Incomplete Sufficient Statistics and Similar Tests

Abstract

For a family of exponential densities a method is given, called "$D$ method," for constructing a class of similar tests in the case that the minimal sufficient statistic is boundedly incomplete. This method also provides a proof of a criterion for bounded incompleteness. Under certain conditions the criterion states that a sufficient statistic for a family of exponential densities is boundedly incomplete if the number of components of the statistic is larger than the number of parameters specifying the distribution. Applications are indicated in the Behrens-Fisher problem, and in the problem of testing the ratio of mean to standard deviation in a normal population. In the latter problem it is shown that the $D$ method generates the whole class of similar tests. Some unsolved problems concerning the existence of an optimal similar test are indicated.