I have closed form moment-generating function and characteristic function of a distribution, which describes waiting time of a continuous univariate random process. However, I cannot analytically invert the characteristic function to get PDF.

Now, I have run simulations of the original process in order to verify that my model adequately describes the process. Because data in this case are simulated, I exactly know model parameters. Therefore, I am not interested in parameter estimation for now.

By eyeballing histogram of simulated data and numerically computed PDF, my model seems to fit the data very well. However, I want to say something more concrete and quantitative. Is there any goodness-of-fit test with arbitrary distribution, whose PDF and CDF are not analytically available? If not, what other approaches can I take?

  • $\begingroup$ What's the matter with numerically inverting the cf? $\endgroup$ – whuber Nov 9 '11 at 2:30
  • $\begingroup$ @whuber thank you for the comment. I tried to use Mathematica to invert the cf. However, it had a problem in integration under some parameter combinations. I then switched to FFT-based approximation, which did not suffer the same problem. However, the latter approach did not give densities at dense enough points. Probably, can I try interpolation to fill the gaps? $\endgroup$ – Seiji Kumagai Nov 9 '11 at 20:23
  • $\begingroup$ For a GoF test you usually don't need the PDF: you only need the CDF at a fairly dispersed set of points. That might be more reliably obtained than the PDF. $\endgroup$ – whuber Nov 9 '11 at 20:43

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