Boundary adjusted, polynomial adaptive, nonparametric kernel conditional density estimation

We consider local polynomial nonparametric kernel estimation of conditional density functions g(y|x) when bounds [a,b] on the response Y may be known or unknown. When bounds are known, we use kernel functions for the response that exploit their presence, while when bounds are unknown we advocate using the empirical support [min(Yi),max(Yi)] in their place. A key feature of this approach is the use of data-driven methods for joint selection of the order of the approximating local polynomial, p, and the bandwidths for the response and predictors, hy and hx. This approach is able to automatically adjust to the presence of unknown bounds and automatically adapt the degree of the approximating polynomial and the bandwidths to the underlying data generating process. Theoretical underpinnings are provided, and simulations indicate that this approach is particularly useful when the distribution of Y is bounded with unknown bounds, while it remains competitive with existing methods in infinite support settings. An application to European patent data reveals that the conditional distribution of patent applications exhibits substantial probability mass at the lower boundary, and our method adapts to this feature in a way that existing approaches cannot. The R package np implements these methods for interested readers.