Interpretting the p-value when inverting the null hypothesis

Question

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?

Answer 1

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.