A note on characterizations of multivariate stable distributions

Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.Research supported, in part, by the Air Force Office of Scientific Research under Contract AFOSR 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

A note on convergence to mixtures of normal distributions

Double arrays of random variables obtained by normalizing a sequence that is asymptotically close to a martingale difference sequence are considered, and conditions ensuring that the row sums converge in distribution to a mixture of normal distributions are found. The main condition is that the sums of squares in each row converge in probability to a random variable.Research sponsored in part by the Office of Naval Research, Contract # N00014-75-C-0809     

A class of weighted multivariate normal distributions and its properties

This article proposes a class of weighted multivariate normal distributions whose probability density function has the form of a product of a multivariate normal density and a weighting function. The class is obtained from marginal distributions of various doubly truncated multivariate normal distributions. The class strictly includes the multivariate normal and multivariate skew-normal. It is useful for selection modeling and inequality constrained normal mean vector analysis. We report on a study of some distributional properties and the Bayesian perspective of the class. A probabilistic representation of the distributions is also given. The representation is shown to be straightforward to specify the distribution and to implement computation, with output readily adapted for the required analysis. Necessary theories and illustrative examples are provided.     

A note on a paper by G. Mastroianni and G. Monegato

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral

where or or and is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

     

A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N(μ, Σ) random variable on R^m, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in R^m with closure $\text{C}$. Let (·,-·) be the Eculidean inner product on R^m, and let μ_c be the conditional expected value of Y given Y ∈ C. For v ∈ R^m and s ≥ 0, let β_s(v) be the expected value of |(v, Y) ? (v, μ)|^s and let γ_s(v) be the conditional expected value of |(v, Y) ? (v, μ_c)|^s given Y ∈ C. For s ≥ 1, γ_s(v) < β_s(v) if and only if $\text{C}\text{+ Σ v ≠}\text{C}$, and γ_s(v) < β_s(v) for all v ≠ 0 if and only if $\text{C}\text{+ v ≠}\text{C}$ for any v ∈ R^m such that v ≠ 0.     

A note on a paper of G. Wegner

G. Wegner has shown that the minimal number of vertices of a graph of girth 5 and valency 5 is 30. In this note, we give a simple proof of this result.     

Efficient algorithms for generating truncated multivariate normal distributions

Sampling from a truncated multivariate normal distribution (TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation (DA) algorithm and a non-iterative inverse Bayes formulae (IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA algorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.     

A note on multivariate Gaussian estimates

An upper bound of multivariate Gaussian probability for a general convex domain D is given based on a geometric observation. The bound is sharper than known ones on multivariate Mills' ratio in many cases.     

A note on Nasr's and Wong's papers

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form

     

A note on a paper by G.Pólya

This note is concerned with the computer implementation of Pólya's paper [8] for determining the zeros of a special class of entire functions. Grau's modification to the Graeffe process [6] is utilised to bound the iterated coefficients.     

On probabilistic constraints induced by rectangular sets and multivariate normal distributions

In this paper, we consider optimization problems under probabilistic constraints which are defined by two-sided inequalities for the underlying normally distributed random vector. As a main step for an algorithmic solution of such problems, we prove a derivative formula for (normal) probabilities of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a similar well-known statement for multivariate normal distribution functions. As an application, we consider a problem from water reservoir management. One of the outcomes of the problem solution is that the (still frequently encountered) use of simple individual probabilistic constraints can completely fail. By contrast, the (more difficult) use of joint probabilistic constraints, which heavily depends on the derivative formula mentioned before, yields very reasonable and robust solutions over the whole time horizon considered.     

Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

A note on a characterization of the Poisson and uniform distributions

It is shown that if U, X are independent random variables, X≥0, U is uniformly distributed in (0,1), and X satisfies the equation UX+2∼U+X, then X−2 has the Poisson distribution with the parameter equal one. The above equation also characterizes the uniform distribution if X−2 is a Poisson random variable. Moreover, a multivariate generalization is given. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

A note on the Sarmanov bivariate distributions

We investigate the maximum correlation for Sarmanov bivariate distributions with fixed marginals and strengthen the existing results in the literature. The improvement in the maximum correlation is significant. A characterization of the Sarmanov distribution via chi-square divergence is also given. This extends Nelsen [13] result about the Farlie-Gumbel-Morgenstern (FGM) distribution.     

A note on conjugate distributions for copulas

A family of conjugated distributions for a given type of copulas is defined in this paper. Those copulas can be written as a mixture of d‐dimensional parameter exponential functions. The generalized Farlie–Gumbel–Morgenstern copula is an example of this representation. This family is used to illustrate the estimation technique with real data. Also, the applicability of Bayesian predictive approach is shown in an education policy issue by defining goals for the number of students per class that leads to improve their performance at school. Copyright © 2014 John Wiley & Sons, Ltd.     

A note on recent papers of Ramachandra and Huxley

The methods of the two authors on the zeros of zeta and L-functions are compared.