This research is concerned with problems of inference in classical multivariate procedures such as multivariate analysis of variance, discriminant analysis , principal components analysis, and canonical correlations analysis. The problems posed are of two main types. One involves problems in estimating canonical correlations and the eigenstructures of parameter matrices arising in multivariate analysis of variance. The investigators propose to apply decision-theoretic concepts and to evaluate the properties and behavior of estimates based on these concepts. The second type of problem to be investigated involves hypothesis testing. Widely used tests in multivariate analysis are ones which hypothesize the equality of a subset of the characteristic roots of a parameter matrix. No small sample properties (unbiasedness, monotonicity of the power functions, admissibility, etc.) are known for common tests, including the likelihood ratio test. The usefulness of the bootstrapping technique in problems of inference in multivariate analysis will be investigated. Classical multivariate statistical procedures are characterized by assumptions of linearity, independence and normality. These procedures are widely used throughout the sciences to analyze experimental and observational data. The statistical procedures for making some of the estimates used in these multivariate analyses are crude and, therefore, improving them may have a significant impact on the conclusions scientists draw about relationships between measurements. Also in this setting, our current knowledge about the performance of statistical decision making procedures applies strictly to experiments that are repeated a very large number of times. Since replicating experiments is usually expensive, it is typically done with low frequency. The research into the performance of statistical decision making procedures when the number of replicates is small is expected to provide valuable information and insight to a broad spectrum of the scientific community.
