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On residual empirical distribution functions in ARCH models with applications to testing and estimation. (English) Zbl 0921.62063
Let $$\{y_i\}$$ be an ARCH(1) model, i.e. $y_i=\sigma_i \varepsilon_i,\quad\sigma^2_i= \alpha_0+ \alpha_1y^2_{i-1}, \quad i\in\mathbb{Z},$ where the $$\{\varepsilon_i\}$$ are i.i.d. random variables with unknown distribution function $$G(x)$$, $$E\varepsilon^2_1=1$$, $$\alpha_0>0$$, $$0\leq\alpha_1 <1$$; $$\alpha= (\alpha_0,\alpha_1)$$ is an unknown parameter vector. The author considers the residuals $$\varepsilon_k(\theta)= y_k(\theta_0+ \theta_1y^2_{k-1})^{-1/2}$$, $$k=1,\dots,n$$, and the residual empirical distribution function $$G_n(x,\theta)= n^{-1}\sum^n_{k=1} I(\varepsilon_k (\theta)\leq x)$$. Then it is proved that $\sup_{x\in R^1, |\tau|\leq \theta}\biggl| n^{1/2}\bigl[ G_n(x,\alpha+ n^{-1/2} \tau)-G_n(x,\alpha) \bigr]-2^{-1}xg(x){\mathbf E}[(\tau_0+ \tau_1 y_1^2)/(\alpha_0+ \alpha_1y^2_1) ]\biggr|=o_p(1).$ This interesting result is used for testing and robust rank estimation of unknown parameters.

##### MSC:
 62G30 Order statistics; empirical distribution functions 62G35 Nonparametric robustness 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G05 Nonparametric estimation