Identification and Selection Procedures Based on Tests

Abstract

Let $X_1, \cdots, X_k$ be independent random variables with distributions $Q_1, \cdots, Q_k$ defined on a common range space $(\mathscr{X}, \mathscr{G})$. At the beginning it is assumed that the $Q_i$ are known but not the pairing with the $X_i$, and the goal is to identify the $X_i$ which comes from $Q_1$. First it is shown that every procedure based on a total ordering in $\mathscr{X}$ can be viewed as being based on (the $p$-value of) a test for deciding between $H_0: \{Q_1\}$ versus $H_1: \{Q_2, \cdots, Q_k\}$. Then the class of procedures based on tests is studied in detail. It is demonstrated how typical properties of tests $\varphi$ (powerfulness, unbiasedness, consistency etc.) transfer to the corresponding procedures $S_\varphi$. The next step is to get free of the assumption that $Q_1, \cdots, Q_k$ are known, thereby passing over from identification to selection procedures. Throughout this paper the objective is to compare procedures (not to establish specified ones), and as one main result it is shown that the asymptotic relative efficiency (Pitman) of one test $\varphi$ with respect to a second test $\psi$ and of $S_\varphi$ with respect to $S_\psi$ are identical.