Skewness-Kurtosis Controlled Higher Order Equivalent Decisions

Asymptotic relative efficiency of one test with respect to (w.r.t.) another is extensively studied in the literature. For an introduction and overview we refer to Nikitin [1]. Several notions of efficiency may be distinguished w.r.t. how the probabilities of first and second kind test errors behave in the case of increasing sample sizes. Roughly spoken, studies of Pitman type are dealing with situations where both kinds of probabilities of test errors stabilize asymptotically at some fixed positive levels while studies of different other types take into consideration that both error probabilities are tending to zero or that one of them stabilizes asymptotically at a positive value and the other one tends to zero. Moreover, one may take into account different speeds of convergence of the two error probabilities.