A Bayesian Analysis of Hypothesis Testing Problems with Skewed Alternatives

We consider hypothesis testing problems with skewed alternatives via a Bayesian decision theoretic formulation. The fundamental problem can be stated as a three- decision problem: vs. or . When is rejected, it is desired to select either or . The statistical methods in the literature give equal preferences on both alternatives. We present different methods of statistical methodology based on the skewed alternatives. We develop a general framework by specifying different loss functions, hierarchical priors, and develop a Bayesian decision theoretic methodology. We demonstrate that the Bayes rules result in better performance when compared to the classical decision approach. We also consider multiple hypothesis testing problems with skewed alternatives and introduce a new concept of directional positive false discovery rates for left and right alternatives. Controlling these directional positive false discovery rates is applicable to the situation when one of left and right alternatives is more important than the other. We give several examples where such situations arise. The Bayesian decision methods for multiple testing problems are applied to microarray data of non-coding genes