# Equivalent of a flipped lognormal distribution

What distribution could represent a "flipped" (skewed left) lognormal distribution?

For ex: what name would you do to the distribution in the figure below? I fitted the histogram with a Beta distribution since the values of regularity are between 0 and 1. Is that correct? Is there a better/more flexible approach? I use Matlab..

• Flipping a log-normal results in a left skewed distribution on the negative numbers, bounded above by zero. Is that what you meant? And if so, why would you need another distribution to "represent" it? – Scortchi - Reinstate Monica May 30 '16 at 15:17
• By adding the picture, you have entirely changed the question. (a) A Lognormal (reflected or otherwise) is not bounded on (0,1). (b) There is a hump (or perhaps discrete mass) in your data at around $X = \frac14$ which suggests there is still more going on here than you describe. – wolfies Jun 1 '16 at 7:11
• yep! that can be neglected...I know what it is... I would like to have a good estimation of the mode... so I fit (at the moment) a beta distribution and compute the mode accordingly... not sure if my approach is correct. ps that bumb is relative to a cycling activity... the other are walks.. – gabboshow Jun 1 '16 at 7:20
• Should I maybe use a bimodal distribution? and take the two means? – gabboshow Jun 1 '16 at 7:23
• I'm puzzled by these extra questions: you can see perfectly well from your plot that the fitted beta distribution gives a low estimate of the major mode. I think at this point a new question would be in order: along the lines of how to estimate the mode(s). You might well be advised to take a further step back & explain the context & what you're really trying to find out. – Scortchi - Reinstate Monica Jun 1 '16 at 13:46

A reversed Lognormal ...

I will use the notation here that is common in defining the Johnson family, since the latter commonly provides a 3 or 4 parameter version of the Lognormal that captures that which you seek.

If $Z \sim N(0,1)$, and $Y=\exp\big({\frac{Z-\gamma}{\delta }}\big)$, then $Y$ has a Lognormal distribution with pdf say $f(y)$:

$$f(y) = \frac{\delta}{y \sqrt{2 \pi }} {\exp\big[{-\frac{1}{2} \big(\gamma +\delta \log (y)\big)^2}\big]} \quad \quad \text{ for } y > 0$$

Applying a second transform $X=\xi -Y$ yields the reversed Lognormal that you seek, with pdf say $g(x)$:

$$g(x) = \frac{\delta}{(\xi -x) \sqrt{2 \pi }} {\exp\big[{-\frac{1}{2} \big(\gamma +\delta \log (\xi -x)\big)^2}\big]} \quad \quad \text{ for } x< \xi$$

Example

The following diagram represents grouped data from Table 1 in Pretorius (1930, p.148). Here, $X$ denotes barometric height (grouped data), while the vertical axis denotes observed frequency.

• The blue square curve represents the grouped data
• The red curve is a fitted reverse Lognormal using the automated JohnsonSL function from the mathStatica package for Mathematica. References

• Pretorius, S. J. (1930), Skew bivariate frequency surfaces, examined in the light of numerical illustrations, Biometrika, 22, 109-223.
• thanks a lot for your detailed answer! Any idea how to fit this family of distribution using Matlab in order to estimate the Mode of the fitted distribution? – gabboshow Jun 1 '16 at 7:09

Independent of scale (min,max of x-values), a beta distribution can have a left tail, and so can a power function distribution. However, if your data did result in a left tail on a histogram, you could probably fit it very well using the stable distribution, which has four parameters. The stable is used in QF (quantitative finance) to fit log-returns of asset prices, and can take on mixtures of distributions like Cauchy, Laplace, non-central Student's t, beta, logistic, normal, etc. Don't look for it in software packages, since it's not that popular of a distribution in statistics.