Cumulant Plots for Assessing the Gamma Distribution

This article introduces graphical procedures for assessing the fit of the gamma distribution. The procedures are based on a standardized version of the cumulant generating function. Plots with bands of 95% simultaneous confidence level are developed by utilizing asymptotic and finite-sample results. The plots have linear scales and do not rely on the use of tables or values of special functions. Further, it is found through simulation, that the goodness-of-fit test implied by these plots compares favorably with respect to power to other known tests for the gamma distribution in samples drawn from lognormal and inverse Gaussian distributions.     

On the Correlation Structure of Gaussian Copula Models for Geostatistical Count Data

We describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero‐inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.     

A Note on Distinguishing Random Trees Populations

ABSTRACT

Several new results are presented for a class of univariate distributions for which the maximum likelihood estimate of the population mean is the sample mean. It is shown that the convolution of any two such distributions also belongs to this class of functions. It is also shown that the marginal distribution for the sample mean captures all of the Fisher information for the population mean contained in the full distribution. Parameters orthogonal to the mean are found for special cases of these distributions. If the distribution is conditioned on the sample mean, the conditional distribution depends on the parameters only through parameters orthogonal to the mean.     

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.     

A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

Abstract

We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Other properties such as reproductivity and infinite divisibility are also given.     

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.     

Multivariate Bayesian variable selection and prediction

The multivariate regression model is considered with p regressors. A latent vector with p binary entries serves to identify one of two types of regression coefficients: those close to 0 and those not. Specializing our general distributional setting to the linear model with Gaussian errors and using natural conjugate prior distributions, we derive the marginal posterior distribution of the binary latent vector. Fast algorithms aid its direct computation, and in high dimensions these are supplemented by a Markov chain Monte Carlo approach to sampling from the known posterior distribution. Problems with hundreds of regressor variables become quite feasible. We give a simple method of assigning the hyperparameters of the prior distribution. The posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infrared absorbances as regressors.     

An extended Lomax distribution

A new five-parameter continuous distribution, the so-called McDonald Lomax distribution, that extends the Lomax distribution and some other distributions is proposed and studied. The model has as special sub-models new four- and three-parameter distributions. Various structural properties of the new distribution are derived, including expansions for the density function, explicit expressions for the moments, generating and quantile functions, mean deviations and Rényi entropy. The score function is derived and the estimation is performed by maximum likelihood. We also obtain the observed information matrix. An application illustrates the usefulness of the proposed model.     

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.     

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.     

The bivariate alpha-skew-normal distribution

In this paper, we propose a new bivariate distribution, namely bivariate alpha-skew-normal distribution. The proposed distribution is very flexible and capable of generalizing the univariate alpha-skew-normal distribution as its marginal component distributions; it features a probability density function with up to two modes and has the bivariate normal distribution as a special case. The joint moment generating function as well as the main moments are provided. Inference is based on a usual maximum-likelihood estimation approach. The asymptotic properties of the maximum-likelihood estimates are verified in light of a simulation study. The usefulness of the new model is illustrated in a real benchmark data.     

A bimodal flexible distribution for lifetime data

ABSTRACT

A four-parameter extended bimodal lifetime model called the exponentiated log-sinh Cauchy distribution is proposed. It extends the log-sinh Cauchy and folded Cauchy distributions. We derive some of its mathematical properties including explicit expressions for the ordinary moments and generating and quantile functions. The method of maximum likelihood is used to estimate the model parameters. We implement the fit of the model in the GAMLSS package and provide the codes. The flexibility of the model is illustrated by means of three real data sets.     

Relative Price Changes and Inequality in the Size Distribution of Various Components of Income

This article uses a comprehensive model of economic inequality to examine the impact of relative price changes on inequality in the marginal distributions of various income components in which the marginal distributions are derived from a multidimensional joint distribution. The multidimensional joint distribution function is assumed to be a member of the Pearson Type VI family; that is, it is assumed to be a beta distribution of the second kind. The multidimensional joint distribution is so called because it is a joint distribution of components of income and expenditures on various commodity groups. Gini measures of inequality are devised from the marginal distributions of the various income components. The inequality measures are shown to depend on the parameters of the multidimensional joint distribution. It is then shown that the parameters of the multidimensional joint distribution depend on the relative prices of various commodity groups and several other specified exogenous variables. Thus, knowledge of how changes in relative prices affect the parameters of the multidimensional joint distribution is deductively equivalent to knowledge of how changes in relative prices affect inequality in the marginal distributions of various components of income. It is found that relative price changes have a statistically significant impact on inequality in various components of income.     

The Kumaraswamy normal linear regression model with applications

ABSTRACT

For any continuous baseline G distribution, Cordeiro and Castro pioneered the Kumaraswamy-G family of distributions with two extra positive parameters, which generalizes both Lehmann types I and II classes. We study some mathematical properties of the Kumaraswamy-normal (KwN) distribution including ordinary and incomplete moments, mean deviations, quantile and generating functions, probability weighted moments, and two entropy measures. We propose a new linear regression model based on the KwN distribution, which extends the normal linear regression model. We obtain the maximum likelihood estimates of the model parameters and provide some diagnostic measures such as global influence, local influence, and residuals. We illustrate the potentiality of the introduced models by means of two applications to real datasets.     

Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of the saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We also demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.     

Number of appearances of events in random sequences: A new approach to non-overlapping runs

ABSTRACT

Distributions of non-overlapping runs in multistate systems are derived using a new generating function (GF) approach, which has been applied previously mainly to exact length runs. By utilizing this method, univariate and multivariate joint distributions of non-overlapping runs are derived in a unified way. The GF approach also leads naturally to the asymptotic Gaussian distributions, with the mean, variance, and covariance linear in the size of the system.     

General properties for the beta extended half-normal model

We formulate and study a four-parameter lifetime model called the beta extended half-normal distribution. This model includes as sub-models the exponential, extended half-normal and half-normal distributions. We derive expansions for the new density function which do not depend on complicated functions. We obtain explicit expressions for the moments and incomplete moments, generating function, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. In addition, the model parameters are estimated by maximum likelihood. We provide the observed information matrix. The new model is modified to cope with possible long-term survivors in the data. The usefulness of the new distribution is shown by means of two real data sets.     

An extended Gompertz-Makeham distribution with application to lifetime data

In this paper, we propose an extension of the Gompertz-Makeham distribution. This distribution is called the transmuted Gompertz-Makeham (TGM). The new model which can handle bathtub-shaped, increasing, increasing-constant and constant hazard rate functions. This property makes TGM is useful in survival analysis. Various statistical and reliability measures of the model are obtained, including hazard rate function, moments, moment generating function (mgf), quantile function, random number generating, skewness, kurtosis, conditional moments, mean deviations, Bonferroni curve, Lorenz curve, Gini index, mean inactivity time, mean residual lifetime and stochastic ordering; we also obtain the density of the ith order statistic. Estimation of the model parameters is justified by the method of maximum likelihood. An application to real data demonstrates that the TGM distribution can provides a better fit than some other very well known distributions.     

An absolutely continuous bivariate Topp–Leone distribution: A useful model on a bounded domain

We introduce an absolutely continuous bivariate generalization of the Topp–Leone distribution, which is a special member of the proportional reversed hazard family using a one-parameter bivariate exchangeable distribution. We show that a copula approach could also be used in defining the bivariate Topp–Leone distribution. The marginal distributions of the new bivariate distribution have also Topp–Leone distributions. We study its distributional and dependence properties. We estimate the parameters by maximum-likelihood procedure, perform a simulation study on the estimators, and apply them to a real data set. Furthermore, we give a way of generating bivariate distributions using the proposed distribution.     

An extended Maxwell distribution: Properties and applications

In this article, we propose an extension of the Maxwell distribution, so-called the extended Maxwell distribution. This extension is evolved by using the Maxwell-X family of distributions and Weibull distribution. We study its fundamental properties such as hazard rate, moments, generating functions, skewness, kurtosis, stochastic ordering, conditional moments and moment generating function, hazard rate, mean and variance of the (reversed) residual life, reliability curves, entropy, etc. In estimation viewpoint, the maximum likelihood estimation of the unknown parameters of the distribution and asymptotic confidence intervals are discussed. We also obtain expected Fisher’s information matrix as well as discuss the existence and uniqueness of the maximum likelihood estimators. The EMa distribution and other competing distributions are fitted to two real datasets and it is shown that the distribution is a good competitor to the compared distributions.