Theory and Methods of Statistics

8.1.4 Asymptotic Comparison of Tests: Pitman’s Approach

Let θ be a numerical characteristic of a population (or a combination of populations), for which we want to test H₀: θ = 0 vs H₊: θ > 0. Typically, a test statistic T_n based on a sample of size n would reject H₀ in favor of H₊ at level α if $T_n>c_n\left(\alpha \right)$. Here we look into the problem of comparing the asymptotic powers of two sequences of tests based on $\left\{T_n^{(1)}\right\}$ and $\left\{T_n^{(2)}\right\}$ with critical values $\left\{c_n^{(1)}\right\}$