# On some tests for exponentiality based on the mean residual life function

## Keywords:

Brownian bridge, limit theorems, Monte Carlo simulations.## Abstract

We build on the work of Aly (1983) and Jammalamadaka & Taufer (2006) to developnew tests for exponentiality based on the mean residual life function. We obtain theasymptotic null distributions of the proposed tests and give approximations for theirlimiting critical values. We also give tables of their finite sample Monte Carlo criticalvalues. We report the results of several Monte Carlo studies conducted to compare theproposed tests with a number of their competitors in terms of power.## References

Ahmad, I.A. & Alwasel, I. (1999) A goodness-of-fit test for exponentiality based on the memoryless

property. Journal of the Royal Statistical Soceity Series B, 61:681-689.

Alwasel, I. (2001) On goodness-of-fit testing for exponentiality using the memoryless property. Journal of

Nonparametric Statistics, 3:23-36.

Aly, E.E. (1983) Some limit theorems for uniform and exponential spacings. Canadian Journal of Statistics,

:211-219.

Aly, E.E. (1988) Strong approximations of quadratic sums of uniform spacings. Canadian Journal of

Statistics, 16:201-207.

Aly, E.E. (1989) On testing exponentiality against IFRA alternatives. Metrika, 36:255-267.

Aly, E.E. (1990a) On testing exponentiality against IFR alternatives. Statistics, 21:217-226.

Aly, E.E. (1990b) Testing exponentiality against IFRA or NBU alternatives with type II censored data.

Statistics & Decisions, 8:265-269.

Aly, E.E. (1992) On testing exponentiality against HNBUE alternatives. Statistics & Decisions, 10:239-

Aly, E.E. & Lu, M.G. (1988) A unified asymptotic theory for testing exponentiality against NBU, IFR or

IFRA alternatives. Statistics & Decisions, 6:261-274.

Aly, E.E., Kochar, S.C. & McKeague, I. (1994) Some tests for comparing incidence functions and causespecific

hazard rates. Journal of the American Statistical Association, 89:994-999.

Baringhaus, l.A & Henze, N. (1992) On adaptive omnibus test for exponentiality. Communications in

Statistics- Theory and Methods, 21:969-978.

Baringhaus, L. & Henze, N. (2000) Tests of fit for exponentiality based on a characterization via the

mean residual life function. Statistical Papers, 41:225-236.

Bergman, B. & Klefsjö, B. (1989) A family of test statistics for detecting monotone mean residual life.

Journal of Statistical Planning and Inference, 21:161-178.

Bhattacharjee, M.C. & Sen, P.K. (1995) Kolmogorov-Smirnov type tests for NB(W)UE alternatives

under censoring schemes. In H.L. Koul and J.V. Deshpande (Eds) Analysis of Censored Data

(Hayward, CA: Institute of Mathematical Statistics), pp. 25-38.

Bandyopadhyay, D. & Basu, A.P. (1990) A class of tests for exponentiality against decreasing mean

residual life alternatives. Communications in Statistics- Theory and Methods, 19:905-920.

Csörgő, M. (1983) Quantile processes with statistical applications. CBMS-NSF Regional Conference

Series in Applied Mathematics, 42:SAIM, Philadelphia.

Csörgő, M. & Horváth, L. (1997) Limit theorems in change point analysis. John Wiley and Sons, New

York.

Deshpande, J.V. (1983) A class of tests for exponentiality against increasing failure rate average

alternatives. Biometrika, 70:514-518.

Grzegorzewski, P. & Wieczorkowski, R. (1999) Entropy-based goodness of fit tests for exponentiality.

Communications in Statistics- Theory and Methods, 28:1183-1202.

Ebrahimi, N., Soofi, E.S. & Habibullah, M. (1992) Testing exponentiality based on Kullback-Leibler

information. Journal of the Royal Statistical Soceity Series B, 54:739-748.

Henze, N. (1993) A new flexible class of omnibus tests for exponentiality. Communications in Statistics-

Theory and Methods, 22:115-133.

Henze, N. & Meintanis, S.G. (2002) Tests of fit for exponentiality based on the empirical Laplace

transform. Statistics, 36:147-162.

Hollander, M. & Proschan, F. (1972) Testing whether new is better than used. The Annals of Mathematical

Statistics, 43:1136-1146.

Jammalamadaka, S.R. & Taufer, E. (2003) Testing exponentiality by comparing the empirical distribution

function of the normalized spacings with that of the original data. Journal of Nonparametric

Statistics, 15:719-729.

Jammalamadaka, S.R. & Taufer, E. (2006) Use of mean residual life in testing departure from

exponentiality. Journal of Nonparametric Statistics, 18:277-292.

Klar, B. (2001) A class of tests for exponentiality against HNBUE alternatives. Statistics and Probability

Letters, 47:199-207.

Koul, H.L. (1978) Testing for new is better than used in expectation. Communications in Statistics- Theory

and Methods, 7:685-701.

Shanbhag, D.N. (1970) The Characterizations for exponential and geometric distributions. Journal of the

American Statistical Association. 65:1256-1259.

Shorack, G.R. & Wellner, J.A. (1986) Empirical processes with applications to statistics. John Wiley and

Sons, New York.

Taufer, E. (2000) A new test for exponentiality against omnibus alternatives. Stochastic Modelling and

Applications, 3:23-36.