Bivariate semi-Pareto distributions and processes

A bivariate semi-Pareto distribution is introduced and characterized using geometric minimization. Autoregressive minification models for bivariate random vectors with bivariate semi-Pareto and bivariate Pareto distributions are also discussed. Multivariate generalizations of the distributions and the processes are briefly indicated.     

Bivariate semi α-Laplace distribution and processes

The Laplace distribution is considered as a better choice for modeling whenever data exhibit high kurtosis and heavier tails than Gaussian tails. Even though this is the case, not much work has been done on bivariate Laplace distribution. In this work, we introduce and study a new class of bivariate distributions called bivariate semi α-Laplace distribution, containing bivariate Laplace distributions. Three characterizations of bivariate semi α-Laplace distribution are obtained. Relation with bivariate semi stable distribution is established. An autoregressive model with bivariate semi α-Laplace marginal distributions is developed.     

Some Related Minima Stability and Minima Infinite Divisibility of the General Multivariate Pareto Distributions

This article studies the minima stable property of the general multivariate Pareto distributions MP^(k)(I), MP^(k)(II), MP^(k)(III), MP^(k)(IV) which can be applied to characterize the MP^(k) distribution via its weighted ordered coordinates minima and marginal distribution. Also, the multivariate semi-Pareto distribution (denoted by MSP) is discerned in the class of geometric minima infinite divisible and geometric minima stable distributions. If the exponent measure is satisfied by some functional equation, then the geometric minima stable property can be used to characterize the MSP distribution. Finally, the finite sample minima infinite divisible property of the MP^(k)(I), (II), and (IV) distributions is also discussed.     

A Note on the Complementary Mixture Pareto II Distribution

We introduce a new survival distribution, of Pareto type, that arises from a cure-mixture frailty model. We describe its properties and demonstrate connections with familiar distributions including the Pareto and exponential. We derive its characteristic function and moments.     

A Marshall–Olkin family of heavy-tailed distributions which includes the lognormal one

Abstract

A new class of heavy-tailed distribution functions,, containing the lognormal distribution as a particular case is introduced. The class thus obtained depends on a set of three parameters, incorporating an additional distribution to the classical lognormal one. This new class of heavy-tailed distribution is presented as an alternative to other useful heavy-tailed distributions, such as the lognormal, Weibull, and Pareto distributions. The density and distribution functions of this new class are given by a closed expression which allows us to easily compute probabilities, quantiles, moments, and related measurements. Finally, some applications are shown as examples.     

Properties and Inference for a New Class of Skew-t Distributions

In this article we introduce a new generalization of skew-t distributions, which contains the standard skew-t distribution, as a special case. This new class of distributions is an adequate model for modeling some dataset rather than the standard skew-t distributions. This kind of distributions can be represented as a scale-shape mixture of the extended skew-normal distributions. The main properties of this family of distributions are studied and a recurrence relation for the cumulative distribution functions (cdf) of them is presented. We derive the distribution of the order statistics from the trivariate exchangeable t-distribution in terms of our distribution and then an exact expression for the cdf of order statistics is derived. Likelihood inference for this distribution is also examined. The method is illustrated with a numerical example via a simulation study.     

On a class of distributions on the simplex

In the present paper we define and investigate a novel class of distributions on the simplex, termed normalized infinitely divisible distributions, which includes the Dirichlet distribution. Distributional properties and general moment formulae are derived. Particular attention is devoted to special cases of normalized infinitely divisible distributions which lead to explicit expressions. As a by-product new distributions over the unit interval and a generalization of the Bessel function distribution are obtained.     

On detecting outliers in the Pareto distribution

In this paper, we introduce two new statistics for detecting outliers in the Pareto distribution. These new statistics are the extension of the statistics for detecting outliers in exponential and gamma distributions. In fact, we compare the power of our test statistics with the other statistics and select the best test statistic for detecting outliers in the Pareto distribution. Finally, numerical examples of different insurance claims are used to see the performance of the test.     

The Kumaraswamy skew-t distribution and its related properties

Skew normal distribution is an alternative distribution to the normal distribution to accommodate asymmetry. Since then extensive studies have been done on applying Azzalini’s skewness mechanism to other well-known distributions, such as skew-t distribution, which is more flexible and can better accommodate long tailed data than the skew normal one. The Kumaraswamy generalized distribution (Kw ? F) is another new class of distribution which is capable of fitting skewed data that can not be fitted well by existing distributions. Such a distribution has been widely studied and various versions of generalization of this distribution family have been introduced. In this article, we introduce a new generalization of the skew-t distribution based on the Kumaraswamy generalized distribution. The new class of distribution, which we call the Kumaraswamy skew-t (KwST) has the ability of fitting skewed, long, and heavy-tailed data and is more flexible than the skew-t distribution as it contains the skew-t distribution as a special case. Related properties of this distribution family such as mathematical properties, moments, and order statistics are discussed. The proposed distribution is applied to a real dataset to illustrate the estimation procedure.     

A skew extension of the slash distribution via beta-normal distribution

In this work we introduce a generalization of the slash distribution using beta-normal distribution. This newly defined generalization is more flexible than the ordinary slash distribution and contains distributions that can be not only symmetric and unimodal, but also asymmetric and bimodal. We study the properties of the new generalized distribution and demonstrate its use on some real data sets considering maximum likelihood estimation procedure.     

Absolute continuous bivariate generalized exponential distribution

Generalized exponential distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma distributions. In this paper we introduce an absolute continuous bivariate generalized exponential distribution by using a simple transformation from a well known bivariate exchangeable distribution. The marginal distributions of the proposed bivariate generalized exponential distributions are generalized exponential distributions. The joint probability density function and the joint cumulative distribution function can be expressed in closed forms. It is observed that the proposed bivariate distribution can be obtained using Clayton copula with generalized exponential distribution as marginals. We derive different properties of this new distribution. It is a five-parameter distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.     

A new flexible hazard rate distribution: Application and estimation of its parameters

We introduce a new class of flexible hazard rate distributions which have constant, increasing, decreasing, and bathtub-shaped hazard function. This class of distributions obtained by compounding the power and exponential hazard rate functions, which is called the power-exponential hazard rate distribution and contains several important lifetime distributions. We obtain some distributional properties of the new family of distributions. The estimation of parameters is obtained by using the maximum likelihood and the Bayesian methods under squared error, linear-exponential, and Stein’s loss functions. Also, approximate confidence intervals and HPD credible intervals of parameters are presented. An application to real dataset is provided to show that the new hazard rate distribution has a better fit than the other existing hazard rate distributions and some four-parameter distributions. Finally , to compare the performance of proposed estimators and confidence intervals, an extensive Monte Carlo simulation study is conducted.     

Canonical expansions,correlation structure,and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.     

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.     

On characterizations of the generalized Pareto distributions based on progressively censored order statistics

It is well-known, in the literature, that most of the characterization results on exponential distribution are based on the solution of Cauchy functional equation and integrated Cauchy functional equation. In the present paper, we consider the functional equation $$F(x) = F(xy) + F(xQ(y)), \quad x, xQ(y) \in [0, \theta),\; y \in [0,1],$$ where F and Q satisfy certain conditions, to give some new characterization results on the generalized Pareto distributions based on progressively Type-II right censored order statistics. We prove the main results without restricting to distributions that are absolutely continuous with respect to Lebesgue measure.     

A new family of lifetime models

We introduce a new family of distributions by adding a parameter to the Marshall–Olkin family of distributions. Some properties of the new family of distributions are derived. A particular case of the family, a three-parameter generalization of the exponential distribution, is given special attention. The shape properties, moments, distributions of the order statistics, entropies and estimation procedures are derived. An application to a real data set is discussed.     

A Sieve model for extreme values

From the class of extreme value distributions, we focus on the set of heavy-tailed distributions which produce low-frequency, high-cost events. The regular Pareto distribution is the basic model of choice, being the simplest heavy-tailed distribution. Real data suggest that modifications of the Pareto distribution may be a better fit; an alternative model is the truncated Pareto distribution (TPD). For further study, this paper proposed a TPD Sieve class of distributions. The properties and estimation on the Sieve class are also discussed. We fit the models to the largest Black Sea bass caught in Buzzard's Bay, MA, USA and the costliest Atlantic hurricanes from 1900 to 2005. Using measures of model adequacy, the TPD Sieve model is generally found to be the best-fitting model.     

Weighted Marshall–Olkin bivariate exponential distribution

Recently, Gupta and Kundu [R.D. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (2009), pp. 621–634] have introduced a new class of weighted exponential (WE) distributions, and this can be used quite effectively to model lifetime data. In this paper, we introduce a new class of weighted Marshall–Olkin bivariate exponential distributions. This new singular distribution has univariate WE marginals. We study different properties of the proposed model. There are four parameters in this model and the maximum-likelihood estimators (MLEs) of the unknown parameters cannot be obtained in explicit forms. We need to solve a four-dimensional optimization problem to compute the MLEs. One data set has been analysed for illustrative purposes and finally we propose some generalization of the proposed model.     

Geeta distribution and its properties

A new discrete distribution defined over all the positive integers and with the name of Geeta distribution is described. It is L-shaped like the logarithmic series distribution, Yule distribution and the discrete Pareto distribution but is far more versatile than them as it has two parameters. It belongs to the classes of location parameter distributions, modified power series distributions, Lagrange series distributions and exponential distributions. Its mean fi, variance a2 and two recurrence formulae for higher central moments are obtained. Convolution theorem and variations in the model with changes in the parameters have been considered. ML estimators, MVU estimators and estimators based of mean and variance and on mean and first frequency have been derived.     

A new four-parameter lifetime distribution

Generalizing lifetime distributions is always precious for applied statisticians. In this paper, we introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley geometric (EPLG) distribution, obtained by compounding EPL and geometric distributions. The new distribution arises in a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the maximum lifetime value among all risks. The distribution exhibits decreasing, increasing, unimodal and bathtub-shaped hazard rate functions, depending on its parameters. It contains several lifetime distributions as particular cases: EPL, new generalized Lindley, generalized Lindley, power Lindley and Lindley geometric distributions. We derive several properties of the new distribution such as closed-form expressions for the density, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Simulation studies are also provided. Finally, two real data applications are given for showing the flexibility and potentiality of the new distribution.