Lalonde, Trent L.
Schaffer, Jay Ryan, 1969-
Shafie Holighi, Khalil
Applied Statistics and Research Methods
University of Northern Colorado
Type of Resources
Place of Publication
University of Northern Colorado
A new exponentiality test was developed by modifying the Lilliefors test of exponentiality for the purpose of improving the power of the test it directly modified. Lilliefors has considered the maximum absolute differences between the sample empirical distribution function (EDF) and the exponential cumulative distribution function (CDF). The proposed test considered the sum of all the absolute differences between the CDF and EDF. By considering the sum of all the absolute differences rather than only a point difference of each observation, the proposed test would expect to be less affected by individual extreme (too low or too high) observations and capable of detecting smaller, but consistent, differences between the distributions. The proposed test statistic is not only easy to understand but also very simple and easy to compute. The proposed test was compared directly to the Lilliefors test (LF-test), the Cramer-Von Mises test (CVM-test), Finkelstein & Schafers test (S-test) and the ̃n test (D-test). The critical values were developed and the accuracy of the intended significance levels was verified for the proposed test. The results showed that all five tests of exponentiality worked very well in terms of controlling the intended significance levels. The proposed test performed very closely to the other four tests of exponentiality in terms of the accuracy of the intended significance levels across all considered sample sizes. The proposed exponentiality test (PML-test) did successfully improve upon the power of the test it directly modified (i.e. LF-test). The actual method employed in the development of the test statistic in this study, achieved its primary goal of improving the power of the LF-test of exponentiality. This study showed that the proposed exponentiality test (PML-test) demonstrated consistently superior power over the S-test, LF-test, CVM-test, and D-test for most of the alternative distributions presented in this study. The D-test, CVM-test, and S-test exhibited similar power for a fixed sample size and significance level. The LF-test consistently showed the lowest power among five exponentiality tests. So, practically speaking the proposed test can hope to replace the other four exponentiality tests discussed throughout this study while maintaining a very simple form for computation and easy to understand for those people who have limited knowledge of statistics. This study has shown that using the sum of all the absolute differences between the two functions (CDF and EDF) will have more power than just using the maximum differences between these two functions (like LF-test) or using the sum of squared differences between these two functions (like Cramer-Von Mises type test). The research presented here has the potential to modify many other tests and / or to develop tests for distributional assumption.
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