Parameter orthogonality transformations in distributional regression models

Distributional regression has become an increasingly popular tool in regression analysis, providing the ability to link distributional parameters beyond the mean to additive predictors. For many response distributions, maximum likelihood estimates (MLEs) of the distributional parameters are correlated and, as a result, misspecifying the regression predictor for one distributional parameter induces a bias in the estimates for the predictors of other parameters, even if these are correctly specified. In this article we develop a framework for the reparametrization of any two-parameter distribution, such that its parameters have asymptotically uncorrelated MLEs i.e., are orthogonal. We propose to retain the location parameter and replace the second parameter of the chosen distribution with a function that provides a numeric solution to an ordinary differential equation (ODE) based on the covariance matrix of the MLEs, creating an orthogonal parametrization. Finally, our proposed method is demonstrated on an extreme rainfall dataset from Tasmania, Australia.