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Recent interest in modeling multivariate responses for members of groups has emphasized the need for testing goodness of fit. Here we describe a way to test the covariance structure of a multivariate distribution parameterized by a vector θ. The idea is to extend this distribution, the “null” distribution, to a more general distribution that depends on θ, an additional scalar γ, and a specific quadratic function of the response vector chosen to capture features of an alternative covariance structure. When γ = 0, the more general distribution reduces to the null one. Standard likelihood theory yields a score test for γ = 0; that is, a test of fit of the null distribution. The score statistic is the standardized difference between observed and expected values of the quadratic function, where the expectation is taken with respect to the null distribution, with θ replaced by its maximum likelihood estimate. Applying the methods to case-control data on familial cancers of the ovary and breast, we illu...

Using both frequentist and Bayesian techniques, predicting densities are derived for future observations from a multivariate linear model with matrix normal error terms. All the candidates belong to a general location-scale family of predicting densities. An analytic comparison is undertaken, using a Kullback—Leibler loss, by citing an optimal member of a subclass including most of these predicting densities as members. The subclass is based on an invariant Student-t random matrix, and the optimal member is the Bayesian predictive density corresponding to a Jeffreys noninformative prior. Information-based numerical comparisons illustrate the nature of the dominance.

Suppose Y′ = (Y′ 1, …, Y′ k ) possesses a multivariate normal distribution with mean vector 0 and positive semidefinite covariance matrix Σ. If Ci ⊂ Rpi denote convex regions symmetric about the origin, then conditions are given such that and/or obtain. These conditions imply that chi-squared random variables defined from a multivariate normal distribution are always positively dependent and nonnegatively correlated. Other applications involve conservative simultaneous confidence regions in a multivariate regression setting.

Through the use of an empirical Bayes argument, a confidence set for the mean of a multivariate normal distribution is derived. The set is a recentered sphere, is easy to compute, and has uniformly smaller volume than the usual confidence set. An exact formula for the coverage probability is derived, and numerical evidence is presented which shows that the empirical Bayes set uniformly dominates the usual set in coverage probability.

The factorial moment generating function (FMGF) of the multivariate Poisson normal (or Hermite) distribution is derived as the limiting form of a multinomial process and as the FMGF of a mixture of independent Poisson distributions when the parameters have a multivariate normal distribution. The FMGF is then used as starting point to derive a limiting form, a series expansion, marginal and conditional distributions as well as for estimating the parameters by using jointly the method of moments and least squaresf or correlated variables.

Multivariate white noise processes arise in a variety of contexts; for example, as error terms in multivariate regression models, as innovations in multiple time series models, or simply as random samples from multivariate normal distribution. Here we study the asymptotic properties of white noise autocorrelations in detail to determine their use in testing the adequacy of fit.

The multivariate acceptance sampling problem is discussed and a solution for a particular sub-class of the general problem developed. This is where the related quality variables have a multivariate normal distribution with known covariance matrix, and where the specification region may be set in the form of a general ellipsoid. Single sampling acceptance procedures by variables for fraction defective are developed. Methods are given for determining the sample size n and acceptance number c so as to achieve desired operating characteristics.

This paper is concerned with the three-parameter logarithmic normal distribution; i.e., the general distribution in which the terminus is unknown. Maximum likelihood equations for estimating population parameters from random samples are derived, and an iterative method for their solution is outlined. Variances and covariances of these estimates are obtained from the information matrix. An illustrative example is included.

A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. This paper presents some meaningful derivations of a multivariate exponential distribution that serves to indicate conditions under which the distribution is appropriate. Two of these derivations are based on “shock models,” and one is based on the requirement that residual life is independent of age. It is significant that the derivations all lead to the same distribution. For this distribution, the moment generating function is obtained, comparison is made with the case of independence, the distribution of the minimum is discussed, and various other properties are investigated. A multivariate Weibull distribution is obtained through a change of variables.

Multiple-stage selection from an infinite population with some known distribution is studied. A method is given for finding the exact moments of the distribution of the true yields of the varieties left after selection with error. Considerable simplification is possible where the original population is normally distributed. This simplification is a result of the use of expressions that are derived for the derivatives of a multivariate normal probability integral. Methods are indicated for numerical calculation of the resulting integrals on a computer when appropriate tables are not available.

Formulas for confidence, prediction, and tolerance regions for the multivariate normal distribution for the various cases of known and unknown mean vector and covariance matrix are assembled for easy reference in this expository paper. Tables are provided for the bivariate case.

The asymptotic joint distribution of an arbitrary number of linear systematic statistics (that is, linear combinations of order statistics), when observations are made on a random vector, is shown to be normal under fairly general conditions. The linear systematic statistics may correspond to the same or to different components of the vector. Formulas for evaluating the parameters of the asymptotic normal distribution are derived. As an illustration, these are applied to the case of trimmed means when the distribution sampled is bivariate normal.

Applying the result of expectations of non-negative random variables, inequalities for the one-sided and two-sided rectangular probabilities of higher-dimensional multivariate normal and multivariate t distributions are obtained. It is shown that their k-dimensional rectangular probabilities is lower bounded by the (k/m)th power of their m-dimensional rectangular probabilities for every k≥ m ≥ 1, and this bound is quite good when k/m is close to one. Applications to multiple decision problems are considered.

Exact expectations and variances have been obtained for the maximum likelihood estimates of the elements of the mean vector and covariance matrix of the multivariate normal distribution when a subset of the variates does not have observations on some sampling units. The biases, variances, and mean square errors of the estimates are compared with those of the usual estimates computed from the complete observation vectors. When the correlations between the complete and incomplete sets of variates are small the multivariate missing value estimates are less efficient in the mean square error sense.

The distribution of a linear function of two independent F variates with respective degrees of freedom n, N-n and m, N-m has been expressed as a quickly-convergent infinite series of incomplete beta functions. This exact distribution is used to determine the error of a simple F approximation to it obtained by equating the first two cumulants. By those results small sample percentage points can be obtained for some tests on the mean vector of a multivariate normal distribution whose covariance matrix has certain patterns.

For rectangular confidence regions for the mean values of multivariate normal distributions the following conjecture of 0. J. Dunn [3], [4] is proved: Such a confidence region constructed for the case of independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates. This result is based on an inequality for the probabilities of rectangles in normal distributions, which permits one to factor out the probability for any single coordinate.

In this paper a method is developed for estimating the parameters in the multivariate normal distribution in which the missing observavations are not restricted to follow certain patterns as in most previous papers. The large sample properties of the estimators are discussed. Equivalence with maximum likelihood estimators has been established for a subclass of problems. The results of some simulation studies are provided to support the theoretical development.