On Efficiency and Optimality of Quadratic Tests

Let Z₁, Z₂, ⋯ be i.i.d. standard normal variables. Results are obtained which relate to the tail behavior as x → ∞ of distributions of the form F(x) = P\∑^∞ₖ₌₁λₖ(Zₖ + aₖ)² - 1 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.