Flexible univariate and multivariate models based on hidden truncation

A broad spectrum of flexible univariate and multivariate models can be constructed by using a hidden truncation paradigm. Such models can be viewed as being characterized by a basic marginal density, a family of conditional densities and a specified hidden truncation point, or points. The resulting class of distributions includes the basic marginal density as a special case (or as a limiting case), but also includes an array of models that may unexpectedly include many well known densities. Most of the well known skew-normal models (developed from the seed distribution popularized by Azzalini [(1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12(2), 171–178]) can be viewed as being products of such a hidden truncation construction. However, the many hidden truncation models with non-normal component densities undoubtedly deserve further attention.     

Finite mixtures of multivariate Poisson distributions with application

In the present paper we examine finite mixtures of multivariate Poisson distributions as an alternative class of models for multivariate count data. The proposed models allow for both overdispersion in the marginal distributions and negative correlation, while they are computationally tractable using standard ideas from finite mixture modelling. An EM type algorithm for maximum likelihood (ML) estimation of the parameters is developed. The identifiability of this class of mixtures is proved. Properties of ML estimators are derived. A real data application concerning model based clustering for multivariate count data related to different types of crime is presented to illustrate the practical potential of the proposed class of models.     

Multivariate Matsumoto–Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2M-X$2M-X$ theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Weso?owski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].     

Stochastic comparisons of multivariate frailty models

A multivariate frailty model in which survival function depends on baseline distributions of components and the frailty random variable is considered. Since misspecification in choice of frailty distribution and/or baseline distribution may affect the distribution of multivariate frailty model, using theory of stochastic orders, we compare multivariate frailty models arising from different choices of frailty distribution.     

IG-symmetry and R-symmetry: Interrelations and applications to the inverse Gaussian theory

The two parameter inverse Gaussian (IG) distribution is often more appropriate and convenient for modelling and analysis of nonnegative right skewed data than the better known and now ubiquitous Gaussian distribution. Its convenience stems from its analytic simplicity and the striking similarities of its methodologies with those employed with the normal theory models. These, known as the G–IG analogies, include the concepts and measures of IG-symmetry, IG-skewness and IG-kurtosis, the IG-analogues of the corresponding classical notions and measures. The new IG-associated entities, although well defined and mathematically transparent, are intuitively and conceptually opaque. In this paper, we first elaborate the importance of the IG distribution and of the G–IG analogies. Then we consider the IG-related root-reciprocal IG (RRIG) distribution and introduce a physically transparent, conceptually clear notion of reciprocal symmetry (R-symmetry) and use it to explain the IG-symmetry. We study the moments and mixture properties of the R-symmetric distributions and the relationship of R-symmetry with IG-symmetry and note that RRIG distribution provides a link, in addition to Tweedie's Laplace transform link, between the Gaussian and inverse Gaussian distributions. We also give a structural characterization of the unimodal R-symmetric distributions. This work further expands the long list of G–IG analogies. Several applications including product convolution, monotonicity of power functions, peakedness and monotone limit theorems of R-symmetry are outlined.     

On characterizations of the logistic distribution

We modify and extend George and Mudholkar's [1981. A characterization of the logistic distribution by a sample median. Ann. Inst. Statist. Math. 33, 125–129] characterization result about the logistic distribution, which is in terms of the sample median and Laplace distribution. Moreover, we give some new characterization results in terms of the smallest order statistics and the exponential distribution.     

Normal approximation to the hypergeometric distribution in nonstandard cases and a sub-Gaussian Berry–Esseen theorem

In this paper, we consider simple random sampling without replacement from a dichotomous finite population. We investigate accuracy of the Normal approximation to the Hypergeometric probabilities for a wide range of parameter values, including the nonstandard cases where the sampling fraction tends to one and where the proportion of the objects of interest in the population tends to the boundary values, zero and one. We establish a non-uniform Berry–Esseen theorem for the Hypergeometric distribution which shows that in the nonstandard cases, the rate of Normal approximation to the Hypergeometric distribution can be considerably slower than the rate of Normal approximation to the Binomial distribution. We also report results from a moderately large numerical study and provide some guidelines for using the Normal approximation to the Hypergeometric distribution in finite samples.     

A generalization of an identity of hoeffding and some applications

Summary We generalize a well-known identity due to Hoeffding and use this generalization to prove a result of Cambanis, Simons and Stout under somewhat different hypotheses and to extend some results of Lehmann concerning bivariate distributions with quadrant dependence.     

A multivariate version of Gini's rank association coefficient

In this paper, we introduce a multivariate generalization of the population version of Gini's rank association coefficient, giving a response to this open question posed in [4]. We also study some properties of this version, present the corresponding results for the sample statistic, and provide several examples.     

A bivariate distribution with Lomax and geometric margins

We develop a stochastic model describing the joint distribution of $(X,N)$, where $N$ has a geometric distribution while $X$ is the sum of $N$ dependent, heavy-tail Pareto components. Models of this form arise in many applications, ranging from hydro-climatology to finance and insurance. We present fundamental properties of this vector, including marginal and conditional distributions, moments, representations, and parameter estimation. We also include an example from finance, illustrating modeling potential of this new bivariate distribution.     

A set of independent sequential residuals for the multivariate regression model

In this paper a set of residuals for the multivariate linear regression model is introduced. These residuals are shown to be independent with known distributions which do not depend on the parameters of the model. Transformations of the mentioned residuals may be used to construct exact α goodness-of-fit tests for the multivariate regression model.     

Characterizations based on record values and order statistics

Consider a sequence of independent and identically distributed random variables {_Xi,i?1}$\{X_i,i?1\}$ with a common absolutely continuous distribution function F  . Let _X1:n?_X2:n???_Xn:n$X_{1:n}?X_{2:n}???X_{n:n}$ be the order statistics of {_X1,_X2,…,_Xn}$\{X_1,X_2,\dots ,X_n\}$ and {_Yl,l?1}$\{Y_l,l?1\}$ be the sequence of record values generated by {_Xi,i?1}$\{X_i,i?1\}$. In this work, the conditional distribution of _Yl$Y_l$ given _Xn:n$X_{n:n}$ is established. Some characterizations of F   based on record values and _Xn:n$X_{n:n}$ are then given.     

Multivariate inverse Gaussian distribution as a limit of multivariate waiting time distributions

Multivariate inverse Gaussian distribution proposed by Minami [2003. A multivariate extension of inverse Gaussian distribution derived from inverse relationship. Commun. Statist. Theory Methods 32(12), 2285–2304] was derived through multivariate inverse relationship with multivariate Gaussian distributions and characterized as the distribution of the location at a certain stopping time of a multivariate Brownian motion. In this paper, we show that the multivariate inverse Gaussian distribution is also a limiting distribution of multivariate Lagrange distributions, which is a family of waiting time distributions, under certain conditions.     

Characterizations of exponentiality within the HNBUE class and related tests

A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential.     

Some properties of the generalized TTT transform

Recently Li and Shaked [2007. A general family of univariate stochastic orders. J. Statist. Plann. Inference 137, 3601–3610] introduced the generalized total time on test (GTTT) transform with respect to a given function ?$?$. In this paper we study some properties of it which are related with stochastic orderings. A concept of Lehmann and Rojo [1992. Invariant directional orderings. Ann. Statist. 20, 2100–2110] is applied to a new setting and the GTTT transform is used to define invariance properties and distances of some stochastic orders. Iterations of the GTTT transforms are also studied and their relations with exponential mixtures of gamma distributions are established.     

Characterizations of distributions through coincidences of semiparametric families

Semiparametric families are families that have both a real parameter and a parameter that is itself a distribution. A number of semiparametric families suitable for lifetime data are introduced: scale, power, frailty (proportional hazards), age, moment, Laplace transform, and convolution parameter families. The coincidence of two families provides a characterization of the underlying distribution. Characterizations of the Weibull, gamma, lognormal, and Gompertz distributions are obtained.     

Some properties of extreme stable laws and related infinitely divisible random variables

In the course of studying the moment sequence {nⁿ:n=0,1,…}$\{n^n:n=0,1,\dots \}$, Eaton et al. [1971. On extreme stable laws and some applications. J. Appl. Probab. 8, 794–801] have shown that this sequence, which is, indeed, the moment sequence of a log-extreme stable law with characteristic exponent γ=1$\gamma =1$, corresponds to a scale mixture of exponential distributions and hence to a distribution with decreasing failure rate. Following essentially the approach of Shanbhag et al. [1977. Some further results in infinite divisibility. Math. Proc. Cambridge Philos. Soc. 82, 289–295] we show that, under certain conditions, log-extreme stable laws with characteristic exponent γ∈[1,2)$\gamma \in [1,2)$ are scale mixtures of exponential distributions and hence are infinitely divisible and have decreasing failure rates. In addition, we study the moment problem associated with the log-extreme stable laws with characteristic exponent γ∈(0,2]$\gamma \in (0,2]$ and throw further light on the existing literature on the subject. As a by-product, we show that generalized Poisson and generalized negative binomial distributions are mixed Poisson distributions. Finally, we address some relevant questions on structural aspects of infinitely divisible distributions, and make new observations, including in particular that certain results appearing in Steutel and van Harn [2004. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York] have links with the Wiener–Hopf factorization met in the theory of random walk.