Functional Least Squares Estimators in an Additive Effects Outliers Model

Consider the additive effects outliers (A.O.) model where one observes Y_j,n = X_j + v_j,n, 0 ≤ j ≤ n, with [formula omitted]. The sequence of r.v.s {X_j,j < n} is independent of {v_j,n 0 < j < n} and v_j,n 0 ≤ j ≤ n, are i.i.d. with d.f. (1 − γ_n)I[x ≥ 0] + γ_nL_n(x), x ∊ R, 0 < γ_n < 1, where the d.f.s L_n, n > 0, are not necessarily known and ε_j's are i.i.d. This paper discusses the asymptotic behavior of functional least squares estimators under the above model. Uniform consistency and uniform strong consistency of these estimators are proven. The weak convergence of these estimators to a Gaussian process and their asymptotic biases are also discussed under the above A.O. model.