Minimum Distance Estimation in an Additive Effects Outliers Model

Abstract

In the additive effects outliers (A.O.) model considered here one observes $Y_{j,n} = X_j + \upsilon_{j,n}, 0 \leq j \leq n$, where $\{X_j\}$ is the first order autoregressive [AR(1)] process with the autoregressive parameter $|\rho| < 1$. The A.O.'s $\{\upsilon_{j,n}, 0 \leq j \leq n\}$ are i.i.d. with distribution function (d.f.) $(1 - \gamma_n)I\lbrack x \geq 0\rbrack + \gamma_n L_n(x), x \in \mathbb{R}, 0 \leq \gamma_n \leq 1$, where the d.f.'s $\{L_n, n \geq 0\}$ are not necessarily known. This paper discusses the existence, the asymptotic normality and biases of the class of minimum distance estimators of $\rho$, defined by Koul, under the A.O. model. Their influence functions are computed and are shown to be directly proportional to the asymptotic biases. Thus, this class of estimators of $\rho$ is shown to be robust against A.O. model.