Goodness of Fit Statistics for the Exponential Distribution When the Data Are Grouped

Abstract

In many industrial and biological experiments, the recorded data consist of the number of observations falling in an interval. In this paper, we develop two test statistics to test whether the grouped observations come from an exponential distribution. Following the procedure of Damianou and Kemp (Damianou, C., Kemp, A. W. (1990 Damianou, C. and Kemp, A. W. 1990. New goodness of statistics for discrete and continuous data. American Journal of Mathematical and Management Sciences, 10: 275–307. [Taylor & Francis Online] , [Google Scholar]). New goodness of statistics for discrete and continuous data. American Journal of Mathematical and Management Sciences 10:275–307.), Kolmogrov–Smirnov type statistics are developed with the maximum likelihood estimator of the scale parameter substituted for the true unknown scale. The asymptotic theory for both the statistics is studied and power studies carried out via simulations.     

A simple test of symmetry about an unknown median

A simple test statistic for testing symmetry of a distribution function about an unknown value is presented. The asymptotic distributions under symmetry and asymmetry are derived. Using the normal as a “calibration” distribution, the critical values of the test are calculated by Monte Carlo methods. Comparisons with other tests indicate that this procedure performs well.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.