The repeated median intercept estimator: Influence function and asymptotic normality
Consider the simple linear regression model \[ Y_i = \alpha + \beta X_i + e_i, \quad i = 1, \dots, n, \tag{1} \] where \(\{{\mathbf Z}_i = (X_i, Y_i)\}\) are the observed vectors and \(\{e_i\}\) represent noise. We assume that the random vectors \((X_i, e_i)\) are i.i.d., and that \(X_i\) and \(e_i\) are mutually independent with continuous distributions \(G\) and \(F\) for which \(F(0) = 1/2 = G(0)\). When \(X_1, \dots, X_n\) are all distinct (an event with probability one), we may define the repeated median estimator of the intercept \(\alpha\), \[ \widehat \alpha_n = \text{med}_i \text{med}_{j,j \neq i} (X_j Y_i - X_i Y_j)/(X_j - X_i) \tag{2} \] introduced by \textit{A. F. Siegel} [Biometrika 69, 242-244 (1982; Zbl 0483.62026)]. Given two points \({\mathbf z}_1 = (x_1, y_1)\) and \({\mathbf z}_2 = (x_2, y_2)\) with \(x_1 \neq x_2\), the kernel function \(h({\mathbf z}_1, {\mathbf z}_2) = (x_2 y_1 - x_1 y_2)/(x_2 - x_1)\) gives the intersection between the \(y\)-axis and the line through \({\mathbf z}_1\) and \({\mathbf z}_2\). Siegel showed that \(\widehat \alpha_n \) has a \(50 \%\) breakdown point, and is a Fisher consistent estimator of \(\alpha\) when \(F\) is a symmetric distribution. We prove asymptotic normality. The main result is that \[ \sqrt n (\widehat \alpha_n - \alpha) @> d >> N \biggl( 0, \bigl( 2f(0) \bigr)^{-2} \biggr) \quad \text{as } n \to \infty. \tag{3} \] The asymptotics are compared with Monte Carlo results. By changing the kernel function in (2), we obtain repeated median estimators of other quantities. For instance, when \(h({\mathbf z}_1, {\mathbf z}_2) = (y_2 - y_1)/(x_2 - x_1)\) we obtain a repeated median estimator of the slope parameter \(\beta\), as introduced by Siegel. Influence function and asymptotic normality for this estimator were proved by \textit{O. Hössjer} et al. [Ann. Stat. 22, No. 3, 1478-1501 (1994; Zbl 0817.62018)]. The techniques used in this paper are based on those used in that paper.
scientific article; zbMATH DE number 718684