A Flexible Continuous AP Distribution: Theory, Estimation and Applications
Ashutosh Pandey *
Department of Statistics, B.N. College, Patna University, Patna, India.
*Author to whom correspondence should be addressed.
Abstract
This study introduces a continuous probability distribution, designated the AP distribution, and examines its theoretical properties and application to observed data. The proposed density is constructed from a positive, integrable kernel and a corresponding normalising constant. Its probability density function and cumulative distribution function are presented, together with the hazard rate function. The model is symmetric about zero, and its density changes with the shape parameter. Analytical derivations address the raw moments, mean, variance, skewness, kurtosis, moment-generating function and characteristic function. In particular, symmetry implies that the odd-order moments vanish. The shape parameter is estimated using maximum likelihood estimation, with numerical optimisation implemented in R because the likelihood equation does not admit an explicit analytical solution. A numerical illustration is provided, followed by an application to 10 observations of glass fibre strength. The proposed model is compared with normal and logistic distributions using log-likelihood, Akaike information criterion, Bayesian information criterion and the one-sample Kolmogorov–Smirnov test. For the reported dataset, the normal distribution has the lowest Akaike and Bayesian information criterion values, whereas the AP distribution has the smallest Kolmogorov–Smirnov statistic. These findings demonstrate the implementation of the proposed framework and show that its comparative performance depends on the goodness-of-fit measure considered. Further applications are required to assess performance across other datasets.
Keywords: AP distribution, statistical distributions, maximum likelihood estimation, goodness-of-fit, kolmogorov–smirnov test, model comparison