Asymptotic normality of Koenker-Basset estimates in nonlinear regression models. (Ukrainian, English) Zbl 1125.62064

Ivanov, O. V.; Orlovs’kyj, I. V.

Teor. Jmovirn. Mat. Stat. 72, 30-41 (2005); translation in Theory Probab. Math. Stat. 72, 33-45 (2006).

A nonlinear regression model is considered of the form \(X_j=g(j,\vartheta)+\varepsilon_j\), where \(X_j\) are observations, g is a known function, \(\vartheta\in\Theta\subseteq R^d\) is an unknown parameter, \(\varepsilon_j\) are i.i.d. with \({\mathbf E}\xi_j=0\) and \({\mathbf P}\{\xi<0\}=\beta\) (known). A generalized least absolute deviations estimate is defined by \(\hat\vartheta_n=\arg\min_{\tau\in\Theta^c} S_\beta(\tau)\), where \[ S_\beta(\tau)=\sum_j \rho_\beta(X_j-g(j,\tau)),\quad \rho_\beta(x)=\beta x{\mathbf 1}_{\{x\geq 0\}}+(\beta-1) x{\mathbf 1}_{\{x\leq 0\}}. \] Conditions of asymptotic normality of \(\widehat\vartheta_n\) are derived.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

62J02	General nonlinear regression
62G08	Nonparametric regression and quantile regression
62G20	Asymptotic properties of nonparametric inference

Keywords:

generalized least absolute deviation estimate; quantile estimate; Koenker-Basset estimate