An inequality for the multivariate normal distribution

Herman Chernoff used Hermite polynomials to prove an inequality for the normal distribution. This inequality is useful in solving a variation of the classical isoperimetric problem which, in turn, is relevant to data compression in the theory of element identification. As the inequality is of interest in itself, we prove a multivariate generalization of it using a different argument.     

A multivariate skew normal distribution

In this paper, we define a new class of multivariate skew-normal distributions. Its properties are studied. In particular we derive its density, moment generating function, the first two moments and marginal and conditional distributions. We illustrate the contours of a bivariate density as well as conditional expectations. We also give an extension to construct a general multivariate skew normal distribution.     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

An inequality for Lp-norms with respect to the multivariate normal distribution

A partial converse of Jensen's inequality for integrals of norms on $\text{R}$^k is proved.     

Another characterization of multivariate normal distribution

We establish a characterization of the multivariate normal based on a maximal property relating Var[g(ζ)] and the gradient of g(·).     

Inference for a truncated positive normal distribution

The main object of this paper is to study an extension of the half normal distribution defined by adding a positive truncation to it. The new model is more flexible than the half-normal distribution and contains the half normal distribution as a special case. Properties of this distribution, such as moments, hazard function and entropy are studied and parameters estimation is dealt with by using moments and maximum likelihood. A real data application indicates good fit performance of the new model when compared to other competitors in literatures.     

Moments and cumulants of the multivariate normal distribution

Expressions in vector notation are given for the central moments, the non–central moments and the cumulants of arbitrary order of the multivariate normal distribution     

Some results on the multivariate truncated normal distribution

This note formalizes some analytical results on the n-dimensional multivariate truncated normal distribution where truncation is one-sided and at an arbitrary point. Results on linear transformations, marginal and conditional distributions, and independence are provided. Also, results on log-concavity, A-unimodality and the MTP₂ property are derived.     

Improved confidence sets for the mean of a multivariate normal distribution

A new class of confidence sets for the mean of a p-variate normal distribution (p3) is introduced. They are neither spheres nor ellipsoids. We show that we can construct our confidence sets so that their coverage probabilities are equal to the specified confidence coefficient. Some of them are shown to dominate the usual confidence set, a sphere centered at the observations. Numerical results are also given which show how small their volumes are.     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

On confidence sequences for the mean vector of a multivariate normal distribution

Let X₁, X₂,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets {R_n(a₁, a₂,…, a_n), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ R_n(X₁, X₂,…, X_n) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence R_n(X₁, X₂, …, X_n) depends on Mahalanobis' or Hotelling's according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given.     

Expansions for the multivariate normal

Mehler gave an expansion for the standard bivariate normal density. Kibble extended it to a multivariate normal density whose covariance is a correlation matrix. We give extensions of these expansions for general covariances.     

Maximum-likelihood estimation of the parameters of a multivariate normal distribution

This paper provides an exposition of alternative approaches for obtaining maximum- likelihood estimators (MLE) for the parameters of a multivariate normal distribution under different assumptions about the parameters. A central focus is on two general techniques, namely, matrix differentiation and matrix transformations. These are systematically applied to derive the MLE of the means under a rank constraint and of the covariances when there are missing observations. Derivations using induction and inequalities are also included to illustrate alternative methods. Other examples, such as a connection with an econometric model, are included. Although the paper is primarily expository, some of the proofs are new.     

A characterization of multivariate normal distribution and its application

Let X₁, X₂, …, X_n be i.i.d. d-dimensional random vectors with a continuous density. Let and . In this paper we find that the distribution of Z_k (or Y_k) can be used for characterizing multivariate normal distribution. This characterization can be employed for testing multivariate normality in terms of the so-called transformation method.     

Improved prediction for a multivariate normal distribution with unknown mean and variance

The prediction problem for a multivariate normal distribution is considered where both mean and variance are unknown. When the Kullback–Leibler loss is used, the Bayesian predictive density based on the right invariant prior, which turns out to be a density of a multivariate t-distribution, is the best invariant and minimax predictive density. In this paper, we introduce an improper shrinkage prior and show that the Bayesian predictive density against the shrinkage prior improves upon the best invariant predictive density when the dimension is greater than or equal to three.     

An algorithm for rescaling a matrix positive definite

For a given square real matrix M, we present a general algorithm which decides the existence of a positive diagonal matrix D such that DM is positive definite and which constructs the D if it exists. It is shown that solving this matrix rescaling problem is equivalent to finding a solution of an infinite system of linear inequalities. The algorithm solves this infinite system of linear inequalities by generating and solving a sequence of linear programs.     

Fiducial inference on the largest mean of a multivariate normal distribution

Inference on the largest mean of a multivariate normal distribution is a surprisingly difficult and unexplored topic. Difficulties arise when two or more of the means are simultaneously the largest mean. Our proposed solution is based on an extension of R.A. Fisher’s fiducial inference methods termed generalized fiducial inference. We use a model selection technique along with the generalized fiducial distribution to allow for equal largest means and alleviate the overestimation that commonly occurs. Our proposed confidence intervals for the largest mean have asymptotically correct frequentist coverage and simulation results suggest that they possess promising small sample empirical properties. In addition to the theoretical calculations and simulations we also applied this approach to the air quality index of the four largest cities in the northeastern United States (Baltimore, Boston, New York, and Philadelphia).