Lilliefors test for exponentiality

While doing research on my thesis, I began to look for various tests of statistical distributions. I am familiar with χ², but this test requires a (seemingly) arbitrary choice of the number and shape of bins. Essentially one takes the observed number of items falling in a range and compares that with the expected number (from the proposed distribution). The width and number of bins is up to the experimenter. Also, I was dealing with very steep curves (they look like an exponential decay). χ² requires the expected value in a bin to be no less than 5 (more the merrier). This severely impacts the number of possible bins for the test given my empirical distributions.

So, I took a trip to the library and found Conover’s book Practical Nonparametric Statistics. In that volume, I found descriptions of non-parametric tests. For example, the Kolmogorov-Smirnov test. This test checks the vertical distance between an empircal cumulative distribution function (ECDF) and a fully specified test distribution.

They key words here are “fully specified”. In this context it means that you can test whether a sample comes from, for example, a exponential distribution with λ=2. If that rate parameter (λ) is computed from the sample, the KS test is invalid.

Lilliefors, on the other hand, modified the KS test for cases where the parameters in the proposed distribution are estimated from the sample. The advantage here is that based simply on the sample data, the distribution can be tested for normality or exponentiality. Unlike χ², the author cannot manipulate the bin widths/numbers; the entire test is computed from the sample. i.e. it is nonparametric.

The original article by Lilliefors gives a table of p-values, and a function for arbitrary sample sizes. It has been given better approximations by Stephens and also by Mason and Bell. It seems this test is still in somewhat active development. The most recent work I’ve found is an examination of its asymptotic behavior (see Nikitin).

References

• Lilliefors, H. W. “On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown.” Journal of the American Statistical Association. Vol. 64, 1969, pp. 387–389. [jstor]
• Conover, W. J. Practical Nonparametric Statistics. Hoboken, NJ: John Wiley & Sons, Inc., 1980. [pub]
• Stephens, M. A. EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 1974, 69(347), pp. 730-737. [jstor]
• Mason, A. L. and Bell, C. B. New Lilliefors and Srinivasan Tables With Applications. Communications in Statistics: Simulation and Computation. 1986, 15(2), pp. 451-477. [doi]
• Nikitin, Y. Y. and Tchirina, A. V. Lilliefors test for exponentiality: large deviation, asymptotic efficiency, and conditions of local optimality. Mathematical Methods of Statistics, 2007, 16(1), pp. 16-24. [doi]