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Estate Khmaladze
MCS, Victoria University of Wellington

Goodness of Fit Testing for the Class of Exponential Distributions

Joint work with John Haywood, MCS, Victoria University of Wellington.

"Local", or better to say contiguous, alternatives are the closest alternatives against which it is still possible to have some power. With this in mind we would like to think about goodness of fit tests as those which have some power against all, or a huge majority, of local alternatives. Tests of that kind are often based on non-linear functionals, with a complicated asymptotic null distribution. Therefore a second desirable property of a goodness of fit test is that its statistic will be asymptotically distribution free.

"Goodness of fit testing of exponentiality" is an area of research which has a very long history and has produced a large number of papers. The analysis of this literature demonstrates, however, that surprisingly many tests are based on asymptotically linear functionals from the empirical process, and hence can not be considered as goodness of fit tests. Such tests will have no asymptotic power against a great majority of local alternatives, although they may have good power against some focused "cone" of alternatives.

We suggest potentially a whole class of goodness of fit tests with both of the desirable properties mentioned above, by constructing a new version of the empirical process that weakly converges to a standard Brownian motion, under the hypothesis of exponentiality. Any statistic based on this process will asymptotically behave as a statistic from the standard Brownian motion and, hence, will be asymptotically distribution free. This new, transformed version of the empirical process was introduced in its general form, as an "innovation martingale", in Khmaladze (1981, Theory of Probability and Its Applications). For the case of exponentiality, the transformation is especially simple and convenient to implement.