Goodness-of-fit tests based on new characterizations of the exponential distribution
Jansen van Rensburg, Helena Maria
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The exponential density is probably one of the most widely used distributions in practice. Due to its importance, many goodness-of-fit tests for exponentiality have been proposed in the literature. The objectives of this research are as follow: 0 to study the importance of the exponential distribution in practical problems, 0 to investigate alternative classes of distributions to the exponential distribution, 0 to present an overview of existing characterizations of the exponential distribution, 0 to evaluate existing goodness-of-fit tests for exponentiality, 0 to develop new goodness-of-fit tests for exponentiality, and 0 to compare the proposed goodness-of-fit tests to existing tests by means of relative efficiencies and simulation studies of the power of the tests. To achieve these objectives, we begin with a brief discussion of the exponential distribution and other parametric families of life distributions, followed by a summary of six well-known nonparametric classes of alternative distributions. A comprehensive literature study of existing characterizations of the exponential distribution and existing goodness-of fit tests for exponentiality are presented. We then propose and prove two new characterizations of the exponential distribution in the class of NBUE life distributions based on properties of order statistics. These characterizations are used to develop a new class of goodness-of-fit tests for exponentiality. The tests are shown to be consistent and the limiting distributions under the null and alternative hypotheses are derived. We show that the new class of test statistics includes two statistics which arc equivalent to the well-known Gini test statistic (Gail and Gastwirth 1978a) and the coefficient of variation test statistic (Borges, Proschan and Rodrigucs 1984). The newly proposed tests are compared to existing goodness-of-fit tests by means of Pitman and approximate Bahadur relative efficiencies. Monte Carlo studies are conducted to compare the various tests with regard to power for small and moderate sample sizes against a wide range of alternative distributions. We recommend three members of the class of test statistics as being very effective testing procedures for exponentiality. In conclusion, practical examples based on real-life data are presented.
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