  Understanding variable dependence, particularly eliciting their statistical
properties given a set of covariates, provides the mathematical foundation in
practical operations management such as risk analysis and decision-making given
observed circumstances. This article presents an estimation method for modeling
the conditional joint distribution of bivariate outcomes based on the
distribution regression and factorization methods. This method is considered
semiparametric in that it allows for flexible modeling of both the marginal and
joint distributions conditional on covariates without imposing global
parametric assumptions across the entire distribution. In contrast to existing
parametric approaches, our method can accommodate discrete, continuous, or
mixed variables, and provides a simple yet effective way to capture
distributional dependence structures between bivariate outcomes and covariates.
Various simulation results confirm that our method can perform similarly or
better in finite samples compared to the alternative methods. In an application
to the study of a motor third-party liability insurance portfolio, the proposed
method effectively estimates risk measures such as the conditional
Value-at-Risk and Expected Shortfall. This result suggests that this
semiparametric approach can serve as an alternative in insurance risk
management.
