Models for Dependent Extremes Using Stable Mixtures

Abstract.  This paper unifies and extends results on a class of multivariate extreme value (EV) models studied by Hougaard, Crowder and Tawn. In these models, both unconditional and conditional distributions are themselves EV distributions, and all lower-dimensional marginals and maxima belong to the class. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential-stable location mixtures (for Gumbel) or as power-stable scale mixtures (for non-Gumbel EV distributions). A third interpretation is through a peaks over thresholds model with a positive stable intensity. The mixing variables are used as a modelling tool and for better understanding and model checking. We study EV analogues of components of variance models, and new time series, spatial and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation.     

Nonparametric Estimation of Distribution Functions of Nonstandard Mixtures

ABSTRACT

Nonstandard mixtures are those that result from a mixture of a discrete and a continuous random variable. They arise in practice, for example, in medical studies of exposure. Here, a random variable that models exposure might have a discrete mass point at no exposure, but otherwise may be continuous. In this article we explore estimating the distribution function associated with such a random variable from a nonparametric viewpoint. We assume that the locations of the discrete mass points are known so that we will be able to apply a classical nonparametric smoothing approach to the problem. The proposed estimator is a mixture of an empirical distribution function and a kernel estimate of a distribution function. A simple theoretical argument reveals that existing bandwidth selection algorithms can be applied to the smooth component of this estimator as well. The proposed approach is applied to two example sets of data.     

Finite normal mixture copulas for multivariate discrete data modeling

A new family of copulas is introduced that provides flexible dependence structure while being tractable and simple to use for multivariate discrete data modeling. The construction exploits finite mixtures of uncorrelated normal distributions. Accordingly, the cumulative distribution function is simply the product of univariate normal distributions. At the same time, however, the mixing operation introduces association. The properties of the new family of copulas are examined and a concrete application is used to show its applicability.     

Gaussian copula distributions for mixed data,with application in discrimination

The construction of a joint model for mixed discrete and continuous random variables that accounts for their associations is an important statistical problem in many practical applications. In this paper, we use copulas to construct a class of joint distributions of mixed discrete and continuous random variables. In particular, we employ the Gaussian copula to generate joint distributions for mixed variables. Examples include the robit-normal and probit-normal-exponential distributions, the first for modelling the distribution of mixed binary-continuous data and the second for a mixture of continuous, binary and trichotomous variables. The new class of joint distributions is general enough to include many mixed-data models currently available. We study properties of the distributions and outline likelihood estimation; a small simulation study is used to investigate the finite-sample properties of estimates obtained by full and pairwise likelihood methods. Finally, we present an application to discriminant analysis of multiple correlated binary and continuous data from a study involving advanced breast cancer patients.     

Computational procedures for deriving sampling distributions of attribute-contaminated random variables

Statistical distributions generated from any J- or U-shaped random variables are cumbersome to derive if not completely indefinable and thus are unavailable analytically because of the singularities at the tails of the basic random variable. This paper presents a computational method for providing a numerical convolution derived from a basic U-shaped random variable composed of a continuous part mixed with (or contaminated by) a discrete part at the tails. The J-shaped sampling distribution case is implied as a special case. Though the computations are based on a background Normal Distribution, it can be generalized on any other distribution.Such distributions will open up an area of sampling distributions of mixed random variables that are not elaborately covered in textbooks dealing with the theory of distributions.     

Reliability properties of generalized mixtures of Weibull distributions with a common shape parameter

Weibull mixtures have been considered in many applied problems, and they have also been generalized by allowing negative mixing weights. In this paper, we study the classification of the aging properties of generalized mixtures of two or three Weibull distributions in terms of the mixing weights, scale parameters and a common shape parameter, which extends the cases of exponential or Rayleigh distributions. We apply these general results to classify the aging properties of the minimum and maximum lifetimes of two-component systems whose component lifetimes follow one of the known bivariate Weibull distributions.     

Importance of the prior mass for agreement between frequentist and Bayesian approaches in the two-sided test

We study distributional properties of generalized order statistics (gos) related by a random shift or scaling scheme in the continuous and discrete case, respectively. In the continuous case, we obtain new characterizations of distributions relating non-neighbouring gos extending some results given in the literature for the neighbouring cases. On the other hand, in the discrete case, we investigate the existence and uniqueness of a discrete parent distribution supported on the integers whose gos are related by a random translation.     

Improved Measures of the Spread of Data for Some Unknown Complex Distributions Using Saddlepoint Approximations

Measures of the spread of data for random sums arise frequently in many problems and have a wide range of applications in real life, such as in the insurance field (e.g., the total claim size in a portfolio). The exact distribution of random sums is extremely difficult to determine, and normal approximation usually performs very badly for this complex distributions. A better method of approximating a random-sum distribution involves the use of saddlepoint approximations.

Saddlepoint approximations are powerful tools for providing accurate expressions for distribution functions that are not known in closed form. This method not only yields an accurate approximation near the center of the distribution but also controls the relative error in the far tail of the distribution.

In this article, we discuss approximations to the unknown complex random-sum Poisson–Erlang random variable, which has a continuous distribution, and the random-sum Poisson-negative binomial random variable, which has a discrete distribution. We show that the saddlepoint approximation method is not only quick, dependable, stable, and accurate enough for general statistical inference but is also applicable without deep knowledge of probability theory. Numerical examples of application of the saddlepoint approximation method to continuous and discrete random-sum Poisson distributions are presented.     

On generalized moment identity and its applications: a unified approach

In this paper, we obtain a generalized moment identity for the case when the distributions of the random variables are not necessarily purely discrete or absolutely continuous. The proposed identity is useful to find the generator which has been used for the approximation of distributions by Stein's method. Apparently, a new approach is discussed for the approximation of distributions by Stein's method. We bring the characterization based on the relationship between conditional expectations and hazard measure in our unified framework. As an application, a new lower bound to the mean-squared error is obtained and it is compared with Bayesian Cramer–Rao bound.     

Application of Dirichlet mixture of normals in growth curve models

In this paper, we present growth curve models with an auxiliary variable which contains an uncertain data distribution based on mixtures of standard components, such as normal distributions. The multimodality of the auxiliary random variable motivates and necessitates the use of mixtures of normal distributions in our model. We have observed that Dirichlet process priors, composed of discrete and continuous components, are appropriate in addressing the two problems of determining the number of components and estimating the parameters simultaneously and are especially useful in the aforementioned multimodal scenario. A model for the application of Dirichlet mixture of normals (DMN) in growth curve models under Bayesian formulation is presented and algorithms for computing the number of components, as well as estimating the parameters are also rendered. The simulation results show that our model gives improved goodness of fit statistics over models without DMN and the estimates for the number of components and for parameters are reasonably accurate.     

Efficient designs for Bayesian networks with sub-tree bounds

We present upper and lower bounds for information measures, and use these to find the optimal design of experiments for Bayesian networks. The bounds are inspired by properties of the junction tree algorithm, which is commonly used for calculating conditional probabilities in graphical models like Bayesian networks. We demonstrate methods for iteratively improving the upper and lower bounds until they are sufficiently tight. We illustrate properties of the algorithm by tutorial examples in the case where we want to ensure optimality and for the case where the goal is an approximate solution with a guarantee. We further use the bounds to accelerate established algorithms for constructing useful designs. An example with petroleum fields in the North Sea is studied, where the design problem is related to exploration drilling campaigns. All of our examples consider binary random variables, but the theory can also be applied to other discrete or continuous distributions.     

On the mean residual life function of a mixture in reliability studies

In reliability studies, the additional life time given that a component has survived until time t is called the Mean residual life function (MRLF). This MRLF determines the distribution function uniquely. There exist many life testing situations which can be best described as mixtures of distributions. In this paper we have considered the general MRLF and have developed a method of obtaining the mixing distribution when the original distribution is exponential. Some examples are discussed, in one of which Morrison’s (1978) result is obtained as a special case.     

Minimally informative distributions with given rank correlation for use in uncertainty analysis

Minimum information bivariate distributions with uniform marginals and a specified rank correlation are studied in this paper. These distributions play an important role in a particular way of modeling dependent random variables which has been used in the computer code UNICORN for carrying out uncertainty analyses. It is shown that these minimum information distributions have a particular form which makes simulation of conditional distributions very simple. Approximations to the continuous distributions are discussed and explicit formulae are determined. Finally a relation is discussed to DAD theorems, and a numerical algorithm is given (which has geometric rate of covergence) for determining the minimum information distributions.     

The Discrete Normal Distribution

Abstract

The normal distribution has been playing a key role in stochastic modeling for a continuous setup. But its distribution function does not have an analytical form. Moreover, the distribution of a complex multicomponent system made of normal variates occasionally poses derivational difficulties. It may be worth exploring the possibility of developing a discrete version of the normal distribution so that the same can be used for modeling discrete data. Keeping in mind the above requirement we propose a discrete version of the continuous normal distribution. The Increasing Failure Rate property in the discrete setup has been ensured. Characterization results have also been made to establish a direct link between the discrete normal distribution and its continuous counterpart. The corresponding concept of a discrete approximator for the normal deviate has been suggested. An application of the discrete normal distributions for evaluating the reliability of complex systems has been elaborated as an alternative to simulation methods.     

Exploring the Equivalence of Two Common Mixture Models for Duration Data

The beta-geometric (BG) distribution and the Pareto distribution of the second kind (P(II)) are two basic models for duration-time data that share some underlying characteristics (i.e., continuous mixtures of memoryless distributions), but differ in two important respects: first, the BG is the natural model to use when the event of interest occurs in discrete time, while the P(II) is the right choice for a continuous-time setting. Second, the underlying mixing distributions (the beta and gamma for the BG and P(II), respectively), are very different—and often believed to be noncomparable with each other. Despite these and other key differences, the two models are strikingly similar in terms of their fit and predictive performance as well as their parameter estimates. We explore this equivalence, both empirically and analytically, and discuss the implications from both a substantive and methodological standpoint.     

Properties and applications of the Poisson-reciprocal inverse Gaussian distribution

The barely known continuous reciprocal inverse Gaussian distribution is used in this paper to introduce the Poisson-reciprocal inverse Gaussian discrete distribution. Several of its most relevant statistical properties are examined, some of them directly inherited from the reciprocal of the inverse Gaussian distribution. Furthermore, a mixed Poisson regression model that uses the reciprocal inverse Gaussian as mixing distribution is presented. Parameters estimation in this regression model is performed via an EM type algorithm. In light of the numerical results displayed in the paper, the distributions introduced in this work are competitive with the classical negative binomial and Poisson-inverse Gaussian distributions.     

Progressively Type-II right censored order statistics from discrete distributions

Progressively Type-II right censored order statistics from continuous distributions have been studied rather extensively in the literature; see Balakrishnan and Aggarwala [2000. Progressive Censoring: Theory, Methods and Applications. Birkhäuser, Boston]. In this paper, we derive the joint and marginal distributions of progressively Type-II right censored order statistics from discrete distributions. We then use these distributions to show the non-Markovian property as well as to discuss some properties in the special case of the geometric distribution.     

On elliptical multilevel models

Multilevel models have been widely applied to analyze data sets which present some hierarchical structure. In this paper we propose a generalization of the normal multilevel models, named elliptical multilevel models. This proposal suggests the use of distributions in the elliptical class, thus involving all symmetric continuous distributions, including the normal distribution as a particular case. Elliptical distributions may have lighter or heavier tails than the normal ones. In the case of normal error models with the presence of outlying observations, heavy-tailed error models may be applied to accommodate such observations. In particular, we discuss some aspects of the elliptical multilevel models, such as maximum likelihood estimation and residual analysis to assess features related to the fitting and the model assumptions. Finally, two motivating examples analyzed under normal multilevel models are reanalyzed under Student-t and power exponential multilevel models. Comparisons with the normal multilevel model are performed by using residual analysis.     

Order statistics from discrete distributions

Several recurrence relations and identities available for single and product moments of order1 statistics in a sample size n from an arbitrary continuous distribution are extended for the discrete case,, Making use of these recurrence relations it is shown that it is sufficient to evaluate just two single moments and (n-l)/2 product moments when n is odd and two single moments and {n-2)/2 product moments when n is even, in order to evaluate the first, second and product moments of order statistics in a sample of size n drawn from an arbitrary discrete distribution, given these moments in samples of sizes n-1 and less.. A series representation for the product moments of order statistics is derived.. Besides enabling us to obtain an exact and explicit expression for the product moments of order statistics from the geometric distribution, it. makes the computation of the product moments of order statistics from other discrete distributions easy too.