Abstract
The consistency of the tests developed by Cox (1961, 1962) and Atkinson (1970) in which null and alternative hypotheses specify different parametric families is discussed. It is shown that Atkinson's procedure is not always consistent. An example is presented. Under the null hypothesis a discussion of the approach to asymptotic theory is provided through the examination of the terms which differentiate the two procedures and some insight into the behaviour of the tests in finite samples is then obtained.
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