Asymptotics for Doubly Flexible Logspline Response Models

Consider a $\mathscr{Y}$-valued response variable having a density function $f(\cdot\mid x)$ that depends on an $\mathscr{X}$-valued input variable $x.$ It is assumed that $\mathscr{X}$ and $\mathscr{Y}$ are compact intervals and that $f(\cdot\mid\cdot)$ is continuous and positive on $\mathscr{X} \times \mathscr{Y}.$ Let $F(\cdot\mid x)$ denote the distribution function of $f(\cdot\mid x)$ and let $Q(\cdot\mid x)$ 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 $f(\cdot \mid \cdot), F(\cdot \mid \cdot)$ and $Q(\cdot \mid \cdot).$ Under mild conditions, if the number of parameters suitably tends to infinity as $n \rightarrow \infty,$ these estimates have optimal rates of convergence. The asymptotic behavior of the corresponding confidence bounds is also investigated.