Reliability of parametrically specified model of measurements (Q918588)

In \textit{Yu. P. Pyt'ev}, Sov. Math., Dokl. 36, No.1, 96-100 (1987); translation from Dokl. Akad. Nauk SSSR 295, 542-545 (1987; Zbl 0645.62109), the reliability of a simple hypothesis vs a simple alternative was defined as \(\alpha (x)=\inf \{\alpha:\phi^{\alpha}(x)=1\}\), where x is a given observation and \(\phi^{\alpha}\) is the UMP test at the significance level \(\alpha\). A more general definition of reliability of a composite hypothesis vs composite alternative was also introduced. In this paper, an approximation of \(\alpha\) (x) for smooth densities is discussed. As examples the reliability of Gaussian models are computed. Comment: compare the reliability \(\alpha\) (x) with the critical level in \textit{E. L. Lehmann}, Testing statistical hypotheses (1959; Zbl 0089.141), p. 150, \([\alpha (x)=\inf \{\alpha:\) \(x\in S_{\alpha}\}\), where \(S_{\alpha}\) is the rejection region] or with p-values in the second edition of this book (1986; Zbl 0608.62020), p. 170.