Inverse Gaussian versus lognormal distribution in earthquake forecasting: keys and clues

In earthquake fault systems, active faults release elastic strain energy in a near-repetitive manner. Earthquake forecasting that basically refers to the assessment of earthquake hazards via probability estimates is crucial for many strategic and engineering planning. As the current need across sciences dominantly grows for conceptualization, abstraction, and application, comparison of lifetime probability distributions or understanding their physical significance becomes a fundamental concern in statistical seismology. Using various characteristic measures derived from density function, hazard rate function, and mean residual life function with its asymptotic (limiting) behavior, the present study examines the similitude of the two most versatile inverse Gaussian and lognormal distributions in earthquake forecasting. We consider three homogeneous and complete seismic catalogs from northeast India, northwest Himalaya, and Kachchh (western India) region for illustration. We employ maximum likelihood and moment methods for parameter estimation, and Fisher information for uncertainty valuation. Using three performance tests based on Akaike information criterion, Kolmogorov-Smirnov criterion, and Anderson-Darling test, we show that the heavy-tailed lognormal distribution performs relatively better in terms of its model fit to the observed data. We envisage that the ubiquitous heavy-tailed property of lognormal distribution helps in capturing desired characteristics of seismicity dynamics, providing better insights to the long-term earthquake forecasting in a seismically active region.