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June, 1990 Large-Sample Inference for Log-Spline Models
Charles J. Stone
Ann. Statist. 18(2): 717-741 (June, 1990). DOI: 10.1214/aos/1176347622


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.


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Charles J. Stone. "Large-Sample Inference for Log-Spline Models." Ann. Statist. 18 (2) 717 - 741, June, 1990.


Published: June, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0712.62036
MathSciNet: MR1056333
Digital Object Identifier: 10.1214/aos/1176347622

Primary: 62G05
Secondary: 62F12

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 2 • June, 1990
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