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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.

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  • $\begingroup$ 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.) $\endgroup$
    – Glen_b
    Commented Dec 18, 2022 at 8:42
  • $\begingroup$ 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 ? $\endgroup$
    – pecer10012
    Commented Dec 18, 2022 at 8:52
  • $\begingroup$ 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 $\endgroup$
    – pecer10012
    Commented Dec 18, 2022 at 9:00
  • $\begingroup$ 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. $\endgroup$
    – Glen_b
    Commented Dec 18, 2022 at 9:32
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    $\begingroup$ 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. $\endgroup$
    – Glen_b
    Commented Dec 18, 2022 at 11:03

1 Answer 1

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If you want to test the hypothesis of the observations being i.i.d. draws from an exponential(1) distribution, yes, you could use a Kolmogorov-Smirnov. Of the tests you mention, a chi-squared test of an exponential(1) is possible (but involves several more choices, such as the number of bins and the locations of the bin boundaries).

Typically, of those two, the Kolmogorov-Smirnov would tend to have better power against a wide range of alternatives of typical interest, but there are many other tests of fully specified distribution (including Cramer-von Mises and Anderson-Darling). Which you might choose depends what sorts of alternatives you seek power against. The Anderson Darling would tend to do better than the Kolmogorov-Smirnov against heavier-tailed alternatives, for example but would often be less effective at picking up a shift of probability in the middle of the distribution.

None of these could be claimed to be "the most" appropriate without an explicit definition of appropriateness. They all have advantages and disadvantages. Like many omnibus tests, all of the tests mentioned here are biased against some alternatives, at least at some sample sizes.

The Lilliefors test for an exponential would test for an exponential with an unspecified parameter. The chi-squared test could be used here as well with the loss of one d.f. for the parameter estimation (if that was based on the binned data; if the estimate was based on unbinned data, the situation is more complicated). There are many other tests for general exponentiality.

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    $\begingroup$ Thanks for the perfect explanation @Glen_b $\endgroup$
    – pecer10012
    Commented Dec 18, 2022 at 10:10
  • $\begingroup$ 2 more points to be able to upvote your answers :) $\endgroup$
    – pecer10012
    Commented Dec 18, 2022 at 10:11

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