Theoretical results on the discrete Weibull distribution of Nakagawa and Osaki

In this paper, we discuss some theoretical results and properties of the discrete Weibull distribution, which was introduced by Nakagawa and Osaki [The discrete Weibull distribution. IEEE Trans Reliab. 1975;24:300–301]. We study the monotonicity of the probability mass, survival and hazard functions. Moreover, reliability, moments, p-quantiles, entropies and order statistics are also studied. We consider likelihood-based methods to estimate the model parameters based on complete and censored samples, and to derive confidence intervals. We also consider two additional methods to estimate the model parameters. The uniqueness of the maximum likelihood estimate of one of the parameters that index the discrete Weibull model is discussed. Numerical evaluation of the considered model is performed by Monte Carlo simulations. For illustrative purposes, two real data sets are analyzed.     

Small‐sphere distributions for directional data with application to medical imaging

We propose novel parametric concentric multi‐unimodal small‐subsphere families of densities for p ? 1 ≥ 2‐dimensional spherical data. Their parameters describe a common axis for K small hypersubspheres, an array of K directional modes, one mode for each subsphere, and K pairs of concentrations parameters, each pair governing horizontal (within the subsphere) and vertical (orthogonal to the subsphere) concentrations. We introduce two kinds of distributions. In its one‐subsphere version, the first kind coincides with a special case of the Fisher–Bingham distribution, and the second kind is a novel adaption that models independent horizontal and vertical variations. In its multisubsphere version, the second kind allows for a correlation of horizontal variation over different subspheres. In medical imaging, the situation of p ? 1 = 2 occurs precisely in modeling the variation of a skeletally represented organ shape due to rotation, twisting, and bending. For both kinds, we provide new computationally feasible algorithms for simulation and estimation and propose several tests. To the best knowledge of the authors, our proposed models are the first to treat the variation of directional data along several concentric small hypersubspheres, concentrated near modes on each subsphere, let alone horizontal dependence. Using several simulations, we show that our methods are more powerful than a recent nonparametric method and ad hoc methods. Using data from medical imaging, we demonstrate the advantage of our method and infer on the dominating axis of rotation of the human knee joint at different walking phases.     

On MLEs of the parameters of a modified Weibull distribution based on record values

The modified Weibull distribution can be used quite effectively to model complex data from mechanical engineering or survival analysis studies that posses a monotonic or a bathtub-shape hazard rate. In this paper, we study the MLEs of the parameters of a modified Weibull distribution model in the presence of upper kth record values. The existence and uniqueness of the MLEs are proven in this case. Real data analysis is performed for illustrative purposes.     

Parametrizing the Kepler exoplanet period-radius distribution with the bivariate normal inverse Gaussian distribution

This paper presents a simple and robust method for obtaining a comprehensive understanding of the joint period and radius distribution in Kepler exoplanets. The proposed method is based on particle swarm optimization and bivariate Normal Inverse Gaussian distribution. Furthermore, in the construction of the probability density function, this study selects planet-host stars with the GK-type. The injecting approach is also employed to solve the survey completeness of sample. The resulting occurrence rate of Earth analogs is 0.025 with a 95% bootstrap confidence interval between 0.023 and 0.032.     

Estimating E-bayesian and hierarchical bayesian of scalar parameter of gompertz distribution under type II censoring schemes based on fuzzy data

In this study, the E-Bayesian and hierarchical Bayesian of the scalar parameter of a Gompertz distribution under Type II censoring schemes were estimated based on fuzzy data under the squared error (SE) loss function and the efficiency of the proposed methods was compared with each other and with the Bayesian estimator using Monte Carlo simulation.     

Ordering results for series and parallel systems comprising heterogeneous exponentiated Weibull components

In this paper, we discuss the usual stochastic and reversed hazard rate orders between the series and parallel systems from two sets of independent heterogeneous exponentiated Weibull components. We also obtain the results concerning the convex transform orders between parallel systems and obtain necessary and sufficient conditions under which the dispersive and usual stochastic orders, and the right spread and increasing convex orders between the lifetimes of the two systems are equivalent. Finally, in the multiple-outlier exponentiated Weibull models, based on weak majorization and p-larger orders between the vectors of scale and shape parameters, some characterization results for comparing the lifetimes of parallel and series systems are also established, respectively. The results of this paper can be used in practical situations to find various bounds for the important aging characteristics of these systems.     

Tempered Mittag-Leffler Lévy processes

In this article, we introduce tempered Mittag-Leffler Lévy processes (TMLLP). TMLLP is represented as tempered stable subordinator delayed by a gamma process. Its probability density function and Lévy density are obtained in terms of infinite series and Mittag-Leffler function, respectively. Asymptotic forms of the tails and moments are given. A step-by-step procedure of the parameters estimation and simulation of sample paths is given. We also provide main results available for Mittag-Leffler Lévy processes (MLLP) and some extensions which are not available in a collective way in a single article. Our results generalize and complement the results available on Mittag-Leffler distribution and MLLP in several directions. Further, the asymptotic forms of the moments of the first-exit times of the TMLLP are also discussed.     

Prediction of future failures times based on Type-I hybrid censored samples of random sample sizes

In this article, we consider the prediction of future failure times based on Type-I hybrid censored samples. Point predictors and prediction intervals using different procedures are discussed for a general model. The exponential and Rayleigh distributions are used as illustrative examples to show the most simplified forms of the so obtained predictors as well as prediction intervals. Intensive simulation study and a real life dataset are presented to illustrate our findings and results.     

A semiparametric approach for modelling multivariate nonlinear time series

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach. The Canadian Journal of Statistics 47: 668–687; 2019 © 2019 Statistical Society of Canada     

The laws of large numbers for Pareto-type random variables with infinite means

In this paper, we consider the laws of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means. Based on the Pareto-Zipf distributions, some weak laws of large numbers for weighted sums of NSD random variables are obtained. Meanwhile, we show that a weak law for Pareto-Zipf distributions cannot be extended to a strong law. Furthermore, based on the two tailed Pareto distribution, a strong law of large numbers for weighed NSD random variables is presented. Our results extend the corresponding earlier ones.