Generalized skew-elliptical distributions and their quadratic forms

This paper introduces generalized skew-elliptical distributions (GSE), which include the multivariate skew-normal, skew-t, skew-Cauchy, and skew-elliptical distributions as special cases. GSE are weighted elliptical distributions but the distribution of any even function in GSE random vectors does not depend on the weight function. In particular, this holds for quadratic forms in GSE random vectors. This property is beneficial for inference from non-random samples. We illustrate the latter point on a data set of Australian athletes.     

Efficient and robust estimation for autoregressive regression models using shape mixtures of skewt normal distribution

Multiple linear regression model based on normally distributed and uncorrelated errors is a popular statistical tool with application in various fields. But these assumptions of normality and no serial correlation are hardly met in real life. Hence, this study considers the linear regression time series model for series with outliers and autocorrelated errors. These autocorrelated errors are represented by a covariance-stationary autoregressive process where the independent innovations are driven by shape mixture of skew-t normal distribution. The shape mixture of skew-t normal distribution is a flexible extension of the skew-t normal with an additional shape parameter that controls skewness and kurtosis. With this error model, stochastic modeling of multiple outliers is possible with an adaptive robust maximum likelihood estimation of all the parameters. An Expectation Conditional Maximization Either algorithm is developed to carryout the maximum likelihood estimation. We derive asymptotic standard errors of the estimators through an information-based approximation. The performance of the estimation procedure developed is evaluated through Monte Carlo simulations and real life data analysis.

     

Characteristic functions of scale mixtures of multivariate skew-normal distributions

We obtain the characteristic function of scale mixtures of skew-normal distributions both in the univariate and multivariate cases. The derivation uses the simple stochastic relationship between skew-normal distributions and scale mixtures of skew-normal distributions. In particular, we describe the characteristic function of skew-normal, skew-t, and other related distributions.     

A Skewed and Heavy-Tailed Latent Random Field Model for Spatial Extremes

This article develops Bayesian inference of spatial models with a flexible skew latent structure. Using the multivariate skew-normal distribution of Sahu et al., a valid random field model with stochastic skewing structure is proposed to take into account non-Gaussian features. The skewed spatial model is further improved via scale mixing to accommodate more extreme observations. Finally, the skewed and heavy-tailed random field model is used to describe the parameters of extreme value distributions. Bayesian prediction is done with a well-known Gibbs sampling algorithm, including slice sampling and adaptive simulation techniques. The model performance—as far as the identifiability of the parameters is concerned—is assessed by a simulation study and an analysis of extreme wind speeds across Iran. We conclude that our model provides more satisfactory results according to Bayesian model selection and predictive-based criteria. R code to implement the methods used is available as online supplementary material.     

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985) [1]-and their multivariate generalizations from Azzalini and Dalla Valle (1996) [4]-have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti (2004) [23], DiCiccio and Monti (2009) [24] and Gómez et al. (2007) [25]) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew-t or skew-exponential power distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? The present paper provides an answer to this question. In very general (possibly multivariate) skew-symmetric models, we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference.     

Are You Normal? The Problem of Confounded Residual Structures in Hierarchical Linear Models

We encounter hierarchical data structures in a wide range of applications. Regular linear models are extended by random effects to address correlation between observations in the same group. Inference for random effects is sensitive to distributional misspecifications of the model, making checks for (distributional) assumptions particularly important. The investigation of residual structures is complicated by the presence of different levels and corresponding dependencies. Ignoring these dependencies leads to erroneous conclusions using our familiar tools, such as Q–Q plots or normal tests. We first show the extent of the problem, then we introduce the fraction of confounding as a measure of the level of confounding in a model and finally introduce rotated random effects as a solution to assessing distributional model assumptions. This article has supplementary materials online.     

带缺失数据的偏正态众数回归模型的参数估计

针对现实生活中大量数据存在偏斜的情况,构建偏正态数据下的众数回归模型.又加之数据的缺失常有发生,采用插补方法处理缺失数据集,为比较插补效果,考虑对响应变量随机缺失情形进行统计推断研究.利用高斯牛顿迭代法给出众数回归模型参数的极大似然估计,比较该模型在均值插补,回归插补,众数插补三种插补条件下的插补效果.随机模拟和实例分...     

Bayesian and likelihood inference from equally weighted mixtures

Equally weighted mixture models are recommended for situations where it is required to draw precise finite sample inferences requiring population parameters, but where the population distribution is not constrained to belong to a simple parametric family. They lead to an alternative procedure to the Laird-DerSimonian maximum likelihood algorithm for unequally weighted mixture models. Their primary purpose lies in the facilitation of exact Bayesian computations via importance sampling. Under very general sampling and prior specifications, exact Bayesian computations can be based upon an application of importance sampling, referred to as Permutable Bayesian Marginalization (PBM). An importance function based upon a truncated multivariatet-distribution is proposed, which refers to a generalization of the maximum likelihood procedure. The estimation of discrete distributions, by binomial mixtures, and inference for survivor distributions, via mixtures of exponential or Weibull distributions, are considered. Equally weighted mixture models are also shown to lead to an alternative Gibbs sampling methodology to the Lavine-West approach.     

Joint modelling of location and scale parameters of the skew-normal distribution

Joint location and scale models of the skew-normal distribution provide useful ex- tension for joint mean and variance models of the normal distribution when the data set under consideration involves asymmetric outcomes. This paper focuses on the maximum likelihood estimation of joint location and scale models of the skew-normal distribution. The proposed procedure can simultaneously estimate parameters in the location model and the scale model. Simulation studies and a real example are used to illustrate the proposed methodologies.     

A Flexible Model for Generalized Linear Regression with Measurement Error

This paper focuses on the question of specification of measurement error distribution and the distribution of true predictors in generalized linear models when the predictors are subject to measurement errors. The standard measurement error model typically assumes that the measurement error distribution and the distribution of covariates unobservable in the main study are normal. To make the model flexible enough we, instead, assume that the measurement error distribution is multivariate t and the distribution of true covariates is a finite mixture of normal densities. Likelihood–based method is developed to estimate the regression parameters. However, direct maximization of the marginal likelihood is numerically difficult. Thus as an alternative to it we apply the EM algorithm. This makes the computation of likelihood estimates feasible. The performance of the proposed model is investigated by simulation study.     

Bayesian Inference in the Noncentral Student-t Model

This article takes up Bayesian inference in linear models with disturbances from a noncentral Student-t distribution. The distribution is useful when both long tails and asymmetry are features of the data. The distribution can be expressed as a location-scale mixture of normals with inverse weights distributed according to a chi-square distribution. The computations are performed using Gibbs sampling with data augmentation. An empirical application to Standard and Poor's stock returns indicates that posterior odds strongly favor a noncentral Student-t specification over its symmetric counterpart.     

Characterization of the skew-normal distribution

Two characterization results for the skew-normal distribution based on quadratic statistics have been obtained. The results specialize to known characterizations of the standard normal distribution and generalize to the characterizations of members of a larger family of distributions. Results on the decomposition of the family of distributions of random variables whose square is distributed as χ ₁ ² are obtained. Research supported by a non-service fellowship at Bowling Green State University.     

On the exact distribution of linear combinations of order statistics from dependent random variables

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.     

Testing for a multivariate generalized Pareto distribution

It has recently been shown by Rootzén and Tajvidi (Bernoulli, 12:917–930, 2006) that modelling exceedances of a random variable over a high threshold (peaks-over-threshold approach [POT]) can also in the multivariate setup be done rationally only by a multivariate generalized Pareto distribution (GPD). The selection of a proper threshold is, however, a crucial problem. The contribution of this paper is twofold: We develop first a non asymptotic and exact level-α test based on the single-sample t-test, which checks whether multivariate data are actually generated by a multivariate GPD. Secondly, this procedure is utilized for the derivation of a t-test based threshold selection rule in multivariate peaks-over-threshold models. The application to a hydrological data set illustrates this approach.      

Extremes of Lévy driven mixed MA processes with convolution equivalent distributions

We investigate the extremal behavior of stationary mixed MA processes for t ≥ 0, where f is a deterministic function and Λ is an infinitely divisible and independently scattered random measure. Particular examples of mixed MA processes are superpositions of Ornstein-Uhlenbeck processes, applied as stochastic volatility models in Barndorff-Nielsen and Shephard (2001a). We assume that the finite dimensional distributions of Λ are in the class of convolution equivalent tails and in the maximum domain of attraction of the Gumbel distribution. On the one hand, we compute the tail behavior of Y(0) and sup _{t ∈ [0,1]} Y(t). On the other hand, we study the extremes of Y by means of marked point processes based on maxima of Y in random intervals. A complementary result guarantees the convergence of the running maxima of Y to the Gumbel distribution. Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged.     

The degree sequence of a random graph. I. The models

We show that the joint distribution of the degrees of a random graph can be accurately approximated by several simpler models derived from a set of independent binomial distributions. On the one hand, we consider the distribution of degree sequences of random graphs with n vertices and ½m edges. For a wide range of values of m, this distribution is almost everywhere in close correspondence with the conditional distribution {(X₁,…,X_n) | ∑ X_i=m}, where X₁,…,X_n are independent random variables, each having the same binomial distribution as the degree of one vertex. We also consider random graphs with n vertices and edge probability p. For a wide range of functions p=p(n), the distribution of the degree sequence can be approximated by {(X₁,…,X>_n) | ∑ X_i is even}, where X₁,…,X_n are independent random variables each having the distribution Binom (n−1, p′), where p′ is itself a random variable with a particular truncated normal distribution. To facilitate computations, we demonstrate techniques by which statistics in this model can be inferred from those in a simple model of independent binomial random variables. Where they apply, the accuracy of our method is sufficient to determine asymptotically all probabilities greater than n^−k for any fixed k. In this first paper, we use the geometric mean of degrees as a tutorial example. In the second paper, we will determine the asymptotic distribution of the tth largest degree for all functions t=t(n) as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 97–117 (1997)     

Adaptive control of stochastic systems with unknown disturbance distribution: discounted criteria

We consider a class of discrete-time stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to the equation x _{t +1}=F(x _t , a _t , ξ _t ), t=0, 1, ..., where the ξ _t are i.i.d. random vectors whose common distribution is unknown. Assuming observability of {ξ _t }, we use the empirical estimator of its distribution to construct adaptive policies which are asymptotically discounted cost optimal .AMS Subject Classification (2000) 93E10, 90C40     

Analysis of three graph parameters for random trees

We consider three basic graph parameters, the node‐independence number, the path node‐covering number, and the size of the kernel, and study their distributional behavior for an important class of random tree models, namely the class of simply generated trees, which contains, e.g., binary trees, rooted labeled trees, and planted plane trees, as special instances. We can show for simply generated tree families that the mean and the variance of each of the three parameters under consideration behave for a randomly chosen tree of size n asymptotically ～μn and ～νn, where the constants μ and ν depend on the tree family and the parameter studied. Furthermore we show for all parameters, suitably normalized, convergence in distribution to a Gaussian distributed random variable. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

This work deals with log‐symmetric regression models, which are particularly useful when the response variable is continuous, strictly positive, and following an asymmetric distribution, with the possibility of modeling atypical observations by means of robust estimation. In these regression models, the distribution of the random errors is a member of the log‐symmetric family, which is composed by the log‐contaminated‐normal, log‐hyperbolic, log‐normal, log‐power‐exponential, log‐slash and log‐Student‐t distributions, among others. One way to select the best family member in log‐symmetric regression models is using information criteria. In this paper, we formulate log‐symmetric regression models and conduct a Monte Carlo simulation study to investigate the accuracy of popular information criteria, as Akaike, Bayesian, and Hannan‐Quinn, and their respective corrected versions to choose adequate log‐symmetric regressions models. As a business application, a movie data set assembled by authors is analyzed to compare and obtain the best possible log‐symmetric regression model for box offices. The results provide relevant information for model selection criteria in log‐symmetric regressions and for the movie industry. Economic implications of our study are discussed after the numerical illustrations.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.