# Goodness-of-fit Tests

I wish to test whether a large number of observations $$X_i$$ follows an exponential distribution with parameter $$\lambda=1$$. I also wish to test this hypothesis exactly, and intend that if the observations follow an exponential distribution with a different parameter, the test should reject the null hypothesis given sufficiently many observations. In addition, I want to have a numeric statistic that I could report and do not want the procedure to involve rounding off observation numbers into bins.

Which of these goodness-of-fit tests would be the most appropriate for this purpose?

• Chi-squared Test
• Kolmogorov-Smirnov Test
• Kolmogorov-Liliefors Test
• Quantile-quantile plots

I'm opting for Kolmogorov-Smirnov Test. Is this the best one?

I could also go for QQ-plots but it's a graphical method and I want to test the hypothesis exactly. Maybe someone with experience on this field could share their thoughts.

• 1. When you say "test the hypothesis exactly" what do you mean? $\,$ 2. Most appropriate by which criteria? (If you like power, you might consider a likelihood ratio test.) Commented Dec 18, 2022 at 8:42
• Yes, you are right, LRT is very powerful. I don't want anything too manual like QQ-Plot nor anything too complex like LRT. The KS test is one of the most useful, exact and general goodness of fit method which is why I thought it will do the job perfectly fine. Out of the 4 I've listed, you think KS is the way to go @Glen_b ? Commented Dec 18, 2022 at 8:52
• With "test the hypothesis exactly" I meant in the sense of having certain conditions and criteria that are met such as, for example, critical value or p value. It's harder to tell by the eye in contrast numbers don't lie if you know what I mean @Glen_b Commented Dec 18, 2022 at 9:00
• Oh, you mean "a formal hypothesis test procedure" rather than "an exact test" in the statistical sense? The Kolmogorov-Smirnov - and many other tests would work "perfectly fine" but your question didn't ask for "perfectly fine", it explicitly sought to optimize something (that word "most" -- which is why I suggested power, since (a) you mentioned you wanted to reject with enough data and more power means you can reject with fewer observations. and it is something people do regularly seek to optimize). It's not clear what basis you'd choose to prefer one test to another. Commented Dec 18, 2022 at 9:32
• Incidentally if you're checking a distributional assumption to decide whether you should hold doubt about the suitability of some inference that relied on the assumption, I'd very much lean toward a graphical assessment over a formal test. Commented Dec 18, 2022 at 11:03