Multivariate Exponential Family Research Articles

An exponential family model for clustered multivariate binary data

This paper focuses on the analysis of clustered multivariate binary data that arise from developmental toxicity studies. In these studies, pregnant mice are exposed to chemicals to assess possible adverse effects on developing fetuses. Multivariate binary outcomes arise when each fetus in a litter is assessed for the presence of malformations and/or low birth weight. We analyse the data using a multivariate exponential family model which is flexible in terms of allowing response rates to depend on cluster size. Maximum likelihood estimation of model parameters and the construction of score tests for dose effect are discussed.

Multivariate exponential families and the Taguchi loss function

Taguchi uses a loss function concept to quantify quality as loss due to functional variation. When there is one variable under study a quadratic loss function is used. When the system under consideration is defined by p-variables which may or may not be correlated a more general loss function should be defined. We use information theory to define a multivariate loss function based on a divergence measure between two probability distributions that belong to the multivariate exponential family. The two distributions are the ideal or factory distribution that our data should follow and the other is the real distribution of the multivariate data. This divergence measure uses the concept of the Kullback—Leibler information number.

Directional tests for one-sided alternatives in multivariate models

Consider one-sided testing problems for a multivariate exponential family model. Through conditioning or other considerations, the problem oftentimes reduces to testing a null hypothesis that the natural parameter is a zero vector against the alternative that the natural parameter lies in a closed convex cone $\mathscr{C}$. The problems include testing homogeneity of parameters, testing independence in contingency tables, testing stochastic ordering of distributions and many others. A test methodology is developed that directionalizes the usual test procedures such as likelihood ratio, chi square, Fisher, and so on. The methodology can be applied to families of tests where the family is indexed by a size parameter so as to enable nonrandomized testing by $p$-values. For discrete models, a refined family of tests provides a refined grid for better testing by $p$-values. The tests have essential monotonicity properties that are required for admissibility and for desirable power properties. Two examples are given.

Parametric Statistical Uncertainty Relations and Parametric Statistical Fundamental Equations

Multivariate parametric statistical uncertainty relations are proved to specify multivariate basic parametric statistical models. The relations are expressed by inequalities. They generally show that we cannot exactly determine simultaneously both a function of observation objects and a parametric statistical model in a compound parametric statistical system composed of observations and a model. As special cases of the relations, statistical fundamental equations are presented which are obtained as the conditions of attainment of the equality sign in the relations. Making use of the result, a generalized multivariate exponential family is derived as a family of minimum uncertainty distributions. In the final section, several multivariate distributions are derived as basic multivariate parametric statistical models.

Exact Slopes of Test Statistics for the Multivariate Exponential Family

The objective of this paper is to investigate exact slopes of test statistics {Tn} when the random vectors X1, ..., Xn are distributed according to an unknown member of an exponential family {Pθ; θ∈Ω. Here Ω is a parameter set. We will be concerned with the hypothesis testing problem of H0θ∈Ω0 vs H1: θ∉Ω0 where Ω0 is a subset of Ω. It will be shown that for an important class of problems and test statistics the exact slope of {Tn} at η in Ω−Ω0 is determined by the shortest Kullback–Leibler distance from {θ: Tn(λ(θ)) = Tn(λ(π))} to Ω0, λθ = Eθ)(X).

On Characterization of Multiparameter Exponential Family and of some Irregular Families of Distributions

A characterization through moments given by Khatri (1959) for p.s.d. and by Jani (1993) for one-parameter exponential family has been extended for the wider class viz. multiparameter and multivariate exponential family of distributions. The same problem has beon studied also for some non-exponential families where the support contains the parameter(s), called irregular families of distributions.

Third order efficient tests in exponential lattice models

In full rank multivariate exponential families of lattice distributions, one sided testing problems are considered. For these testing problems the uniformly most powerful unbiased tests are not uniformly efficient of order o( n −1) in the class T 2 ∗ of all tests which are asymptotically similar of order o( n −1) ( Hipp, C. (1983). J. Multivariate Anal. 13 67–108). In this note, better asymptotically similar tests are constructed which—for one specified value of the nuisance parameter—have maximal power in the class T 2 ∗. The power of tests is computed up to terms of order o( n −1), at contiguous alternatives, and for the specified value of the nuisance parameter.

Third-order efficiency of conditional tests in exponential models: The lattice case

As is well known, in full rank multivariate exponential families, tests of Neyman structure are uniformly most powerful unbiased for one-sided problems. For the case of lattice distributions, the power of these tests—evaluated at contiguous alternatives—is approximated by asymptotic expansions up to errors of order o( n −1). Surprisingly the tests with Neyman structure are not third-order efficient in the class of all asymptotically similar tests unless the problem is univariate.

Chernoff Efficiency and Deficiency

In 1952 Chernoff introduced a measure of asymptotic efficiency for tests. Comparison in the sense of Chernoff is concerned with fixed alternatives. In contrast to Bahadur's approach, where the probabilities of first and second kind are treated in an unbalanced way, in Chernoff's approach both probabilities go to zero. For the calculation of Chernoff efficiencies one has to develop large deviation theorems both under the null hypothesis and under the alternative hypothesis. In this paper some basic properties are mentioned and the concept of Chernoff deficiency is introduced in a manner analogous to the Pitman and Bahadur case. It is shown that in typical testing problems in multivariate exponential families, the likelihood ratio test is Chernoff deficient of order $\mathscr{O}(\log n)$. Many of the results agree with corresponding results in the Bahadur case.

On the equivalence of some test criteria based on BAN estimators for the multivariate exponential family

Abstract For the general multivariate exponential family of distributions it is shown that Rao's test criterion based on efficient scores is algebraically identical to the general chi-squared criterion based on maximum likelihood estimates and, similarly, that the Wald statistic is algebraically identical to the general minimum modified chi-squared statistic using linearization; these results are valid also for the multisample versions. Thus, these are extensions to the general exponential family of the findings due to Silvey (1970) and Bhapkar (1966), respectively, for the special case of the multinomial family. It is also shown that the general forms of the chi-squared and modified chi-squared criteria reduce to their respective well-known forms for the multivariate symmetric power series distribution. This finding is, thus, an extension of results noted by Ferguson (1958) and Clickner (1976) for the special case of the multinomial distribution.

Bahadur deficiency of likelihood ratio tests in exponential families

For many testing problems several different tests may have optimal exact Bahadur slope. The introduction of Bahadur deficiency provides further information about the performance of such tests. Roughly speaking a sequence of tests is deficient in the sense of Bahadur of order o ( h n ) at a fixed alternative θ if the additional number of observations necessary to obtain the same power as the optimal test at θ is of order o ( h n ) as the level of significance tends to zero. In this paper it is shown that in typical testing problems in multivariate exponential families the LR test is deficient in the sense of Bahadur of order o (log n).

Generalized Bayes Estimators in Multivariate Problems

Several problems involving multivariate generalized Bayes estimators are investigated. First, a characterization of admissible estimators as generalized Bayes estimators is developed for certain multivariate exponential families and quadratic loss. The problem of verifying whether or not an estimator is generalized Bayes is also considered. Next, an important class of estimators for a multivariate normal mean is considered. (The class includes many minimax, empirical Bayes, and ridge regression estimators of current interest.) Necessary conditions are developed for an estimator in this class to be "nearly" generalized Bayes, in the sense that if it were properly smoothed, it would be generalized Bayes. An application to adaptive ridge regression is given. The paper concludes with the development of an asymptotic approximation to generalized Bayes estimators for general losses and location vector densities. Using this approximation, weakened versions of the above results are obtained for general losses and densities.

The Geometry of Exponential Families

There are two important spaces connected with every multivariate exponential family, the natural parameter space and the expectation parameter space. We describe some geometric results relating the two. (In the simplest case, that of a normal translation family, the two spaces coincide and the geometry is the familiar Euclidean one.) Maximum likelihood estimation, within one-parameter curved subfamilies of the multivariate family, has two simple and useful geometric interpretations. The geometry also relates to the Fisherian question: to what extent can the Fisher information be replaced by $-\partial^2/\partial\theta^2\lbrack\log f_\theta(x)\rbrack\mid_{\theta=\hat{\theta}}$ in the variance bound for $\hat{\theta}$, the maximum likelihood estimator?

The essentially complete class of rules in multinomial identification

The problem of identification within two groups is considered, when the space 𝔛 of the possible values of the observed random variable X is finite. Using an essentially complete class of tests hypothesis ξ=0 (ξis real) for the multivariate exponential family, an essentially complete class of sample-based identification rules has been found. Comparison of the rules to those derived from density estimators has shown that the latter constitute a subclass of the former.

Correction: Correction to Tests of Composite Hypotheses for the Multivariate Exponential Family

Correction: Correction to Tests of Composite Hypotheses for the Multivariate Exponential Family

Tests of Composite Hypotheses for the Multivariate Exponential Family

This paper is concerned with testing the hypothesis that the parameter in a multivariate exponential distribution lies in a linear subspace of the natural parameter space. Our main result characterizes a complete class of tests which is independent of the particular exponential distribution. This class is, in fact, complete relative to the stronger ordering among tests which compares conditional power, given a certain statistic, pointwise. The conclusion holds without any restriction on the exponential distribution. Many of the tests are admissible, but examples show that although the class is essentially the smallest class complete relative to all exponential distributions, it is not in general minimally complete. Some special cases where the class is minimally complete are discussed.

Effect on the Minimal Complete Class of Tests of Changes in the Testing Problem

on the minimal complete class of tests is investigated. The effects of such changes are found to be different for the two families of distributions considered: The discrete multivariate exponential family and the continuous multivariate exponential family. In Section 2, it is shown that with respect to the discrete exponential family, the minimal complete class of tests for a standard testing problem is minimal complete for a wide variety of related problems. In Section 3, an example is given showing that with respect to the continuous exponential family, on the other hand, the minimal complete class of tests for a standard problem is not necessarily minimal complete for a slight variation of this problem. Tests that are admissible for the standard problem are not necessarily admissible for the variation. Partly in a general decision theoretic framework and partly with respect to specific examples, Hoeffding [2] has discussed the effect of changes in the family of probability distributions on the minimax solution and other optimal solutions. He has also giVen key references to the extensive literature on the performance of standard procedures for families of probability distributions not satisfying all the assumptions under which the standard procedures were derived. Workers in this area have primarily concentrated on the effect of changes in the probability model on a single solution rather than on a class of solutions, for example, the class of admissible solutions, as we do here. We recall some basic ideas. Consider the probability structure (DC, a, P, Q) where DC and Q are sets, a, is a a-field of subsets of DC, and for each 0 in Q, Pe is a probability measure on (t. Relative to the above structure, a testing problem is an ordered pair (woo, w,) of disjoint subsets of U. A test (s is a function from DC into [0, 1] measurable with respect to at. The test So is used in the following way: A random element X with values in D having Po as its probability distribution is observed. If x is the outcome then the hypothesis H: 0 E wO is rejected with probability (p(x) in favor of the alternative A: 0 c w. If the test sp is used and 0 is the parameter, then the probability that H is rejected is Eep fc (p(x) dPe(x). If sp and so* are tests, then sp is at least as good as so* if

The Admissibility of Hotelling's $T^2$-Test

In Section 3 we shall prove a theorem based on a method of A. Birnbaum [1] and E. Lehmann concerning the admissibility of certain tests of simple hypotheses in multivariate exponential families. In Section 4 we compute the supporting hyperplanes of the convex acceptance region in some of the most common applications of Hotelling's $T^2$-test and show that the theorem of Section 3 implies the admissibility of this test. In Section 5 we point out some of the limitations of the method of this paper.