Stochastic Processes with Modified Lognormal Distribution Featuring Flexible Upper Tail

TL;DR

The κ-lognormal framework introduces a three-parameter deformation of the lognormal via κ-exponential and κ-logarithm, yielding lighter-than-lognormal right tails and possible bimodality to better capture skewed natural and engineering data. The authors derive closed-form expressions for the κ-lognormal PDF, CDF, quantiles, and hazard rate, along with moment bounds and power-series moment expansions, and establish a scaling relation for moments. They formulate κ-lognormal stochastic processes through Jacobi's multivariate theorem, derive joint densities, and develop warped Gaussian process regression with κ-logarithm warping for prediction in latent and observation spaces. Estimation techniques including marginal NLL optimization, gradient/Hessian expressions, and profile likelihood for joint dependence are proposed, with applications to simulated data, Berea sandstone permeability, time-series forecasting, and spatial regression, demonstrating improved tail control and flexibility over the standard lognormal. The work provides a unifying, practically relevant modeling toolkit for skewed, heavy-tailed, and potentially bimodal phenomena across time and space, with explicit formulas enabling efficient inference and prediction.

Abstract

Asymmetric, non-Gaussian probability distributions are often observed in the analysis of natural and engineering datasets. The lognormal distribution is a standard model for data with skewed frequency histograms and fat tails. However, the lognormal law severely restricts the asymptotic dependence of the probability density and the hazard function for high values. Herein we present a family of three-parameter non-Gaussian probability density functions that are based on generalized kappa-exponential and kappa-logarithm functions and investigate its mathematical properties. These kappa-lognormal densities represent continuous deformations of the lognormal with lighter right tails, controlled by the parameter kappa. In addition, bimodal distributions are obtained for certain parameter combinations. We derive closed-form analytic expressions for the main statistical functions of the kappa-lognormal distribution. For the moments, we derive bounds that are based on hypergeometric functions as well as series expansions. Explicit expressions for the gradient and Hessian of the negative log-likelihood are obtained to facilitate numerical maximum-likelihood estimates of the kappa-lognormal parameters from data. We also formulate a joint probability density function for kappa-lognormal stochastic processes by applying Jacobi's multivariate theorem to a latent Gaussian process. Estimation of the kappa-lognormal distribution based on synthetic and real data is explored. Furthermore, we investigate applications of kappa-lognormal processes with different covariance kernels in time series forecasting and spatial interpolation using warped Gaussian process regression. Our results are of practical interest for modeling skewed distributions in various scientific and engineering fields.

Figures (25)

• Figure 1: Plots of the difference $\exp_{\kappa}(x)-\exp(x)$ for $\kappa=0.1, 0.5, 0.9$ based on the $\kappa$-exponential definition \ref{['eq:exp-kappa']} for $x \ge 0$, which confirm that the natural exponential is the upper bound of the $\kappa$-exponential, namely $\exp(x) \ge \exp_{\kappa}(x)$, for $x \ge 0$ (cf. Proposition \ref{['propo:bounds-kpe']}). For $x<0$ (not shown) the sign of the difference is reversed, that is, $\exp_{\kappa}(x) \ge \exp(x)$, and the natural exponential is a lower bound of the $\kappa$-exponential.
• Figure 2: Plots of the difference $\exp_{\kappa}(y)-\exp(y)$ for $\kappa=0.1, 0.5, 0.9$ based on the $\kappa$-exponential definition \ref{['eq:exp-kappa']} for $y \ge 0$, which confirm that the natural exponential is the upper bound of the $\kappa$-exponential, namely $\exp(y) \ge \exp_{\kappa}(y)$, for $y \ge 0$ (cf. Proposition \ref{['propo:bounds-kpe']}). For $y<0$ (not shown) the sign of the difference is reversed, that is, $\exp_{\kappa}(y) \ge \exp(y)$, and the natural exponential is a lower bound of the $\kappa$-exponential.
• Figure 5: Moments of order $\ell \in \{1, 2, \ldots, 10\}$ for the $\kappa$-lognormal distribution with different $\kappa$ values. All curves are obtained by numerical evaluation of the integral \ref{['eq:moments-integral']} for $\mu=0$ and $\sigma=1$. The horizontal axis shows the moment order $\ell$. The vertical axis uses a logarithmic scale to display $m_{X;\ell}^{1/\ell}(\kappa)$ where $m_{X;\ell}(\kappa) \triangleq m_{X;\ell}(\kappa;0,1)$; the $\ell$-th root of the order-$\ell$ moments is used in order to maintain a common scale for different $\ell$.
• Figure 6: Left: Mean of the lognormal distribution (continuous line, blue online) with $\mu=5$ and $\sigma=2$ calculated by means of numerical integration (cf. \ref{['eq:moments-integral']}) and lower bound of the first-order moment based on \ref{['eq:moments-lb']} (circle markers, red online). Right: Root of order $\ell$ of the expectation $m_{X;\ell}(\kappa;\mu,\sigma)$ versus $\ell$ for $\mu=5$ and $\sigma=2$. Continuous lines represent the numerical integral \ref{['eq:moments-integral']} while markers denote the lower bounds \ref{['eq:moments-lb']}. Four different values of $\kappa$ are used: $\kappa \in \{0.4, 0.5, 0.75, 0.95 \}$.
• Figure 7: Left: Mean of the $\kappa$-lognormal distribution (continuous line) with $\mu=-2$ and $\sigma=2$ versus $\kappa$ (continuous line, blue online). The expectation is calculated by means of numerical integration (cf. \ref{['eq:moments-integral']}). The lower bound of the mean as a function of $\kappa$, based on \ref{['eq:moments-lb']}, is shown using circle markers (red online). Right: Root of order $\ell$ of the moment $m_{X;\ell}(\kappa;\mu,\sigma)$ versus $\ell$ for $\mu=-2$ and $\sigma=2$. Continuous lines represent the numerical integral \ref{['eq:moments-integral']} while markers denote the lower bounds \ref{['eq:moments-lb']}. Four different values of $\kappa$ are used: $\kappa \in \{0.4, 0.5, 0.75, 0.95 \}$.

Theorems & Definitions (22)

• Proposition 1: Bounds of $\kappa$-exponential
• Proposition 2: $\kappa$-logarithm increases faster than the natural logarithm
• Proposition 3: $\kappa$-logarithm convexity
• Definition 1: Lognormal stochastic process
• Definition 2: $\kappa$-lognormal distribution
• Theorem 1: Modes of $\kappa$-lognormal PDF
• Remark 3: Number of Modes of $\kappa$-Lognormal PDF
• Theorem 2: Statistical moments of integer order
• Remark 4: On moment scaling
• Theorem 3: Moment bounds