Statistical procedures based on exponential scores

Abstract

Many statistical models have been proposed in which underlying exponential distributions are assumed. Applications of such models occur in a large number of areas, for example in reliability and life-testing investigations and in the study of the pattern of intervals between point events in a series of events when a Poisson process is often postulated. The methods of statistical inference for such models are usually simple to apply but unfortunately are sensitive to departures from the exponential form. In a significance testing situation this leads to a difficulty in interpretation since the true observed significance level may differ appreciably from that calculated on the assumption of an underlying exponential distribution, if the assumption is incorrect. To overcome this drawback, distribution-free tests have been proposed in which the observations are first ranked and the ranks then replaced by exponential scores which are the expected values of the order statistics in a sample from the standard exponential distribution. This guarantees the validity of the test, whatever the form of the underlying distribution. In addition there is no loss of efficiency in very large samples when the underlying distributions are exponential, and often more generally, when the distributions belong to a Lehmann family which includes the Weibull distributions with common power parameter and hence the exponential distribution as a special case. In this report, we describe a number of statistical tests based on exponential scores, some new, some well-known, many of which have been proposed and evaluated within the last ten years. The purpose of this is to demonstrate the wide area of application of exponential scores procedures. The procedures which are described deal with goodness of fit tests for the exponential distribution, the comparison of two samples with and without censoring, and the comparison of k>2 samples, and finally tests for trend and serial dependence alternatives against a renewal process for the intervals in a series of events. Some of the research findings related to powers of some of the tests are discussed, these findings being based on a series of investigations made by the author over the period 1975-1978.