1990/06/01 by Charles J. Stone · 4 citations
Mathematics · Computer Science · #Statistical Methods and Inference #Bayesian Methods and Mixture Models #Statistical Methods and Bayesian Inference
paper · pdf · doi:10.1214/aos/1176347622
openalex publication_date 1990/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let f be a continuous and positive unknown density on a known compact interval \mathscrY. 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, F and Q 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 F and Q are asymptotically equal to those of the usual empirical distribution function and empirical quantile function.