Showing papers in "Journal of Statistical Distributions and Applications in 2016"

Compounding of distributions: a survey and new generalized classes

Abstract: Generalizing distributions is an old practice and has ever been considered as precious as many other practical problems in statistics. It simply started with defining different mathematical functional forms, and then induction of location, scale or inequality parameters. The generalization through induction of shape parameter(s) started in 1997, and the last two decades were full of such practices. But to cope with the complex situations under series and parallel structures, the art of mixing discrete and continuous started in 1998. In this article, we present a survey on compounding univariate distributions, their extensions and classes. We review several available compound classes and propose some new ones. The recent trends in the construction of generalized and compounding classes are discussed, and the need for future works are addressed.

The generalized Cauchy family of distributions with applications

Abstract: A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. The family of distributions is generated using the quantile functions of uniform, exponential, log-logistic, logistic, extreme value, and Frechet distributions. Several general properties of the T-Cauchy{Y} family are studied in detail including moments, mean deviations and Shannon’s entropy. Some members of the T-Cauchy{Y} family are developed and one member, gamma-Cauchy{exponential} distribution, is studied in detail. The distributions in the T-Cauchy{Y} family are very flexible due to their various shapes. The distributions can be symmetric, skewed to the right or skewed to the left.

Generalized log-logistic proportional hazard model with applications in survival analysis

Abstract: Proportional hazard (PH) models can be formulated with or without assuming a probability distribution for survival times. The former assumption leads to parametric models, whereas the latter leads to the semi-parametric Cox model which is by far the most popular in survival analysis. However, a parametric model may lead to more efficient estimates than the Cox model under certain conditions. Only a few parametric models are closed under the PH assumption, the most common of which is the Weibull that accommodates only monotone hazard functions. We propose a generalization of the log-logistic distribution that belongs to the PH family. It has properties similar to those of log-logistic, and approaches the Weibull in the limit. These features enable it to handle both monotone and nonmonotone hazard functions. Application to four data sets and a simulation study revealed that the model could potentially be very useful in adequately describing different types of time-to-event data.

Exponentiated Marshall-Olkin family of distributions

Abstract: We study general mathematical properties of a new class of continuous distributions with three extra shape parameters called the exponentiated Marshal-Olkin family of distributions. Further, we present some special models of the new class and investigate the shapes and derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions and probability weighted moments. We discuss the estimation of the model parameters by maximum likelihood and show empirically the potentiality of the family by means of two applications to real data.

Extended Conway-Maxwell-Poisson distribution and its properties and applications

Abstract: A new four parameter extended Conway-Maxwell-Poisson (ECOMP) distribution which unifies the recently proposed COM-Poisson type negative binomial (COM-NB) distribution [Chakraborty, S. and Ong, S. H. (2014): A COM-type Generalization of the Negative Binomial Distribution, Accepted in Communications in Statistics-Theory and Methods] and the generalized COM-Poisson (GCOMP) distribution [Imoto, T. :(2014) A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution, Applied Mathematics and Computation, 247, 824–834] is proposed. The additional parameter allows this distribution to have longer (shorter) tail compared to COM-NB and GCOMP. The proposed distribution can be formulated as an exponential combination of negative binomial and COM-Poisson distribution and also arises from a queuing system with state dependent arrival and service rates and belongs to exponential family when one of the parameter is considered as nuisance. Important distributional, reliability and stochastic ordering properties along with asymptotic approximations for the normalizing constant and the mean of this distribution is investigated. Method of parameter estimation and three comparative data fitting applications are also discussed.

The gamma extended Weibull distribution

Abstract: The two-parameter Weibull has been the most popular distribution for modeling lifetime data. We propose a four-parameter gamma extended Weibull model, which generalizes the Weibull and extended Weibull distributions, among several other models. We obtain explicit expressions for the ordinary and incomplete moments, generating and quantile functions and mean deviations. We employ the method of maximum likelihood for estimating the model parameters. We propose a log-gamma extended Weibull regression model with censored data. The applicability of the new models is well justified by means of two real data sets.

Exact inference on scaling parameters in norm and antinorm contoured sample distributions

Abstract: Exact distributions of generalized Chi-square and Fisher statistics are used to derive confidence intervals and significance tests for inferring on one or two scaling parameters, respectively, under non-standard assumptions w.r.t. the multivariate sample distribution. The latter may have convex or radially concave density level sets and heavy or light distribution centers and tails. Independent and l n,p -dependent sample variables are considered.

An extended-G geometric family

Abstract: We introduce and study the extended-G geometric family of distributions, which contains as special models some important distributions such as the XTG (Xie et al. 2002) geometric, Weibull geometric, Chen (Chen 2000) geometric, Gompertz geometric, among others. This family not only includes distributions with bathtub and unimodal failure rate functions but provides a broader class of monotone failure rates. Its density function can be expressed as a linear mixture of extended-G densities. We derive explicit expansions for the ordinary and incomplete moments, generating function, mean deviations and Renvy entropy. The density of the order statistics can also be given as a linear mixture of extended-G densities. The model parameters are estimated by maximum likelihood. The potentiality of the new family is illustrated by means of an application to real data. 60E05, 62P10, 62P30

pTAS distributions with application to risk management

Abstract: The family of positive tempered α-stable (pTAS) or sometimes also tempered one-sided α-stable distributions dates back to Tweedie (1984) and Hougaard (1986) who discussed it in the context of frailty distribution in life table methods for heterogenous populations. The pTAS family generalizes the well-known gamma distribution and allows for heavier tails depending on the parameter α. Because of this property, pTAS distributions appear to be useful in the context of risk management. Against this background, the contribution of his work is three-fold: Firstly, we summarize the properties of the pTAS family. Secondly, we describe its numerical implementation and illustrate the functions by means of R examples in the Appendix. Thirdly, we empirically demonstrate that this family can be successfully applied in risk management. Concretely, applications to credit and operational risk are given.

An R package for modeling and simulating generalized spherical and related distributions

Abstract: A flexible class of multivariate generalized spherical distributions with star-shaped level sets is developed To work in dimension above two requires tools from computational geometry and multivariate numerical integration An algorithm to approximately simulate from these star-shaped distributions is developed; it also works for simulating from more general tessellations These techniques are implemented in the R package gensphere

Skewness-kurtosis adjusted confidence estimators and significance tests

Abstract: First and second kind modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be asymptotically exactly controlled. The orders of convergence to zero of both types of probabilities are assumed to be suitably bounded below according to an Osipov-type condition and the sample distribution is assumed to satisfy a corresponding tail condition due to Linnik. Analogous considerations are presented for the power function when testing a hypothesis concerning the expectation both under the assumption of a true hypothesis as well as under a modified local alternative. A limit theorem for large deviations by S.V. Nagajev/V.V. Petrov applies to prove the results. Applications are given for exponential families.

A class of continuous bivariate distributions with linear sum of hazard gradient components

Abstract: The main purpose of this article is to characterize a class of bivariate continuous non-negative distributions such that the sum of the components of underlying hazard gradient vector is a linear function of its arguments It happens that this class is a stronger version of the Sibuya-type bivariate lack of memory property Such a class is allowed to have only certain marginal distributions and the corresponding restrictions are given in terms of marginal densities and hazard rates We illustrate the methodology developed by examples, obtaining two extended versions of the bivariate Gumbel’s law

Flexible modelling of survival curves for censored data

Abstract: This article outlines flexible strategies to model survival curves for censored data and find parametric confidence intervals using generalised lambda distributions. Owing to the rich shapes of generalised lambda distributions, these distributions are well suited to the problem of estimating survival curves. This article presents three useful techniques in estimating survival curves: matching partial probability weighted moments (PWM), maximum likelihood estimation (MLE) and simulation-refitting (SR) methods. The performance of these techniques are examined using right skewed, left skewed, symmetric bell curved and extreme value simulated data with varying degrees of censoring and sample sizes. Applications of the proposed methods in the context of multi-stage disease modelling and competing risks are also provided. Under controlled simulated experiments, PWM and MLE estimation tend to exhibit more precise estimates for survival curves than the SR method, however, the SR method tends to perform better in practice. The methods proposed in this article are very general and can be used to fit a wide range of empirical survival curves. Compared to the standard Kaplan Meier survival curve, the methods in this article have the added benefits of producing smoother survival curves and more consistent statistical estimates where all the statistical information of the survival curve can be obtained directly under one parametric model.

Nonparametric rank based estimation of bivariate densities given censored data conditional on marginal probabilities

Abstract: In this note we develop a new Kaplan-Meier product-limit type estimator for the bivariate survival function given right censored data in one or both dimensions. Our derivation is based on extending the constrained maximum likelihood density based approach that is utilized in the univariate setting as an alternative strategy to the approach originally developed by Kaplan and Meier (1958). The key feature of our bivariate survival function is that the marginal survival functions correspond exactly to the Kaplan-Meier product limit estimators. This provides a level of consistency between the joint bivariate estimator and the marginal quantities as compared to other approaches. The approach we outline in this note may be extended to higher dimensions and different censoring mechanisms using the same techniques.