Large-Sample Inference for Log-Spline Models

Let f be a continuous and positive unknown density on a known compact interval 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 f, F̂ and Q̂ of $f, F$ and Q, respectively. Under mild conditions, if the number of parameters tends to infinity in a suitable manner as n → ∞, 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 F̂ and Q̂ are asymptotically equal to those of the usual empirical distribution function and empirical quantile function.