Local admissibility and local unbiasedness in hypothesis testing problems (Q1193354)

The paper investigates, extends and unifies local power optimality criteria under (local) side constraints for a simple null hypothesis \(\theta=0\). Classical examples include locally most powerful unbiased tests (type A), tests maximizing the local power among unbiased tests with constant local power on an ellipse (type C), type D tests using maximum local curvature etc. Necessary and sufficient conditions are given for local unbiasedness of a test \(\varphi\) (defined by \(E_ \theta(\varphi)\geq E_ 0(\varphi)\) for \(|\theta|<\varepsilon\) for some \(\varepsilon>0\)) and the authors present a definition of local admissibility [a modification of \textit{H. Kudô}, Bull. Inst. Internat. Statist. 38, 217-234 (1961); ibid., No. 4, 173-180 (1961; Zbl 0111.334)]. Theorem 3.1 shows that a test is locally admissible among locally unbiased tests iff it is locally admissible and locally unbiased. The proof uses the second order Taylor expansion of the power function \(E_ \theta(\varphi)\). Optimum tests reject \(H_ 0\) if \(\lambda'l(x)+\text{tr}(M\cdot V(x))/2>c\) (with a suitable vector \(\lambda\) and a matrix \(M\geq 0\)) where \[ R_ \theta(x)=f_ \theta(x)/f_ 0(x), \quad l(x)=\text{grad}_ \theta R_ \theta(x) \quad\text{and}\quad V(x)=D_ \theta^ 2 R_ \theta(x). \] Theorem 4.1 states that if 0 is in the interior of the domain \(\Theta\) of \(\theta\), this is just the class of locally most mean power unbiased tests considered by \textit{A. Sen Gupta} and \textit{L. Vermeire} [J. Am. Stat. Assoc. 81, 819-825 (1986; Zbl 0635.62020)]. A series of examples is presented relating, e.g., to one-parameter families, exponential families, locally pointed alternatives, half-space alternatives, normal linear models, ordered alternatives, and tests for independence under normality.