On the multivariate predictive distribution of multi-dimensional effective dose: a Bayesian approach

We propose a Bayesian procedure to sample from the distribution of the multi-dimensional effective dose. This effective dose is the set of dose levels of multiple predictive factors that produce a binary response with a fixed probability.We apply our algorithms to parametric and semiparametric logistics regression models, respectively. The graphical display of random samples obtained through Markov chain Monte Carlo can provide some insight into the predictive distribution.     

Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts

A new method to calculate the multivariate t-distribution is introduced. We provide a series of substitutions, which transform the starting q-variate integral into one over the (q—1)-dimensional hypercube. In this situation standard numerical integration methods can be applied. Three algorithms are discussed in detail. As an application we derive an expression to calculate the power of multiple contrast tests assuming normally distributed data.     

Markov chain models for multivariate repeated binary data analysis

Repeated categorical outcomes frequently occur in clinical trials. Muenz and Rubinstein (1985) presented Markov chain models to analyze binary repeated data in a breast cancer study. We extend their method to the setting when more than one repeated outcome variable is of interest. In a randomized clinical trial of breast cancer, we investigate the dependency of toxicities on predictor variables and the relationship among multiple toxic effects.     

Large Sample Properties of ML Estimator of the Parameters of Multivariate O–U Random Fields

Computer modeling is having a profound effect on scientific research, replacing direct physical experimentation by computer simulation of complex models. In this research, the computer output, X(t), is assumed to be a multivariate, three-dimensional (time) Ornstein-Uhlenbeck (O–U) process with parametric covariance function. It is shown that the ML estimates of the parameters are strongly consistent and asymptotically normal when the observations are taken from a complete lattice, not necessarily equally spaced.     

A Bivariate Generalization of the Noncentral Negative Binomial Distribution

This article proposes a bivariate generalization of the noncentral negative binomial distribution which arises as a model in photon and neural counting. This bivariate generalization is derived as a mixed shifted bivariate negative binomial distribution. Various properties and parameter estimation, especially by a minimum distance method based on the probability generating function, are considered. To show the practical usefulness of the bivariate distribution proposed, an application to model low-flux astronomical images is discussed and a real data set has been analyzed.     

Semiparametric Regression Estimation in Copula Models

We consider some methods of semiparametric regression estimation in multivariate models when the common distribution function is represented using a copula and the marginals satisfy a generalized regression model using a transfer functional. Sufficient conditions for consistency and joint asymptotic normality of the finite-dimensional parameters are obtained.     

Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant random matrix

The Euler characteristic heuristic has been proposed as a method for approximating the upper tail probability of the maximum of a random field with smooth sample path. When the random field is Gaussian, this method is proved to be valid in the sense that the relative approximation error is exponentially smaller. However, very little is known about the validity of the method when the random field is non-Gaussian. In this paper, as a milestone to developing the general theory about the validity of the Euler characteristic heuristic, we examine the Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant non-Gaussian random matrix. In this particular example, if the probability density function of the random matrix converges to zero sufficiently fast at the boundary of its support, the approximation error of the Euler characteristic heuristic is proved to be small and the approximation is valid. Moreover, for several standard orthogonally invariant random matrices, the approximation formula for the distribution of the largest eigenvalue and its asymptotic error are obtained explicitly. Our formulas are practical enough for the purpose of numerical calculations.     

Unbiased estimation of the common mean of a multivariate normal distribution

The problem of unbiased estimation of the common mean of a multivariate normal population is considered. An unbiased estimator is proposed which has a smaller variance than the usual estimator over a large part of the parameter space.     

Geometric representation of high dimension, low sample size data

Summary.  High dimension, low sample size data are emerging in various areas of science. We find a common structure underlying many such data sets by using a non-standard type of asymptotics: the dimension tends to ∞ while the sample size is fixed. Our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex. Essentially all the randomness in the data appears only as a random rotation of this simplex. This geometric representation is used to obtain several new statistical insights.     

Structural learning with time-varying components: tracking the cross-section of financial time series

Summary.  When modelling multivariate financial data, the problem of structural learning is compounded by the fact that the covariance structure changes with time. Previous work has focused on modelling those changes by using multivariate stochastic volatility models. We present an alternative to these models that focuses instead on the latent graphical structure that is related to the precision matrix. We develop a graphical model for sequences of Gaussian random vectors when changes in the underlying graph occur at random times, and a new block of data is created with the addition or deletion of an edge. We show how a Bayesian hierarchical model incorporates both the uncertainty about that graph and the time variation thereof.