Conditional distributions of generalized order statistics and some characterizations

Generalized order statistics have been introduced in Kamps (1995a). They enable a unified approach to several models of ordered random variables, e.g. (ordinary) order statistics, record values, sequential order statistics, record values from non-identical distributions. The purpose of this paper is to develop conditional distributions of one generalized order statistic given another and to characterize the underlying continuous distribution by different conditional expectations. Well-known results for ordinary order statistics and record values are extended to generalized order statistics. Received: July 1997     

On two recent characterizations of multivariate normal distribution

Summary In this note we recall two recent characterizations of the multivariate normal distribution and give a few related remarks and examples.     

Characterizations of multivariate geometric distributions in terms of conditional distributions

In this paper some characterizations of a multivariate geometric distribution based on conditional distributions have been obtained. The usefulness of the characterization results is illustrated.     

Matricvariate and matrix multivariate T distributions and associated distributions

Several distributions are studied, simultaneously in the real, complex, quaternion and octonion cases. Specifically, these are the central, nonsingular matricvariate and matrix multivariate T and beta type II distributions and the joint density of the singular values are obtained for real normed division algebras.     

Conditional distributions and the bivariate normal distribution

Summary An alaysis of the extent to which conditional distributions of a bivariate vector characterize bivariate normality is given.     

Limit theorems for multivariate discrete distributions

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.     

Nonparametric covariance estimation in multivariate distributions

The estimation problem of the unknown covariance matrix of a multivariate distribution with the known mean is studied under a matrix-valued quadratic loss function. The conditions on the sample sizes for the best unbiased estimator to have a smaller risk than the sample covariance matrix is established. The former estimator is completely (without exceptional sets of Lebesgue measure zero) characterized by its expectation in the class of all multivariate distributions with zero mean and finite fourth moments. Received: November 1998     

Testing multivariate distributions in GARCH models

In this paper, we consider testing distributional assumptions in multivariate GARCH models based on empirical processes. Using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions, we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution functions. We then apply the Khmaladze's martingale transformation (K-transformation) to the empirical process in the presence of estimated parameters. The K-transformation eliminates the effect of parameter estimation, allowing a distribution-free test statistic to be constructed. We show that the K-transformation takes a very simple form for testing multivariate normal and multivariate t-distributions. The procedure is applied to a multivariate financial time series data set.     

L-statistics from multivariate unified skew-elliptical distributions

We study here the distributions of order statistics and linear combinations of order statistics from a multivariate unified skew-elliptical distribution. We show that these distributions can be expressed as mixtures of unified skew-elliptical distributions, and then use these mixture forms to study some distributional properties and moments.     

On characterizations of distributions by regression of adjacent generalized order statistics

Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .     

Some characterizations of distributions by regression models for ordinal response data

Summary Let a random variableX be classified intok classes. By doing so, a new random variable is obtained, measured on ordinal scale. If this variable is a response variable in certain regression models for ordinal response data, the distribution ofX is characterized by the models. In this paper, characterizations of the distribution ofX by the proportional odds model and the proportional hazards model are given.     

Moment inequalities for order statistics with applications to characterizations of distributions

General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.     

Sampling distributions associated with the multivariate t distribution

The known sampling distributions and simulation methods associated with multivariate t distributions are reviewed. We believe that this review will serve as an important reference and encourage further research activities in the area.     

Estimation for multivariate stable distributions with generalized empirical likelihood

This paper considers the generalized empirical likelihood (GEL) method for estimating the parameters of the multivariate stable distribution. The GEL method is considered to be an extension of the generalized method of moments (GMM). The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. We treat the spectral measure, which summarizes scale and asymmetry, by discretization. In order to estimate all the model parameters simultaneously, we apply the estimating function constructed by equating empirical and theoretical characteristic functions. The efficacy of the proposed GEL method is demonstrated in Monte Carlo studies. An illustrative example involving daily returns of market indexes is also included.     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

Conditional distributions of earnings,wages and hours for blacks and whites

This paper provides new evidence on the conditional distributions of earnings, wages and hours for white and black males in the University of Michigan's Panel Study of Income Dynamics. Conditional hours and ln wages are approximately normal for both races. Conditional earnings are approximately normal for blacks while earnings are well approximated for whites by the product of normal hours and log-normal wages. The distribution of this product is derived here for the first time. Treating the marginal earnings distribution as the average of the conditional distributions, we use these results to predict poverty and affluence rates in our sample.     

Characteristic properties of multivariate survival functions in terms of residual life distributions

This paper discusses the relationships among some characteristic properties of the multivariate survival function based on the residual life distribution, and provides the conditions for their equivalence. In the meanwhile, the corrected version of Ma (1996, Theorem 1 (ii) and (iii)) is given.     

Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

On characterizations of distributions by expected values of order statistics and record values with gap

In this paper, we obtain some recurrence relationships for conditional expectations of nonadjacent order statistics and record values when the distribution function is absolutely continuous, and we prove that the distribution function is uniquely determined by the distribution of conditioned record values and by the expected values of these records. Further, different distributions are characterized by these relationships.