On compounded geometric distributions and their applications

Here, we introduce two-parameter compounded geometric distributions with monotone failure rates. These distributions are derived by compounding geometric distribution and zero-truncated Poisson distribution. Some statistical and reliability properties of the distributions are investigated. Parameters of the proposed distributions are estimated by the maximum likelihood method as well as through the minimum distance method of estimation. Performance of the estimates by both the methods of estimation is compared based on Monte Carlo simulations. An illustration with Air Crash casualties demonstrates that the distributions can be considered as a suitable model under several real situations.     

The poisson-inverse gaussian disiribuiion as a model for species abundance

An inverse Gaussian mixture of Poisson distributions(the P-IG distribution) is considered as a model for species abundance data,, Minimum chi-square and maximum likelihood methods of estimation for the zero-truncated P-IG distribution are developed, Ihe performance of the P-IG distribution is illustrated and discussed for several well-known sets of insect abundance data.     

A note on an integer valued time series model with Poisson–negative binomial marginal distribution

Abstract

This paper proposes a new model for autoregressive time series of counts in terms of a convolution of Poisson and negative binomial random variables, known as Poisson–negative binomial (PNB) distribution. The corresponding first-order integer valued time series models are developed and their properties are discussed. The geometric PNB and the geometric semi PNB distributions are also introduced and studied.     

Bivariate inverse Weibull distribution

ABSTRACT

Recently it is observed that the inverse Weibull (IW) distribution can be used quite effectively to analyse lifetime data in one dimension. The main aim of this paper is to define a bivariate inverse Weibull (BIW) distribution so that the marginals have IW distributions. It is observed that the joint probability density function and the joint cumulative distribution function can be expressed in compact forms. Several properties of this distribution such as marginals, conditional distributions and product moments have been discussed. We obtained the maximum likelihood estimates for the unknown parameters of this distribution and their approximate variance– covariance matrix. We perform some simulations to see the performances of the maximum likelihood estimators. One data set has been re-analysed and it is observed that the bivariate IW distribution provides a better fit than the bivariate exponential distribution.     

A surplus process involving a compound Poisson counting process and applications

Abstract

In this paper, we introduce a surplus process involving a compound Poisson counting process, which is a generalization of the classical ruin model where the claim-counting process is a homogeneous Poisson process. The incentive is to model batch arrival of claims using a counting process that is based on a compound distribution. This reduces the difficulty of modeling claim amounts and is consistent with industrial data. Recursive formula, some properties and relevant main ruin theory results are provided. Further, we consider applications involving zero-truncated negative binomial and zero-truncated binomial batch arrivals when the claim amounts follow exponential or Erlang distribution.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

THE LOG-EIG DISTRIBUTION: A NEW PROBABILITY MODEL FOR LIFETIME DATA

ABSTRACT

In this paper, we propose a new probability model called the log-EIG distribution for lifetime data analysis. Some important properties of the proposed model and maximum likelihood estimation of its parameters are discussed. Its relationship with the exponential inverse Gaussian distribution is similar to that of the lognormal and the normal distributions. Through applications to well-known datasets, we show that the log-EIG distribution competes well, and in some instances even provides a better fit than the commonly used lifetime models such as the gamma, lognormal, Weibull and inverse Gaussian distributions. It can accommodate situations where an increasing failure rate model is required as well as those with a decreasing failure rate at larger times.     

Slustep size probabilities for generalized poisson distributions

A general theorem is given relating the cluster-size distribution to the Fisher-type limiting form of the distribution resulting from a Poisson distribution of such clusters. The theorem provides a new method for obtaining the cluster-size probabilities. It also yields a necessary and sufficient condition for the infinite divisibility of a distribution on the nonnegative integers. The ethod and the criterion are illustrated using the shifted logarithmic and the shifted zero-truncated Poisson distributions.     

On left-truncating and mixing Poisson distributions

ABSTRACT

The distributions obtained by left-truncating at k a mixed Poisson distribution, denoted kT-MP, and those obtained by mixing previously left-truncated Poisson distributions, denoted M-kTP, are characterized by means of their probability generating function. The main consequence is that every kT-MP distribution is a M-kTP distribution, but not the other way around.     

Additive Transformation Models for Multivariate Interval-Censored Data

ABSTRACT

We present a new estimator of extreme quantiles dedicated to Weibull tail distributions. This estimate is based on a consistent estimator of the Weibull tail coefficient. This parameter is defined as the regular variation coefficient of the inverse cumulative hazard function. We give conditions in order to obtain the weak consistency and the asymptotic distribution of the extreme quantiles estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.     

Discrete Weibull geometric distribution and its properties

In this article, the discrete analog of Weibull geometric distribution is introduced. Discrete Weibull, discrete Rayleigh, and geometric distributions are submodels of this distribution. Some basic distributional properties, hazard function, random number generation, moments, and order statistics of this new discrete distribution are studied. Estimation of the parameters are done using maximum likelihood method. The applications of the distribution is established using two datasets.     

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

An asymmetric multivariate weibull distribution

Abstract

A class of multivariate laws as an extension of univariate Weibull distribution is presented. A well known representation of the asymmetric univariate Laplace distribution is used as the starting point. This new family of distributions exhibits some similarities to the multivariate normal distribution. Properties of this class of distributions are explored including moments, correlations, densities and simulation algorithms. The distribution is applied to model bivariate exchange rate data. The fit of the proposed model seems remarkably good. Parameters are estimated and a bootstrap study performed to assess the accuracy of the estimators.     

The beta-Weibull geometric distribution

We propose a new distribution, the so-called beta-Weibull geometric distribution, whose failure rate function can be decreasing, increasing or an upside-down bathtub. This distribution contains special sub-models the exponential geometric [K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], beta exponential [S. Nadarajah and S. Kotz, The exponentiated type distributions, Acta Appl. Math. 92 (2006), pp. 97–111; The beta exponential distribution, Reliab. Eng. Syst. Saf. 91 (2006), pp. 689–697], Weibull geometric [W. Barreto-Souza, A.L. de Morais, and G.M. Cordeiro, The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], generalized exponential geometric [R.B. Silva, W. Barreto-Souza, and G.M. Cordeiro, A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comput. Statist. Data Anal. 54 (2010), pp. 935–944; G.O. Silva, E.M.M. Ortega, and G.M. Cordeiro, The beta modified Weibull distribution, Lifetime Data Anal. 16 (2010), pp. 409–430] and beta Weibull [S. Nadarajah, G.M. Cordeiro, and E.M.M. Ortega, General results for the Kumaraswamy-G distribution, J. Stat. Comput. Simul. (2011). DOI: 10.1080/00949655.2011.562504] distributions, among others. The density function can be expressed as a mixture of Weibull density functions. We derive expansions for the moments, generating function, mean deviations and Rénvy entropy. The parameters of the proposed model are estimated by maximum likelihood. The model fitting using envelops was conducted. The proposed distribution gives a good fit to the ozone level data in New York.     

A spatial scan statistic for survival data based on generalized life distribution

ABSTRACT

For many years, detection of clusters has been of great public health interest and widely studied. Several methods have been developed to detect clusters and their performance has been evaluated in various contexts. Spatial scan statistics are widely used for geographical cluster detection and inference. Different types of discrete or continuous data can be analyzed using spatial scan statistics for Bernoulli, Poisson, ordinal, exponential, and normal models. In this paper, we propose a scan statistic for survival data which is based on generalized life distribution model that provides three important life distributions, viz. Weibull, exponential, and Rayleigh. The proposed method is applied to the survival data of tuberculosis patients in Nainital district of Uttarakhand, India, for the year 2004–05. The Monte Carlo simulation studies reveal that the proposed method performs well for different survival distributions.     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.     

The Poisson-X family of distributions

ABSTRACT

Recently, Risti? and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets.     

The Effect of Mis-Specification Between the Lognormal and Weibull Distributions on the Interval Estimation of a Quantile for Complete Data

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.