On Efficiency and Optimality of Quadratic Tests

Abstract

Let $Z_1, Z_2, \cdots$ be i.i.d. standard normal variables. Results are obtained which relate to the tail behavior as $x \rightarrow \infty$ of distributions of the form $F(x) = P\{\sum^\infty_{k=1}\lambda_k\lbrack(Z_k + a_k)^2 - 1\rbrack \leqslant x\}$. For test statistics which have such limiting distributions $F$, asymptotic relative efficiency measures are discussed. One of these is the limiting approximate Bahadur efficiency. Applications are to tests of fit and tests of symmetry.