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Is anyone aware of an approximation to the density function for the studentized range distribution https://en.wikipedia.org/wiki/Studentized_range_distribution ? I've found a fast approximation for quantiles made in excel https://www.ars.usda.gov/ARSUserFiles/60540520/RapidCalculationOfQ.pdf And I've found Fortran code for the CDF and another function in the jStat library for cdf values; however I'm curious if there are any coarse approximations for the pdf that are out there - this is to assist in fast plotting of the density curve- it's just a visualization and doesn't need to be too precise.

I appreciate any suggestions anyone has. Thanks!

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    $\begingroup$ jstor.org/stable/2347300 Of course, you can always numerically differentiate the CDF. $\endgroup$
    – whuber ♦
    Commented Mar 22, 2023 at 21:33
  • $\begingroup$ Thanks - Looks like some straight forward algorithms for approximating the cdf and the inverse cdf. $\endgroup$
    – Brian Powers
    Commented Mar 22, 2023 at 21:39
  • $\begingroup$ But nothing to give an approximate density function unfortunately. $\endgroup$
    – Brian Powers
    Commented Mar 23, 2023 at 1:31
  • $\begingroup$ I don't follow--if you have an accurate computation of the CDF, you're good to go. $\endgroup$
    – whuber ♦
    Commented Mar 23, 2023 at 1:32
  • $\begingroup$ Yes, you're right - I can try to adapt the code - I'll dig into it and see if I can find a speedy algorithm; The goal is to create something in javascript that can generate an approximate density curve in a few milliseconds. I guess the thing to do is to just calculate F(x) and F(x+delta) and say f(x)=[F(x+delta)-F(x)]/delta $\endgroup$
    – Brian Powers
    Commented Mar 23, 2023 at 1:59

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