The Annals of Statistics

A Regression Type Problem

Yannis G. Yatracos

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Abstract

Let $X_1, \cdots, X_n$ be random vectors that take values in a compact set in $R^d, d = 1, 2$. Let $Y_1, \cdots, Y_n$ be random variables (the responses) which conditionally on $X_1 = x_1, \cdots, X_n = x_n$ are independent with densities $f(y \mid x_i, \theta(x_i)), i = 1, \cdots, n$. Assuming that $\theta$ lies in a sup-norm compact space $\Theta$ of real-valued functions, an $L_1$-consistent estimator (of $\theta$) is constructed via empirical measures. The rate of convergence of the estimator to the true parameter $\theta$ depends on Kolmogorov's entropy of $\Theta$.

Article information

Source
Ann. Statist., Volume 17, Number 4 (1989), 1597-1607.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347383

Digital Object Identifier
doi:10.1214/aos/1176347383

Mathematical Reviews number (MathSciNet)
MR1026301

Zentralblatt MATH identifier
0694.62018

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G30: Order statistics; empirical distribution functions

Keywords
Minimum distance estimation empirical measures nonparametric regression rates of convergence Kolmogorov's entropy

Citation

Yatracos, Yannis G. A Regression Type Problem. Ann. Statist. 17 (1989), no. 4, 1597--1607. doi:10.1214/aos/1176347383. https://projecteuclid.org/euclid.aos/1176347383


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