The problem of identification of parameters by the distribution of the maximum random variable

Suppose that X₁, X₂,…, X_n are independently distributed according to certain distributions. Does the distribution of the maximum of {X₁, X₂,…, X_n} uniquely determine their distributions? In the univariate case, a general theorem covering the case of Cauchy random variables is given here. Also given is an affirmative answer to the above question for general bivariate normal random variables with non-zero correlations. Bivariate normal random variables with nonnegative correlations were considered earlier in this context by T. W. Anderson and S. G. Ghurye.     

Identification of parameters by the distribution of the maximum random variable: The general multivariate normal case

Summary LetX ₁,X ₂, ...,X _r ber independentn-dimensional random vectors each with a non-singular normal distribution with zero means and positive partial correlations. Suppose thatX _i =(X _i1 , ...,X _in ) and the random vectorY=(Y ₁, ...,Y _n ), their maximum, is defined byY _j =max{X _ij :1ir}. LetW be another randomn-vector which is the maximum of another such family of independentn-vectorsZ ₁,Z ₂, ...,Z _s . It is then shown in this paper that the distributions of theZ _i 's are simply a rearrangement of those of theZ _j 's (and of course,r=s), whenever their maximaY andW have the same distribution. This problem was initially studied by Anderson and Ghurye [2] in the univariate and bivariate cases and motivated by a supply-demand problem in econometrics.     

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.     

Analysis of multivariate skew normal models with incomplete data

We establish computationally flexible methods and algorithms for the analysis of multivariate skew normal models when missing values occur in the data. To facilitate the computation and simplify the theoretic derivation, two auxiliary permutation matrices are incorporated into the model for the determination of observed and missing components of each observation. Under missing at random mechanisms, we formulate an analytically simple ECM algorithm for calculating parameter estimation and retrieving each missing value with a single-valued imputation. Gibbs sampling is used to perform a Bayesian inference on model parameters and to create multiple imputations for missing values. The proposed methodologies are illustrated through a real data set and comparisons are made with those obtained from fitting the normal counterparts.     

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.     

Minimax and admissible minimax estimators of the mean of a multivariate normal distribution for unknown covariance matrix

Let X be a p-variate (p ≥ 3) vector normally distributed with mean μ and covariance Σ, and let A be a p × p random matrix distributed independent of X, according to the Wishart distribution W(n, Σ). For estimating μ, we consider estimators of the form δ = δ(X, A). We obtain families of Bayes, minimax and admissible minimax estimators with respect to the quadratic loss function (δ ? μ)′ Σ^?1(δ ? μ) where Σ is unknown. This paper extends previous results of the author [1], given for the case in which the covariance matrix of the distribution is of the form σ²I, where σ is known.     

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.     

On the conditional probability density functions of multivariate uniform random vectors and multivariate normal random vectors

It is shown that the conditional probability density function of Y₁ given (1/n) Σ_i=1ⁿ Y_i=1Y_i^t = Σ, where Y₁, Y₂,…, Y_n are i.i.d, p-variate uniform random vectors with mean 0 equals to that of Y₁ given (1/n) Σ_i=1ⁿ Y_iY_i^t,…, Y_n are i.i.d, p-variate normal random vectors with mean 0 and covariance matrix Σ.     

Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions

The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given.     

A distribution map for the one-median location problem on a network

In this paper we consider the Weber (one-median) location problem on a network. The weights at the nodes are drawn from a multivariate normal distribution. We find the probability that the optimal location is at each of the nodes. This set of probabilities is the distribution map. The methodology is illustrated on two networks of 20 nodes each.     

On the identifiability of multivariate survival distribution functions

Let (T₁, T₂) be a non-negative random vector which is subjected to censoring random intervals [X₁, Y₁] and [X₂, Y₂]. The censoring mechanism is such that the available informations on T₁ and T₂ are expressed by a pair of random vectors W=(W₁, W₂) and δ=(δ₁, δ₂), where W_i=max(min(Y_i, T_i), X_i) and In this paper we will show that under some mild conditions the joint survival function of T₁ and T₂ can be expressed uniquely as functional of observable joint survival functions. Our results extend recent works on the randomly right censored bivariate data case and on the univariate problem with double censoring to the bivariate data with double censoring.     

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.     

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.     

Definition and characterization of multivariate negative binomial distribution

The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s₁,…,s_n)]^?k where β is a polynomial of degree n being linear in each s_i and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x₁,…,x_n)$\text{exp}\text{(}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{θ}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)}\text{f(θ}{}_1\text{,…,θ}{}_{\mathrm{n}}\text{)}$ is negative binomial if and only if its univariate marginals are negative binomial. Let S_t, t = 1,…, m, be subsets of {s₁,…, s_n} with empty ∩_t=1^mS_t. Then an n-variate pgf is of a negative binomial if and only if for all s in S_t being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t.     

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 characterization of the multivariate normal distribution by using the hazard gradient

We give a general result to characterize a multivariate distribution from a relationship between the left truncated mean function and the hazard gradient function. This result allows us to obtain new characterizations of multivariate distributions. In particular, we show that, for the multivariate normal distribution, the simple relationship, obtained in standardized form by McGill (1992,Communications in Statistics. Theory Methods,21(11), 3053–3060), actually characterizes the multivariate normal distribution. Supported by Ministerio de Ciencia y Tecnologia under grant BFM2000-0362.     

Sequential estimation of the minimum of a random variable

The paper presents a sequential method of fixed-width asymptotically consistent estimation of the minimal value μ of a random variable in the sense μ = inf{ x: F(x) > 0} where F is a distribution function of the considered variable. The theory of extremal order statistics is used.     

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.     

An identity on the maximum of a set of random variables

It is shown that if X₁, X₂, …, X_n are symmetric random variables and max(X₁, …, X_n)⁺ = max(0, X₁, …, X_n), then E[max(X₁,…,X_n)⁺]=[max(X₁,X₁,+X₂,+X₁,+X₃,…X₁,+X_n)⁺], and in the case of independent identically distributed symmetric random variables, E[max(X₁, X₂)⁺] = E[(X₁)⁺] + (1/2)E[(X₁ + X₂)⁺], so that for independent standard normal random variables, E[max(X₁, X₂)⁺] = (1/√2π)[1 + (1/√2)].