It is rarely a good idea to try to infer distribution parameters from deep tail percentiles. Those points deep in the tails would be determined from only a few data points. As such, they are very poorly known. And that means your estimates of the distribution parameters will be highly variable, highly sensitive to noise.
Even worse,by the use of two quantiles very near to each other, you are effectively using only a very few points way into the fringes, and they are the same data points used for both quantiles.
Instead you will be far better off using a quantiles that would be closer to the mode of the distribution. For example, you might have chosen the 50 and 75% points.
Is it a great idea to use only two quantiles to try to infer the distribution parameters? Probably not. But if you are going to do something, you want to choose a scheme that is robust to noise.
As an example, I'll choose a random sampling of 1000 numbers from a lognormal distribution. Then compute several sets of quantiles, and see how well I can recover the original distribution parameters for the lognormal. Next, I'll do this operation multiple times, and we can see how robust is the operation, to infer distribution parameters from two quantiles, and see how well the computation performs, depending on the actual quantiles chosen. Surely this is a fair test.
The actual distribution parameters for the lognormal are irrelevant, since that does nothing to change such a test. As such, I'll choose a standard lognormal for my computations. A standard lognormal has distribution parameters of mu=0, sigma=1. Again, all that matters is how well I can predict those numbers.
For the computational engine, I used the code written by @the cyclist, which seems reasonable (though really slow, since it uses vpasolve). fsolve would be faster, but vpasolve will do entirely well for my purposes. First, does the code work? X = lognrnd(0,1,[1,1000]);
[muest,sigest] = lognest(P,X)
As you can see, this code seems to work. It reproduces the population parameters reasonably well.
Next, a simulation, to test the performance of the scheme, using various quantile sets. I'll choose two quantile sets, one deep in the tails, and the second using quantiles that will live where the distribution has significant mass.
Xsim = lognrnd(0,1,[1,1000]);
Xset1 = prctile(Xsim,Qset1);
Xset2 = prctile(Xsim,Qset2);
[muest(1,i),sigest(1,i)] = lognest(Qset1,Xset1);
[muest(2,i),sigest(2,i)] = lognest(Qset2,Xset2);
title('Histogram of estimates for mu')
title('Histogram of estimates for sigma')
In each histogram, you can see the two sets of estimates for the parameters mu and sigma. In blue are the estimates when the 50% and 75% quantiles were used to infer the lognormal parameters. In yellow, you see the estimates of those same two parameters, but this time using the 97.5% and the 99% quantiles from the data.
function [muest,sigest] = lognest(perc,x)
eqn1 = erfc(-(log(x(1))-m)/(s*sqrt(2)))/2 == perc(1)/100;
eqn2 = erfc(-(log(x(2))-m)/(s*sqrt(2)))/2 == perc(2)/100;
[muest,sigest] = vpasolve(eqn1,eqn2,[m,s]);
You need ot understand that the scheme you want to use to infer the lognormal parameters using deep tail quantiles is a poor one. It is highly sensitive to random perturbations from only a few data points. As such, it is not at all robust. Worse, it will also be highly sensitive to deviations from a true lognormal distritribution, since it is deep in the tails that a lognormal distribution will be most different from some other distribution.
Finally, using only two quantiles to infer the population parameters is itself arguably a poor scheme, since you are not really using the entire set of data. A better method would be maximum likelihood estimation, where all of the data can be employed, instead of effectively only a few data points.
