Bayesian decision theoretic approach to hypothesis problems with skewed alternatives

Many hypothesis problems in practice require the selection of the left side or the right side alternative when the null is rejected. For parametric models, this problem can be stated as _H0:θ=_θ0$H_0:\theta ={\theta }_0$vs.  _H−:θ<_θ0$H_{-}:\theta <{\theta }_0$ or _H+:θ>_θ0$H_{+}:\theta >{\theta }_0$. Frequentists use Type-III error (directional error) to develop statistical methodologies. This approach and other approaches considered in the literature do not take into account the situations where the selection of one side may be more important or when one side may be more probable than the other. This problem can be tackled by specifying a loss function and/or by specifying a hierarchical prior structure with allowing the skewness in the alternatives. Based on this, we develop a Bayesian decision theoretic methodology and show that the resulted Bayes rule perform better in the side of the alternatives which is more probable. The methodology can be also used in a frequentist's framework when it is desired to discover an alternative that is more important. We also consider the multiple hypotheses problem and develop new false discovery rates for the selection of the left and the right sides of alternatives. These discovery rates would be useful in the situations when one side of the alternatives are more important or more probable than the other.