**Sankhya:
The Indian Journal of Statistics**

2003, Volume 65, Pt. 1, 139--157

**Testing Exponentiality Against
Likelihood Ratio Behaviour Using Kernel Methods**

By

I.A. Ahmad, University of Central
Florida, Orlando, USA

H.M. Al-Nachawati and M.I. Hendi King, Saud University, Riyadh, Saudi Arabia

SUMMARY. In this work, testing exponentiality against monotone likelihood ratio is taken up as well as testing exponentiality in a goodness-of-fit setting. The procedures are based on the celebrated ``kernel" density estimation of probability density functions and some of its derivatives. The limiting null and nonnull distributions of the test statistics are normal and the null variances are calculated exactly. Small samples null critical values are obtained via simulation. The efficacies of the test statistic used for testing against monotone likelihood are calculated for some common alternatives and are compared to some other procedures. The powers of test statistic used for the goodness-of-fit testing are obtained for some well-known alternatives via simulations and are shown to compare favorably against other more involved tests.

*AMS (1991) subject classification. *62G10.

*Key words and phrases. *Testing exponentiality, monotone likelihood,
Polya frequency distribution, goodness of fit, asymptotic normality, critical
values, Pitman asymptotic efficacy, power of tests, Monte Carlo methods.