Application of Univariate Probability Distributions Fitting using Monte Carlo Simulation
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In this study, we showcased a univariate probability distribution through the application of the three sub- and super-exponential heavier-longer and lighter-shorter tails fitting. The univariate family includes the Lognormal, Gamma, and Weibull distributions. The adequacy of distribution tails was determined using Adequate Fitting Tests (ADTs) and a descriptive criterion. For this purpose, this study investigated the logarithm population data sets of the Karachi region from 1729 to 1946 and again from 1951 to 2018. The data contained both irregular and regular lengths and peaks. We considered three Lognormal, Gamma, and Weibull distributions for tails fitting, statistical methods for validation, and normality tests to verify stochastic behavior in both data sets. Weibull and Lognormal distribution tails were found to possess heavier distribution using two validation tests (maximum likelihood estimation and probability of correct selection). Furthermore, the univariate probability distributions were applied to Monte Carlo simulation in order to generate the actual population data. The results indicated that heavy-tailed Lognormal and Weibull distributions were adequate as compared to the more commonly used lighter tailed Gamma distribution. Hence, it was determined that the Monte Carlo Simulation performs appropriate Lognormal and Weibull distributions for irregular and regular data. Our results indicated that Lognormal-Weibull distributions are suitable for the prediction of both long-term and short-term forecasting.
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