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My talk will be addressed primarily to students. However, teachers also may find it beneficial. We shall discuss some of the following, given restriction of time.
A. When a family of distributions of non-negative random variables enjoys reproductive property, monotonicity of tail probability of the statistic follows almost immediately. This result includes several standard examples for which advanced or distribution specific tools are required. The results are easily generalized to conditional distributions and multivariate distributions as well, yielding new proofs of several results whose proofs are otherwise nontrivial.
B. We use matrix algebra to obtain alternative and illuminating, and possibly less messy, than standard ones, proofs in the following situations: (1) formula for covariance of two quadratic forms of a multivariate normal variable centered at the zero vector, (2) finding expression for maximum likelihood estimate of the mean vector and dispersion matrix when the mean vector is assumed to lie in translate of a flat, (3) expression for the likelihood ratio test for a two-sample problem, (4) expression for maximum likelihood estimates of regression parameters subject to linear restriction. The known proofs rely on brute-force calculation or calculus. |