Lognormaljakaumia

Lognormaljakaumia is a hypothetical probability distribution used in statistical modeling to describe positive-valued quantities that exhibit lognormal-like variability with occasional multiplicative shocks.

Definition: If Y is normally distributed with mean mu and variance sigma^2, written Y ~ N(mu, sigma^2), and J is an independent positive random variable representing shocks with distribution F_J, then X = exp(Y) * J is said to have the lognormaljakaumia distribution. In the special case where J equals 1 almost surely, X reduces to the standard lognormal distribution.

Properties: The distribution preserves the base lognormal form but introduces extra variability through the shock factor J. It can be viewed as a lognormal mixture and may exhibit heavier tails or multiple modes depending on F_J. Moments exist when E[J^k] is finite, and closed forms for the density or moment generating function are generally not available except in specific subcases.

Estimation and inference: Parameters mu and sigma, together with the distribution of J, are typically estimated by maximum likelihood or Bayesian methods. Inference often requires numerical integration or Monte Carlo techniques, and identifiability can be challenging when J and the lognormal component interact closely. Model checking uses diagnostic plots and tail-index comparisons.

Applications: The concept has been proposed for modeling financial assets with random scaling, environmental measurements subject to episodic events, and reliability data where products experience variable operating conditions. It is used mainly in theoretical discussions and as a tool for exploring how multiplicative shocks affect lognormal variability.

See also: lognormal distribution, mixture models, product distributions, jump diffusion processes.