A Robust Point Estimator in a Generalized Regression Model

Abstract

The objective of the present paper is to define a robust point estimator of the parameter $\beta$ in the model \begin{equation*} \tag{1.1} y_j = \alpha + g_j(\beta) + z_j,\quad j = 1, 2, \cdots, n,\end{equation*} where $\alpha$ and $\beta$ are unknown parameters, $g_1, g_2, \cdots, g_n$ are real-valued functions of real variable satisfying suitable conditions and $z_1, z_2, \cdots, z_n$ are independent identically distributed random variables having a distribution function belonging to a specified class. An important special case of (1.1) is the regression model obtained by taking $g_j(\beta) = \beta x_j, j = 1,2, \cdots, n$, where the $x$'s are known constants. Robust point estimators of $\beta$ in this case have already been given by Adichie [1], who followed the method of Hodges and Lehmann and by Brown and Mood [7], who considered the so-called "median" estimators. The estimator presented in this paper, which is also a Hodges-Lehmann type estimator, therefore, provides a third alternative for the regression model. In Section 2 of this paper, a robust point estimator for $\beta$ in the model (1.1) under some suitable regularity conditions on $g_j$ and $z_j$ is defined. In Section 3, a simple computational technique for the calculation of this estimator is given. A small sample property of the estimator is given in Section 4, and in Section 5, asymptotic normality is established under some regularity conditions. In Section 6, some special cases of the model are considered.