and compare the performance of the estimators. The proposed least squares and the Bayesian methods produce better point estimates for the CV_p. The highest posterior density intervals outperform other interval estimates in many cases. The methodologies are also applied to a real dataset to demonstrate the performance of the estimators.
A modified version of the discrete gamma distribution was recently developed to model probabilistic damage accumulation in the context of weapon–target interactions. In this article, we present some newly discovered properties of the modified discrete gamma distribution. In particular, we derive new closed-form expressions for the distribution's mean and non-trivial central moments in terms of special functions, namely, the regularized upper incomplete gamma function and the polylogarithm of negative order. These special functions are exploited to establish asymptotic properties of the mean, variance and higher-order moments of the distribution via Faulhaber's formula. Finally, we show how the construction of the modified discrete gamma distribution can be used to compute specific limiting properties of a family of polylogarithmic integrals.
In recent years, there has been a notable increase in the study of matrix-variate distributions and their applications. Significant progress has been made in understanding the properties and statistical inference of these distributions. In this paper, we introduce two alternative extensions of the univariate Value-at-Risk (VaR) within a matrix-variate context: the matrix upper VaR and the matrix lower VaR. These extensions are obtained as the zeroes of the Gauss hypergeometric function with a matrix argument, thereby providing valuable tools for risk assessment in a variety of fields, particularly in finance and capital allocation. In this paper, we provide the univariate VaR for the generalized beta and F distributions, as well as the matrix-variate VaR for these distributions. Moreover, we derive the beta-Kotz VaR based on a general family of distributions, which includes the classical Gaussian model. Furthermore, new integrals and results involving zonal polynomials are derived. This paper advances the understanding of matrix-variate VaR extensions, opening new avenues for their application in various disciplines. By bridging the gap between matrix-variate distributions and VaR, we aim to stimulate further research and practical implementations in financial risk management and capital optimization.
