Random distributional response model based on spline method

The aim of this paper is to develop a random distribution on scalar regression model framework to account for the entire subject-specific distribution of the outcome, and to relate these distributions to a set of covariates. We develop a basis representation by approximating quantile functions of distributions measured at the grid points in [0,1], based on the non-decreasing basis function. Then, we estimate the basis coefficients of covariates by using an empirical risk minimization method, with the restriction that all subject-specific quantile functions have their monotone increasing property in the original data space. Finally, we conduct a test procedure to identify the association between the distribution and a set of covariates. We systematically investigate the rate of convergence of our estimator and its asymptotic distribution under the nullity. We also show our estimation and test procedures through simulations and an analysis of Canadian weather data obtained from a historical database.