## The Annals of Statistics

### Weighted Median Regression Estimates

Friedrich-Wilhelm Scholz

#### 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

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