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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 \(E|\varepsilon_j|^{\alpha-1}\text{sign}(\varepsilon_j)=0\), \(E|\varepsilon_j|^{3(\alpha-1)}<\infty\).
62J02 General nonlinear regression
62F12 Asymptotic properties of parametric estimators
62J99 Linear inference, regression