Bivariate Conditional Weibull-Lognormal Distribution Model and Its Application

Abstract

This study proposes a new Bivariate Conditional Weibull-Lognormal Distribution (BCWLD), in which the marginal distribution of the explanatory variable follows a Weibull distribution while the conditional distribution of the response variable follows a Lognormal distribution. The joint probability density function was derived using the conditional distribution approach. The marginal density, posterior density, and theoretical joint moments of the proposed model were also established. Model parameters were estimated using the Maximum Likelihood Estimation method, with numerical optimization performed through the L-BFGS-B algorithm. Numerical results confirmed the validity of the marginal and posterior distributions, while goodness-of-fit assessment using the Cramer-von Mises statistic supported the suitability of the Weibull and Lognormal marginal assumptions. These findings demonstrate that the BCWLD provides a more flexible and accurate framework for modelling dependence between non-negative continuous variables and represents a useful addition to the family of conditional bivariate distributions.