# Which distribution creates an exponential-like pattern in x log scale?

I've been reading this very nice paper by Baltrunas et al. and I would like to have a distribution that looks as much as possible like the empirical data the authors found in the figure below:

I don't have access to the data and it doesn't have to be exact, so I've modeled the green, purple and black lines with a lognormal distribution (s=2, loc=0, scale=10^3) which looks like this:

However, I can't find a distribution that behaves like the blue line. The line looks like an exponential CDF pattern but the x axis is in log scale. Is there a known distribution which I can use as reference for this case?

• You need to fit the logs of $x$ against $y.$ Try a Box-Cox parameter of $-1$; that is, look for a relation of the form $y=\alpha+\beta / ({\log}_{10}x).$ A value of $-1/2$ will work better for smaller $x$ but not as well for larger $x:$ use your judgment to fit the data within the most useful range for your application.
We can approximate the blue line by a quarter-ellipse with center at $$(10^6,0)$$, major axis going through $$(10^{0.5}, 0)$$ and minor axis going through $$(10^6, 1)$$. This leads to the equation $$F(x)=\sqrt{1- \left(\frac{\log(10^6/x)}{\log(10^{5.5})}\right) ^{\!\!2}}$$ Using $$\log$$s in base $$10$$, this leads to a pdf of $$f(x)= \frac{(6-\log x)/5.5}{x\sqrt{5.5^2 - (6-\log x)^2}}$$ for $$\sqrt{10} < x < 10^6$$, and zeroes outside that range.