Asymptotics for Doubly Flexible Logspline Response Models

Consider a Y-valued response variable having a density function f (x) that depends on an X-valued input variable x. It is assumed that X and Y are compact intervals and that f () is continuous and positive on X Y. Let F (x) denote the distribution function of f (x) and let Q (x) denote its quantile function. A finite-parameter exponential family model based on tensor-product B-splines is constructed. Maximum likelihood estimation of the parameters of the model based on independent observations of the response variable at fixed settings of the input variable yields estimates of f (), F () and Q (). Under mild conditions, if the number of parameters suitably tends to infinity as n, these estimates have optimal rates of convergence. The asymptotic behavior of the corresponding confidence bounds is also investigated.