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Thanks, Dave. I think I requested the Biometrika and I'll look into the 2nd one when I get back to home base next week.
Yes, I see why checking delta F(x) and delta F(x-) at each sample datum x solves the problem. I wonder though whether the Lilliefors thresholds (for example, in their paper) were generated with the statistic defined in this way. Or is it the case that nowadays there are a plethora of threshold tables, each with their own particular definition, ideally documented to this level of detail.
I think the treatment of the KS statistic was well-established well before Lilliefors in 1967.
Modern treatments tend to present results for the critical points in a different way,.... via an adjustment formula using the sample size to give an adjusted statistic to refer to a simple table (one value for each percentage point). Results have been published for several types for distribution being fitted, including "no-fitting". The last case would enable a check that the simulation results for large samples agree with the theoretical asymptotic distribution and hence a check on the programming of the statistic.