Asymptotic Expansions for Power Functions

Abstract

For i = 0,1 let $p_i^{(n)}$, n∈ℕ, be sequences of p-measures approximating P⁽ⁿ⁾. Let $p_i^{(n)}$ denote a density of $p_i^{(n)}$. It is well known that tests based on the test statistic log($p_1^{(n)} /p_o^{(n)}$) are most powerful for $p_o^{(n)}$ against $p_1^{(n)}$. In this section it will be shown that sequences of test statistics log($\left( {p_1^{(n)} /p_o^{(n)} } \right) + R_N$), n ∈ ℕ, are most powerful of order o(n^−1/2) under relatively weak conditions on the remainder sequence R_n, n ∈ ℕ.