The Annals of Statistics

Weighted Median Regression Estimates

Friedrich-Wilhelm Scholz

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Abstract

In the simple linear regression problem $\{Y_i = \alpha + \beta x_i + e_i i = 1,\cdots, n, e_i$ i.i.d. $\sim F$ continuous, $x_1 \leqq \cdots \leqq x_n$ known, $\alpha, \beta$ unknown$\}$ we investigate the following type of estimator: To each $s_{ij} = (Y_j - Y_i)/(x_j - x_i)$ with $x_i < x_j$ attach weight $w_{ij}$ and as estimator for $\beta$ consider the median of this weight distribution over the $s_{ij}$. A confidence interval for $\beta$ is found by taking certain quantiles of this weight distribution. The asymptotic behavior of both is investigated and conditions for optimal weights are given.

Article information

Source
Ann. Statist., Volume 6, Number 3 (1978), 603-609.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344204

Digital Object Identifier
doi:10.1214/aos/1176344204

Mathematical Reviews number (MathSciNet)
MR468054

Zentralblatt MATH identifier
0388.62036

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G15: Tolerance and confidence regions 62G20: Asymptotic properties 62J05: Linear regression 62G35: Robustness

Keywords
Estimation confidence interval linear regression efficiency

Citation

Scholz, Friedrich-Wilhelm. Weighted Median Regression Estimates. Ann. Statist. 6 (1978), no. 3, 603--609. doi:10.1214/aos/1176344204. https://projecteuclid.org/euclid.aos/1176344204


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