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I have a collection of (200) points that are well-fitted by a (negatively-skewed) beta distribution with parameters $\alpha=3.1$ and $\beta= 1.9$. This is all fine and good, of course, but I am interested in possible theoretical underpinnings/explanations of the curve, so I would appreciate any suggestions for alternative distributions/mechanisms to examine. (The 200 points are, in fact, residuals from the fit of the square of a function [https://arxiv.org/pdf/1610.01410.pdf, eq. (9)] involving--after performing the indicated integration--hyperbolic tangents and polylogarithms.)

Actually, I was initially trying to ask a broad question and not get into too many technicalities/specifics. But, in response to the comment of whuber, let me note that, in fact, the residuals are well-fitted by the indicated beta distribution SCALED by 1/45. An initial plot of the residuals was displayed in Fig. 6 of https://arxiv.org/pdf/1701.01973.pdf (but an updated/corrected plot [can I attach the pdf?] has no negative values).

Again, I must admit that my question is really not a statistical one per se, I was only looking for possible functional forms (maybe somehow related to hyperbolic trigonometric/logarithmic ones) of interest to fit the curve of residuals.

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  • $\begingroup$ Check en.wikipedia.org/wiki/Kumaraswamy_distribution $\endgroup$
    – Tim
    Commented Jan 19, 2017 at 19:18
  • $\begingroup$ I cannot match that distribution to your function unless I (a) change $\alpha$ to approximately $2.3$ and (b) divide it by approximately $47$. You seem only to be attempting to approximate a function of $\varepsilon$ by a formula of the form $C\varepsilon^{\alpha-1}(1-\varepsilon)^{\beta-1}$--but what does that have to do with "distributions," "mechanisms," or anything else statistical? It appears that for us to be able to address your question you would (at a minimum) have to explain what your point collection means. $\endgroup$
    – whuber ♦
    Commented Jan 19, 2017 at 19:18
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    $\begingroup$ Aren't you actually trying to fit your data rather than approximate the distribution that approximates the data (these residuals you mention)? (And if so, why are you trying to do that? What is the use to which you'll put the fitted distribution?) $\endgroup$
    – Glen_b
    Commented Jan 20, 2017 at 3:37
  • $\begingroup$ Thanks, Glen-b. Well, I was speculating that the residuals might have some underlying structure--that since they are all nonnegative on [0,1]--might, at least, be proportional to one of the known probability distributions on that interval. In any case, this is all quite exploratory/tentative in nature. $\endgroup$
    – Paul B. Slater
    Commented Jan 20, 2017 at 22:58

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