The nonexistence of procedures with bounded performance characteristics in certain parametric inference problems (Q1265222)

Let \(X\) be a random vector with distribution \(P_\theta\), \(\theta \in\Theta\), where \(\Theta\) is a parameter space. Consider the following two goals of making inferences about a function \(\tau=\tau (\theta)\). Bounded risk. Let \(L(\theta,a)= \omega(| a-\tau (\theta)|)\) be a loss function, where \(\omega(u)\) \((\geq 0)\) is a given function nondecreasing on \([0,\infty)\) and \(\sup_{u\geq 0} \omega(u)= M\) \((\leq\infty)\). Then the goal is to find if there exists an estimator \(\hat i(X)\) of \(\tau\) such that \(E_\theta\{L (\theta, \hat i(X))\}\leq W\), where \(W\;(0<W<M)\) is a given constant. Hypothesis testing. Consider a hypothesis testing \(H_0:\tau=\tau_0\) against \(H_1:\tau= \tau_1\) \((\tau_0 \neq\tau_1)\). Then the goal is to find if there exists a critical function \(\varphi(X)\) such that the probability of error of first kind is less then \(\alpha\) \((0< \alpha<1)\) and the power is greater than \(1-\beta\) \((0<\beta<1)\), where \(\alpha\) and \(\beta\) are given constants. The author derives a condition, which implies the nonexistence of \(\hat i(X)\) or \(\varphi(X)\). Examples for which such a condition is satisfied are considered.