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I am looking for a very flexible bell shape function, with asymmetry on both sides of the bell, also with the possibility that the left arm of the bell had a milder slope while the right had a steep fall. Any hints, please?

whuber
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Przemyslaw Remin
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  • For a non-negative variable only? (since you mentioned "log-normal" in the title but you didn't clarify this in the question itself). – Alecos Papadopoulos Oct 30 '14 at 10:25
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    Flexibility sounds good, like motherhood and apple pie, but a family of distributions wide enough to do what you want is likely to be more difficult to work with than you really want. There are many good reasons why people tend to go for distributions with no more than about 3 parameters. Any way, the Pearson system of distributions may be a candidate for what you seek. I'd drop the "function" key tag. It has no followers. – Nick Cox Oct 30 '14 at 10:47
  • Perhaps you could disclose the reason you are seeking such a function? What would you use it to model? – whuber Oct 30 '14 at 13:57
  • I would like test if such cumulated distributions which are sigmoid functions are good enough to predict market saturation levels. I have some real data and I definitely see that the Bass diffusion model is useless here. By flexibility I mean keeping one of the arm of the bell untouched while having full control on the other arm's steepness. – Przemyslaw Remin Oct 31 '14 at 09:50
  • See https://stats.stackexchange.com/questions/554057/is-there-a-beta-distribution-over-the-entire-real-line/554077#554077 for a possible answer – kjetil b halvorsen Oct 02 '22 at 00:33

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For non-negative random variables, a Weibull distribution can exhibit a "steeper fall" to the right, with appropriate values for its shape parameter (while the scale parameter controls the values it will concentrate). For example with shape parameter $k=8$ and scale $\lambda =20$ we get

enter image description here

while with shape parameter $k=8$ and scale $\lambda =2$ we get

enter image description here

Same shape -different "main" range.

For random variables that take on positive and negative values, the Skew-normal distribution (and the numerous variations-extensions that have sprung out of it) could be a candidate, with appropriate values for its shape (or "skew") parameter.

Alecos Papadopoulos
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  • Thank you for your answer. Weibull is not flexible enough. While I want to keep the pick in the same place, let's say almost X=20 like in your first picture, I still want to control steepness of left and right arm of the bell separately. For example I want to leave left arm unchanged like in your picture, and I want to make right arm more steeper. I will not be able to do that with Weibull. The second propose of Skew-normal distribution looks like what I need. However, it has pretty complex function including integral. Could you propose function that does not contain integral? – Przemyslaw Remin Oct 31 '14 at 09:41
  • I have no other suggestion. The integral in the skew normal density is the cumulative distribution function of the standard normal, which is implemented and computed directly in all software packages, so why would it give you trouble? I saw in a comment that you moreover require a closed-form cumulative distribution function -that would be difficult to find. – Alecos Papadopoulos Oct 31 '14 at 10:33