Comparison of tests in the multiparameter case I. Second-order power

In a multiparameter setting, considering a very large class of tests it is seen that under contiguous alternatives, unlike in the one-parameter case, identity of power up to the first order may not imply that up to the second order. It is, therefore, possible to discriminate among tests in terms of their second-order power in a multiparameter set-up. An explicit and easily applicable formula for second-order power difference has been obtained. It is also seen that identity of power up to the first order implies identity of "average" power up to the second order. The use of a new kind of polynomials, analogous to Hermite polynomials, is helpful in the derivation of the results.