One and two side generalisations of the log-Normal distribution by means of a new product definition

In this manuscript we introduce a generalisation of the log-Normal distribution that is inspired by a modification of the Kaypten multiplicative process using the $q$-product of Borges [Physica A \textbf{340}, 95 (2004)]. Depending on the value of q the distribution increases the tail for small (when $q<1$) or large (when $q>1$) values of the variable upon analysis. The usual log-Normal distribution is retrieved when $q=1$. The main statistical features of this distribution are presented as well as a related random number generators and tables of quantiles of the Kolmogorov-Smirnov. Lastly, we illustrate the application of this distribution studying the adjustment of a set of variables of biological and financial origin.

💡 Research Summary

The paper introduces a novel generalization of the log‑Normal distribution by replacing the ordinary product in the multiplicative stochastic process with the q‑product (⊗q) proposed by Borges (2004). In the classic log‑Normal model, a variable Y is obtained as Y = exp(∑{i=1}^N X_i) where the X_i are independent Gaussian variables. By substituting the ordinary sum with a q‑product and the ordinary exponential with the q‑exponential, the authors define

 Y_q = exp_q(⊗_{i=1}^N X_i)

and derive the corresponding probability density function

 f_q(y) =