Remarks on Pitman deficiency (Q690154)

Let the distribution \(P_ \theta\) be indexed by the parameter \(\theta \in \mathbb{R}^ 1\). Consider the testing problem \(H\): \(\theta=\theta_ 0\) against \(K\): \(\theta> \theta_ 0\). Let \(\{T_{1n}\}\) and \(\{T_{2n}\}\) be two tests. Under the conditions \(\alpha_{in} \equiv P_{\theta_ 0}(T_{in} \geq C_{in}) \to \alpha\), \(i=1,2\), as \(n \to \infty\) and \[ \beta_{in} (\theta)\equiv P_ \theta(T_{in} \geq C_{in})=A_ i(\alpha,\theta)+n^{-1/2}B_ i(\alpha,\theta)+n^{-1}C_ i(\alpha,\theta)+ O(n^{-3/2}) \] the author suggests formulas for Pitman deficiency between the tests \(\{T_{1n}\}\) and \(\{T_{2n}\}\).