Estimation of the regression slope by means of Gini's cograduation index

TL;DR

This paper introduces a new estimator for the regression slope in a simple linear model using Gini's rank association coefficient, analyzing its properties and efficiency compared to traditional least squares estimators.

Contribution

The paper proposes a novel slope estimator based on Gini's cograduation index and studies its statistical properties and efficiency relative to existing methods.

Findings

The estimator is consistent and asymptotically normal.

It has competitive efficiency compared to least squares in certain conditions.

Confidence intervals based on the estimator are also derived.

Abstract

The simple linear model $$Y_i = \alpha + \beta \, x_i + \epsilon_i \qquad i=1,2, \ldots,N \geq 2$$ is considered, where the $x_i$'s are given constants and $\epsilon_1, \epsilon_2 , \ldots, \epsilon_N$ are iid with continuous distribution function $F$. An estimator of $\beta$ is proposed, based on Gini's rank association coefficient $G(\underline y;b)$ and defined as $\tilde{\beta} = \frac 12 \, \left\{ \sup (b: G(\underline y;b) >0) + \right. $ $ \left. \inf (b: G(\underline y;b) <0) \right\}.$ The properties of $\tilde{\beta}$ and of the related confidence interval are studied. Some comparisons are given, in terms of asymptotic relative efficiency, with other estimators of $\beta$ including that obtained with the method of least squares.