# Anderson Darling exponential distribution

I need a goodness of fit test for the exponential distribution. I understand that Kolmogorov-Smirnov is not generally regarded as very powerful and that Anderson-Darling is regarded as superior. However I have two problems.

• None of the literature I can find specifically talks about goodness of fit for exponential distributions. Anderson-Darling is almost always discussed wrt the normal distribution.
• Is there an R (or python) package to do an Anderson-Darling goodness of fit for the exponential distribution? I can find them for other distributions.
• Is there a better test (which exists in R or python)?
• Is this a completeley specified distribution or are we estimating parameters? – Glen_b Aug 15 '14 at 8:08
• @Glen_b In some case I will have to estimate the parameters which I know causes another problem. A method which had a correction factor for that case would be even better of course! – felix Aug 15 '14 at 8:28
• When you say A-D's more powerful than K-S, what kind of departures from exponentiality are you thinking of? – Scortchi Aug 15 '14 at 9:58
• @Scortchi I am thinking for example if the tail is too long. A test just for that case would be great. – felix Aug 15 '14 at 18:17
• I'd guess the LRT for Weibull shape parameter less than one would have good power against long tailed alternatives. – Scortchi Aug 18 '14 at 9:54

The same considerations apply as to the distribution of the Kolmogorov–Smirnov test statistic discussed here. The Anderson–Darling test statistic (for a given sample size) has a distribution that (1) doesn't depend on the null-hypothesis distribution when all parameters are known, & (2) depends only on the functional form of the null-hypothesis distribution when location & scale parameters are estimated. I don't know of an R implementation of the A–D test specifically for the exponential distribution with estimated rate parameter, but you could quickly make a function to calculate the test statistic by adapting the ad.test function from the nortest package: change the distribution function from the best-fit normal, pnorm((x - mean(x))/sd(x)), to the best-fit exponential,pexp(x/mean(x)). Then get critical values for any desired significance level & sample size by simulation.
• I'm saying that rather than write a function to calculate the A-D statistic from scratch, you can modify the code used in ad.test. (Type ad.test to see it.) – Scortchi Aug 18 '14 at 9:52
• @Scortchi, how did you know to change pnorm((x - mean(x))/sd(x)) for a normal distribution to pexp(x/mean(x)) for an exponential distribution? I am looking to test weibull and lognormal distributions thorough the AD statistic – lukeg Oct 11 '15 at 17:36