## Minimum distance estimation in an additive effects outliers model.(English)Zbl 0741.62082

Summary: In the additive effects outliers (A.O.) model considered here one observes $$Y_{j,n}=X_ j+v_{j,n}$$, $$0\leq j\leq n$$, where $$\{X_ j\}$$ is the first order autoregressive $$[AR(1)]$$ process with the autoregressive parameter $$| \rho|<1$$. The A.O.’s $$\{v_{j,n}, 0\leq j\leq n\}$$ are i.i.d. with distribution function (d.f.) $(1- \gamma_ n)I[x\geq 0]+\gamma_ nL_ n(x),\quad x\in \mathbb{R},\quad 0\leq \gamma_ n\leq 1,$ where the d.f.’s $$\{L_ n, n\geq 0\}$$ are not necessarily known. This paper discusses existence, asymptotic normality and biases of the class of minimum distance estimators of $$\rho$$, defined by H. L. Koul [ibid. 14, 1194-1213 (1986; Zbl 0607.62101)] 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.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference

Zbl 0607.62101
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