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I'm performing an analysis by fitting data points to an exponential distribution and testing the adequacy of this fit using the Kolmogorov-Smirnov test. The standard ks test null hypothesis: the data is indeed derived from an exponential distribution. The ks test statistic is compared with a critical value and the null hypothesis is rejected if it exceeds this critical value. I used this paper to find the critical values. I find the test statistic falls below this threshold, even for a significance level as high as 0.20.

Now here is my question. I've read that a smaller p-value provides stronger evidence supporting an alternative hypothesis, and that one usually looks to 0.05 as a threshold to reject the null hypothesis. However, no significance level is considered to support the null hypothesis. Is there any way to interpret the conventional p-value as supporting the null hypothesis? Is there anything wrong with this interpretation?

AdamO
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pvalue_null_hypothesis
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    It may help you to read this: [Why do statisticians say a non-significant result means “you can't reject the null” as opposed to accepting the null hypothesis?](https://stats.stackexchange.com/q/85903/7290) – gung - Reinstate Monica Jun 01 '18 at 18:54
  • It also depends on the specific kind of test. For example the one-way ANOVA's *F* statistic *p*-value does have an interpretation as supporting an omnibus test for equivalence (see Wellek, S. (2010). *Testing Statistical Hypotheses of Equivalence and Noninferiority*. Chapman and Hall/CRC Press, second edition, chapter 7), but the same is definitely not true of *t* tests which [require a different formulation of the null and test statistic](https://stats.stackexchange.com/tags/tost/info). – Alexis Jun 01 '18 at 20:19
  • You can't use a KS test to test with a fitted parameter -- it's for a fully specified distribution; you need a Lilliefors test (in this case the version for the exponential). – Glen_b Jun 04 '18 at 14:56

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In this case, you would never be able to interpret the p-value in terms of the "flipped" null hypothesis because it's possible for a non-exponential distribution to be arbitrarily close to the actual exponential distribution. That means such a test has 0 power always. In fact, considering all possible alternatives, non-exponential distributions (specifically the empirical distribution function) will always fit the data as good if not better than the exponential maximum likelihood estimate.

AdamO
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