Maxmin $C(\alpha)$ Tests Against Two-Sided Alternatives

Abstract

Let $\{X_n\}$ be a sequence of i.i.d. random variables, each with probability density function $p(x \mid \theta, \xi)$ subject to certain regularity conditions. Here, $\theta$ is an $s$-dimensional vector of nuisance parameters, and $\xi \in (-r, r)$ is the parameter under test. The first $N$ members of the sequence $\{X_n\}$ are to be used for testing the hypothesis, $H_0: \xi = 0$, against the alternative, $H_1: \xi\neq 0$, while $\theta$ remains unspecified. The particular case considered is that in which the left-hand and right-hand derivatives, with respect to $\xi$, of the logarithm of the density function are unequal. It is shown that the class of $C(\alpha)$ tests based on linear combinations of the left and right derivatives, is an essentially complete class of these tests. The asymptotic power functions of these tests depend upon the coefficients of the linear combination. The maxmin test is deduced and compared with strongly symmetric and weakly symmetric tests. The motivation for the study is the vague notion of "fair" tests which do not arbitrarily favor detection of "positive" or "negative" alternatives.