I created a new test for uniformity, but so far, I've had no luck finding its critical values analytically, I could only obtain them by Monte Carlo simulation. What's worse is that histograms show that the null distribution does not approach normal distribution even at large n, so I cannot approximate it with mean and standard deviation.

Is there any sort of "standard procedure" for deriving null distribution of a test statistic? Or at least approximating it with an analytical expression?

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