Misspecified T2 tests. i. location and scale

Properties of Hotelling's (1931) T ² are studied under model misspecification in the model for a multivariate experiment. Stochastic bounds on T ² and further properties of the T ² test are studied under misspecified location and scale. The bounds are evaluated numerically in selected cases.     

On the convolution of normal and t random variables

The need for the convolution of normal and Student's t random variables arises in many areas. Since the 1930s, various authors have attempted to derive closed-form expressions for the probability density function (pdf) of the convolution, but with little success. Here, general closed-form expressions are derived for the pdf.     

Shannon Entropy and Mutual Information for Multivariate Skew‐Elliptical Distributions

Abstract. The entropy and mutual information index are important concepts developed by Shannon in the context of information theory. They have been widely studied in the case of the multivariate normal distribution. We first extend these tools to the full symmetric class of multivariate elliptical distributions and then to the more flexible families of multivariate skew‐elliptical distributions. We study in detail the cases of the multivariate skew‐normal and skew‐t distributions. We implement our findings to the application of the optimal design of an ozone monitoring station network in Santiago de Chile.     

An algorithn for generating a basis of the trades on t-designs

For the purpose of constructing t-designs (a special family of incomplete block designs) from a given design, a trade off method has been introduced and studied [Hedayat and Li (1979)]. In this paper, we present a computer algorithm for generating a basis of the trades on t-designs utilizing the results of Graham, Li and Li (1980).     

On the Bumpy Road to the Dominant Mode

Abstract. Maximum likelihood estimation in many classical statistical problems is beset by multimodality. This article explores several variations of deterministic annealing that tend to avoid inferior modes and find the dominant mode. In Bayesian settings, annealing can be tailored to find the dominant mode of the log posterior. Our annealing algorithms involve essentially trivial changes to existing optimization algorithms built on block relaxation or the EM or MM principle. Our examples include estimation with the multivariate t distribution, Gaussian mixture models, latent class analysis, factor analysis, multidimensional scaling and a one‐way random effects model. In the numerical examples explored, the proposed annealing strategies significantly improve the chances for locating the global maximum.     

Bayesian forecasting with changing linear models

This investigation considers a general linear model which changes parameters exactly once during the observation period. Assuming all the parameters are unknown and a proper prior distribution, the Bayesian predictive distribution of the future observations is derived.

It is shown that the predictive distribution is a mixture of multivariate t distributions and that the mixing distribution is the marginal posterior mass function of the change point parameter.     

Comparative Analyses of Pretest-Posttest Research Designs

Two common methods of analyzing data from a two-group pretest-posttest research design are (a) two-sample t test on the difference score between pretest and posttest and (b) repeated-measures/split-plot analysis of variance. The repeated-measures/split-plot analysis subsumes the t test analysis, although the former requires more assumptions to be satisfied. A numerical example is given to illustrate some of the equivalences of the two methods of analysis. The investigator should choose the method of analysis based on the research objective(s).     

A Retrievable Recipe for Inverse t

Although the percentage points of the Student-t distribution have been widely tabulated, a simple approximation is given and derived in this article. The approximation can be re-derived easily, since it is based on the percentage points from the Gaussian distribution, and can thus be used for applications requiring non-integer degrees of freedom (e.g., Welch's two-sample t test) and for arbitrary significance levels (e.g., for Bonferroni multiple comparison procedures). Comparisons between this approximation and others suggested in the literature indicate three-digit accuracy for even small degrees of freedom and tail areas.     

A Note on Stagewise Regression

Suppose one estimates the coefficient β₂ in E[Y] = β₀ + β₁ X ₁ + β₂ X ₂ by stagewise regression. That is, first the model E[Y] ≌ β₀ + β₁ X ₁ is fit using simple linear regression followed by a simple linear regression of the residuals from this model on X ₂ to yield the estimator β₂. The ratio of the squared t statistic for the estimate b ₂ from multiple regression to the squared t statistic for β₂ is greater than or equal to 1.0 and is shown to be a convenient function of correlation coefficients among Y, X ₁, and X ₂. Examination of stagewise regression can provide useful insights when introducing concepts of multiple regression.     

A linear algebraic algorithm for reducing the support size of t-designs and to generate a basis for trades

A simple linear algebraic algorithm to generate a basis of the null space of a given integral matrix is utilized to present a computer algorithm, which in general, is used to reduce the support size of a given design as in a theorem of FoodyHedayat (Theorem 4.1, 1977), and in particular, it is used to produce a basis for trades. The computations based on this algorithm is of order of a polynomial function.