Asymptotic normality of \(L_ 1\)-estimators in nonlinear regression (Q1898405)

We are concerned with the following nonlinear regression model: \[ y_i = f(x_i, \vartheta) + e_i,\quad i = 1,\dots, N,\tag{1} \] where \(\vartheta\) is the unknown parameter vector in \(R^m\) and \(e_i\)'s are random errors. The \(L_1\)-estimator of \(\vartheta\) is the optimal solution of the minimization problem \[ \text{Min}_\vartheta \sum^N_{i=1} |y_i - f(x_i, \vartheta)|. \tag{2} \] Let \(\vartheta_0\) be the true value of \(\vartheta\). Substituting model (1) into (2), we get another form of problem (2): \[ \text{Min}_\vartheta \sum^N_{i=1} |e_i + f(x_i, \vartheta_0) - f(x_i, \vartheta)|. \tag{3} \] Denote by \(\vartheta_N(e)\) the optimal solution of problem (3), where \(e = (e_1,\dots, e_N)\). Our aim is to study the asymptotic behavior of \(N^{1/2} (\vartheta_N (e) - \vartheta_0)\). So it is natural to introduce a new variable \(z = N^{1/2} (\vartheta - \vartheta_0)\) and to form a new equivalent problem: \[ \text{Min}_z \sum^N_{i=1} [|e_i + f_i(z)|- |e_i|],\tag{4} \] where \[ f_i(z) = f(x_i, \vartheta_0) - f(x_i, \vartheta_0 + N^{-1/2} z). \] Denote the objective function and the optimal solution of problem (4) by \(F_N(e,z)\) and \(z_N(e)\), respectively. Clearly we have \(z_N(e) = N^{1/2}(\vartheta_N(e) - \vartheta_0)\). We will work with problem (4) and the main task of this paper is to establish asymptotic normality of \(z_N(e)\).