Large-Sample Inference for Log-Spline Models

Abstract

Let $f$ be a continuous and positive unknown density on a known compact interval $\mathscr{Y}$. Let $F$ denote the distribution function of $f$ and let $Q = F^{-1}$ denote its quantile function. A finite-parameter exponential family model based on $B$-splines is constructed. Maximum-likelihood estimation of the parameters of the model based on a random sample of size $n$ from $f$ yields estimates $\hat{f, F}$ and $\hat{Q}$ of $f, F$ and $Q$, respectively. Under mild conditions, if the number of parameters tends to infinity in a suitable manner as $n \rightarrow \infty$, these estimates achieve the optimal rate of convergence. The asymptotic behavior of the corresponding confidence bounds is also investigated. In particular, it is shown that the standard errors of $\hat{F}$ and $\hat{Q}$ are asymptotically equal to those of the usual empirical distribution function and empirical quantile function.