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On the asymptotic normality of $$l_\alpha$$-estimators of a parameter of a nonlinear regression model. (English. Ukrainian original) Zbl 0947.62041
Theory Probab. Math. Stat. 60, 1-11 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 1-10 (1999).
The authors consider a nonlinear regression model of the form $X_j=g(j,\vartheta)+\varepsilon_j,\;j=1,\dots,n,$ where $$g(j,\vartheta)$$ is a nonrandom function, $$\varepsilon_j$$ are i.i.d. random errors, and $$\vartheta\in R^q$$ is an unknown parameter. In this model an $$l_\alpha$$ estimator $$\hat\vartheta_n$$ for $$\vartheta$$ is defined as the minimizer of $$S_\alpha(\tau)=\sum_{j=1}^n|X_j-g(j,\tau)|^\alpha.$$ Asymptotic normality of $$d_n(\vartheta)(\hat\vartheta_n-\vartheta)$$ is demonstrated for $$\alpha\in(1,2)$$, where $$d_n(\tau)$$ is the diagonal matrix with entries $$d_{ii}^2=\sum_{j=1}^n\left(\partial g(j,\tau)/\partial \tau_i\right)^2.$$ The authors use some smoothness and contrast conditions on $$g(j,\vartheta)$$ and suppose that $$\varepsilon_j$$ have a density bounded in some neighborhood of zero, and $$E|\varepsilon_j|^{\alpha-1}\text{sign}(\varepsilon_j)=0$$, $$E|\varepsilon_j|^{3(\alpha-1)}<\infty$$.
##### MSC:
 62J02 General nonlinear regression 62F12 Asymptotic properties of parametric estimators 62J99 Linear inference, regression
##### Keywords:
robust estimators; asymptotic normality; functional model