The Logistic-Gompertz-Law Distribution: A Statistical Framework for Modeling Legal Case Durations and Judicial Process Efficiency in Contemporary African Judiciary Systems

This extensive study presents the new four-parameter probability distribution Logistic-Gompertz-Law (LGL) which is an original distribution that is designed to deal with the intricate time-dependence nature of the legal process in modern situations of the African courts system. Based on the classical Gompertz and Logistic distributions, the LGL distribution, a hybrid of sigmoidal growth dynamics and flexible hazard functions, will provide a potent statistical approach to the modeling of the legal case duration, the probability of a settlement, as well as the judicial efficiency metrics in various jurisdictional settings. We offer derivations of its statistical properties, such as closed-form expressions of probability density and cumulative distribution functions, hazard rates, and moment properties which have been proved up by long analytical methods. With extensive Monte Carlo simulations of three different legal cases that are common in African courts, that is, efficient, complex, and standard case proceedings, we reveal that the distribution performs well in terms of exhibiting the traditional S-curve trend of legal cases long observed by legal practitioners but previously immeasurable. Estimation procedures based on both maximum likelihood and Bayesian models are derived and proven. In goodness-of-fit, the LGL distribution always outweighs the traditional ones (Weibull, Gamma, Log-Normal), with the KS value of 0.1901 to 0.3263 and correlation coefficients greater than 0.95 in all cases, and indicating its greater applicability to the law-related temporal data. Applications to field Practical uses in case duration prediction, judicial efficiency measures, and legal resources optimization in the African judicial setting are well elaborated making the LGL distribution a significant legal analytics, court management and evidence-based judicial policy development tool in the developing judicial environment.