## The Annals of Statistics

### Large-Sample Inference for Log-Spline Models

Charles J. Stone

#### 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.

#### Article information

Source
Ann. Statist., Volume 18, Number 2 (1990), 717-741.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347622

Digital Object Identifier
doi:10.1214/aos/1176347622

Mathematical Reviews number (MathSciNet)
MR1056333

Zentralblatt MATH identifier
0712.62036

JSTOR

Subjects
Primary: 62G05: Estimation
Secondary: 62F12: Asymptotic properties of estimators

#### Citation

Stone, Charles J. Large-Sample Inference for Log-Spline Models. Ann. Statist. 18 (1990), no. 2, 717--741. doi:10.1214/aos/1176347622. https://projecteuclid.org/euclid.aos/1176347622