The folded t distribution

Measurements are frequently recorded without their algebraicsign. As a consequence the underlying distribution of measurements is replaced by a distribution of absolute measurements. When the underlying distribution is t the resulting distribution is called the “folded-t distribution”. Here we study this distribution, we find the relationship between the folded-t distribution and a special case of the folded normal distribution and we derive relationships of the folded-t distribution to other distributions pertaining to computer generation. Also tables are presented which give areas of the folded-t distribution.     

A generalized skew slash distribution via gamma-normal distribution

In this article, we introduce a generalization of the slash distribution via the gamma-normal distribution. We define the new slash distribution by relation of a gamma-normal random variable with respect to a power of a uniform random variable. The newly defined distribution generalizes the slash distribution and is more flexible in terms of its kurtosis and skewness than the slash distribution. Basic properties of the new distribution are studied. We derive the maximum likelihood estimators of its parameters and apply the distribution to a real dataset.     

On nonparametric Bayesian inference for the distribution of a random sample

The nonparametric Bayesian approach for inference regarding the unknown distribution of a random sample customarily assumes that this distribution is random and arises through Dirichlet-process mixing. Previous work within this setting has focused on the mean of the posterior distribution of this random distribution, which is the predictive distribution of a future observation given the sample. Our interest here is in learning about other features of this posterior distribution as well as about posteriors associated with functionals of the distribution of the data. We indicate how to do this in the case of linear functionals. An illustration, with a sample from a Gamma distribution, utilizes Dirichlet-process mixtures of normals to recover this distribution and its features.     

The effect of mixing-distribution misspecification in conjugate mixture models

Parametric mixture models are commonly used in the analysis of clustered data. Parametric families are specified for the conditional distribution of the response variable given a cluster-specific effect, and for the marginal distribution of the cluster-specific effects. This latter distribution is referred to as the mixing distribution. If the form of the mixing distribution is misspecified, then Bayesian and maximum-likelihood estimators of parameters associated with either distribution may be inconsistent. The magnitude of the asymptotic bias is investigated, using an approximation based on infinitesimal contamination of the mixing distribution. The approximation is useful when there is a closed-form expression for the marginal distribution of the response under the assumed mixing distribution, but not under the true mixing distribution. Typically this occurs when the assumed mixing distribution is conjugate, meaning that the conditional distribution of the cluster-specific parameter given the response variable belongs to the same parametric family as the mixing distribution.     

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.     

An alternative multivariate skew Laplace distribution: properties and estimation

A special case of the multivariate exponential power distribution is considered as a multivariate extension of the univariate symmetric Laplace distribution. In this paper, we focus on this multivariate symmetric Laplace distribution, and extend it to a multivariate skew distribution. We call this skew extension of the multivariate symmetric Laplace distribution the “multivariate skew Laplace (MSL) distribution” to distinguish between the asymmetric multivariate Laplace distribution proposed by Kozubowski and Podgórski (Comput Stat 15:531–540, 2000a) Kotz et al. (The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Chap. 6. Birkhäuser, Boston, 2001) and Kotz et al. (An asymmetric multivariate Laplace Distribution, Working paper, 2003). One of the advantages of (MSL) distribution is that it can handle both heavy tails and skewness and that it has a simple form compared to other multivariate skew distributions. Some fundamental properties of the multivariate skew Laplace distribution are discussed. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of the multivariate skew Laplace distribution is given. Some examples are provided to demonstrate the modeling strength of the skew Laplace distribution.     

A Lagrangian Non Central Negative Binomial Distribution of the First Kind

A Lagrangian probability distribution of the first kind is proposed. Its probability mass function is expressed in terms of generalized Laguerre polynomials or, equivalently, a generalized hypergeometric function. The distribution may also be formulated as a Charlier series distribution generalized by the generalizing Consul distribution and a non central negative binomial distribution generalized by the generalizing Geeta distribution. This article studies formulation and properties of the distribution such as mixture, dispersion, recursive formulas, conditional distribution and the relationship with queuing theory. Two illustrative examples of application to fitting are given.     

Bivariate discrete generalized exponential distribution

In this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper.     

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.     

Power-normal distribution

Recently, Gupta and Gupta [Analyzing skewed data by power-normal model, Test 17 (2008), pp. 197–210] proposed the power-normal distribution for which normal distribution is a special case. The power-normal distribution is a skewed distribution, whose support is the whole real line. Our main aim of this paper is to consider bivariate power-normal distribution, whose marginals are power-normal distributions. We obtain the proposed bivariate power-normal distribution from Clayton copula, and by making a suitable transformation in both the marginals. Lindley–Singpurwalla distribution also can be used to obtain the same distribution. Different properties of this new distribution have been investigated in detail. Two different estimators are proposed. One data analysis has been performed for illustrative purposes. Finally, we propose some generalizations to multivariate case also along the same line and discuss some of its properties.     

The complementary beta distribution

The complementary beta distribution is proposed as a new distribution on the unit interval. It results from reversing the roles of the distribution and quantile functions of the beta distribution. It has some attractive properties that are complementary to those of the beta distribution. In particular, the complementary beta distribution is much more amenable than the beta distribution to exact computations involving expectations of order statistics, including L-moments. At least for a wide range of parameter values, complementary beta and beta distributions with parameters that are reciprocals of the other's parameters are good approximations to one another. We also note the position of the complementary beta distribution in a wider family of distributions defined through the same simple form for their quantile density functions.     

The power of the circular cone test: A noncentral chi-bar-squared distribution

Pincus (1975) derived the null distribution of the likelihood-ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood-ratio test statistic has a chi-bar-squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood-ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi-bar-squared” distribution. To our knowledge, this is the first order-restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear-model setting. That is, the distribution of the likelihood-ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.     

The Folded Logistic Distribution

ABSTRACT

Physical measurements like dimensions, including time, and angles in scientific experiments are frequently recorded without their algebraic sign. The directions of those physical quantities measured with respect to a frame of reference in most practical applications are considered to be unimportant and are ignored. As a consequence, the underlying distribution of measurements is replaced by a distribution of absolute measurements. When the underlying distribution is logistic, the resulting distribution is called the “folded logistic distribution”. Here, the properties of the folded logistic distribution will be presented and the techniques for estimating parameters will be given. The advantages of using this folded logistic distribution over the folded normal distribution will be discussed and some examples will be cited.     

Modified slash distribution

In this paper we introduce a modified slash distribution obtained by modifying the usual slash distribution. This new distribution is based on the quotient of two independent random variables, whose distributions are the normal and the power of an exponential distribution of scale parameter equals to two, respectively. In this way, the result is a new distribution whose kurtosis values are greater when compared with that of the slash distribution. We study the density, some properties, moments, kurtosis and make inferences by the method of moments and maximum likelihood. We introduce a multivariate version of this new distribution. Moreover, we provide two illustrations with real data showing that the new distribution fits better the data than the ordinary slash distribution.     

A LIMIT THEOREM FOR THE GALTON-WATSON PROCESS WITH IMMIGRATION

It is difficult, in general, to optain an explicit expression for the limiting-stationary distribution, when such a distribution exists, of the process in which teh individuals reproduce as in a Galton-Wastson process, but are also subject to an independent immigration component at each generation. The main result of this paper is a limit theorem which suggests a means of approximating this distribution by a gamma density, when the mean of the offspring distribution is less than, but close to, unity. Following along the same lines, it is easy to show that a similar limit theorem holds for the asymptotic conditional limit distribution of an ordinary subcritical Galton-Watson process, whereby this distribution approaches the exponential as the offspring mean approaches unity.     

Marshall-Olkin extended weibull distribution and its application to censored data

In this paper we show that the Marshall-Olkin extended Weibull distribution can be obtained as a compound distribution with mixing exponential distribution. In addition, we provide simple sufficient conditions for the shape of the hazard rate function of the distribution. Moreover, we extend the considered distribution to accommodate randomly right censored data. Finally, application of the extended distribution to a data set representing the remission times of bladder cancer patients is given and its goodness-of-fit is demonstrated.     

Power binomial exponential distribution: Modeling,simulation and application

ABSTRACT

The binomial exponential 2 (BE2) distribution was proposed by Bakouch et al. as a distribution of a random sum of independent exponential random variables, when the sample size has a zero truncated binomial distribution. In this article, we introduce a generalization of BE2 distribution which offers a more flexible model for lifetime data than the BE2 distribution. The hazard rate function of the proposed distribution can be decreasing, increasing, decreasing–increasing–decreasing and unimodal, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties and parameters estimation of the distribution are investigated. Three different algorithms are proposed for generating random data from the new distribution. Two real data applications regarding the strength data and Proschan's air-conditioner data are used to show that the new distribution is better than the BE2 distribution and some other well-known distributions in modeling lifetime data.     

A full likelihood procedure of exchangeable negative binomials for modelling correlated and overdispersed count data

This paper introduces an exchangeable negative binomial distribution resulting from relaxing the independence of the Bernoulli sequence associated with a negative binomial distribution to exchangeability. It is demonstrated that the introduced distribution is a mixture of negative binomial distributions and can be characterized by infinitely many parameters that form a completely monotone sequence. The moments of the distribution are derived and a small simulation is conducted to illustrate the distribution. For data analytic purposes, two methods, truncation and completely-monotone links, are given for converting the saturated distribution of infinitely many parameters to parsimonious distributions of finitely many parameters. A full likelihood procedure is described which can be used to investigate correlated and overdispersed count data common in biomedical sciences and teratology. In the end, the introduced distribution is applied to analyze a real clinical data of burn wounds on patients.     

Distribution of the sum of independent generalized logarithmic series variables

Jain and Gupta (1973) have given a generalized logarithmic series distribution which, for β = 1, reduces to the logarithmic series distribution. In this note we obtain the distribution of the sum of independent generalized logarithmic series variables. This distribution conforms, in a special case, to the First-type Stirling distribution (Patil and Wani, 1965) and would be useful in estimation theory.     

On a Skew Bimodal Normal–Normal distribution fitted to the Old-Faithful geyser data

The Bimodal Normal distribution introduced by Alavi (2011) is a symmetric distribution where its variance is three times the variance of the corresponding normal distribution. Azzalini (1985) introduced the univariate Skew Normal distribution to model asymmetry data. In this paper the Skew Bimodal Normal–Normal distribution is introduced as a skew-symmetric distribution generated by the cumulative function of standard normal. Some properties of the distribution and some methods for generating data from this distribution are introduced. The maximum likelihood estimation of parameters is obtained. The distribution is fitted to the Old Faithful Geyser data.