I'm working on samples that I'm trying to fit into log-normal distributions. In some cases, Kolmogorov-Smirnov test statistics is something like D = 0.0056 with an associated p-value of 0. Hence, my sample shows very small departures from a theoretical log-normal distribution, but looking at the p-value I reject the null hypothesis that that my sample is drawn from the reference distribution (log-normal).
KS-test is performed through R code:
sample.z <- std(sample) # I standardize data to allow for comparisons LN <- rlnorm(1e5, 0, 1) # theoretical lognormal with mean = 0 and sigma = 1 ks.test(sample.z, LN)
Looking at the QQ-plot I see significant departures in the tails of my distributions. Thus, I was thinking that maybe I'm getting these results because of Pareto tails in my approximately log-normal distributions. Indeed,
confirms this hypothesis by identifying a Pareto tail after a certain lower bound (xmin) with a certain alpha.
Now, I would like to show the fit of my Pareto tail in the QQ-plot. How can I do that? Can you suggest other data-visualization methods to stress out the presence of Pareto tails in log-normal distributions?