\(\psi\)-correct decision for selection and elimination
Let \(X_ i\) \((i=1,2,...,k)\) be k independent r.v.'s from populations \(\pi_ i\) with continuous distribution function \(F(\cdot;\theta_ i)\) which are stochastically increasing in \(\theta_ i\). The ordered \(\theta_ i\) are given by \[ \theta_{\rho (1)}\leq \theta_{\rho (2)}\leq...\leq \theta_{\rho (k-t)}<\theta_{\rho (k-t+1)}\leq...\leq \theta_{\rho (k)} \] where \(\rho\) (\(\cdot)\) is an unknown parametric function. The ordered \(X_ i\) are given by \(X_{R(1)}<X_{R(2)}<...<X_{R(k)}\) where R(\(\cdot)\) is a random function. Define the following two subsets of indices: \[ G=\{R(k),\quad R(k-1),...,R(k-t+1)\}\text{ and } \gamma =\{\rho (k), \rho(k- 1),...,\rho(k-t+1)\}. \] Then the usual correct selection, CS, is defined as \(CS=\{G=\gamma \}\). Let \(\psi\) (\(\cdot)\) be an increasing function with \(\psi(\theta)>0\). Define \[ \Omega_{\psi}=\{{\underset \tilde{} \theta}: \psi(\theta_ j)\leq \theta_ i,\quad i\in \gamma,\quad j\in \gamma^ c\} \] to be the preference zone. The \(\psi\)-correct decision, \(\psi\)-CD, is defined as \[ \psi -CD=\{\min_{i\in G}\psi (\theta_ i)>\max_{j\in G^ c}\theta_ j\}. \] The paper shows that under certain conditions on the distribution functions and the \(\psi\) function, \[ \inf_{\theta \in \Omega}P(\psi -CD)=\inf_{\theta \in \Omega_{\psi}}P(CS). \] For location and scale parameters, the above result is true for any \(\psi(\cdot)\).
scientific article; zbMATH DE number 4048844