$M$-estimators of structural parameters in pseudolinear models

Liese, Friedrich, and Vajda, Igor. "$M$-estimators of structural parameters in pseudolinear models." Applications of Mathematics 44.4 (1999): 245-270. <http://eudml.org/doc/33033>.

@article{Liese1999,
abstract = {Real valued $M$-estimators $\hat\{\theta \}_n:=\min \sum _1^n\rho (Y_i-\tau (\theta ))$ in a statistical model with observations $Y_i\sim F_\{\theta _0\}$ are replaced by $\mathbb \{R\}^p$-valued $M$-estimators $\hat\{\beta \}_n:=\min \sum _1^n\rho (Y_i-\tau (u(z_i^T\,\beta )))$ in a new model with observations $Y_i\sim F_\{u(z_i^t\beta _0)\}$, where $z_i\in \mathbb \{R\}^p$ are regressors, $\beta _0\in \mathbb \{R\}^p$ is a structural parameter and $u:\mathbb \{R\}\rightarrow \mathbb \{R\}$ a structural function of the new model. Sufficient conditions for the consistency of $\hat\{\beta \}_n$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” $\hat\{\theta \}_n$. The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If $F_\theta $ has a natural exponential density $\mathrm \{e\}^\{\theta x-b(x)\}$ then our pseudolinear model with $u=(g\circ \mu )^\{-1\}$ reduces to the well known generalized linear model, provided $\mu (\theta )= \{\mathrm \{d\}\}b(\theta )/\{\mathrm \{d\}\}\theta $ and $g$ is the so-called link function of the generalized linear model. General results are illustrated for special pairs $\rho $ and $\tau $ leading to some classical $M$-estimators of mathematical statistics, as well as to a new class of generalized $\alpha $-quantile estimators.},
author = {Liese, Friedrich, Vajda, Igor},
journal = {Applications of Mathematics},
keywords = {$M$-estimator; generalized linear models; pseudolinear models; M-estimator; pseudo-linear models},
language = {eng},
number = {4},
pages = {245-270},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$M$-estimators of structural parameters in pseudolinear models},
url = {http://eudml.org/doc/33033},
volume = {44},
year = {1999},
}

TY - JOUR
AU - Liese, Friedrich
AU - Vajda, Igor
TI - $M$-estimators of structural parameters in pseudolinear models
JO - Applications of Mathematics
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 44
IS - 4
SP - 245
EP - 270
AB - Real valued $M$-estimators $\hat{\theta }_n:=\min \sum _1^n\rho (Y_i-\tau (\theta ))$ in a statistical model with observations $Y_i\sim F_{\theta _0}$ are replaced by $\mathbb {R}^p$-valued $M$-estimators $\hat{\beta }_n:=\min \sum _1^n\rho (Y_i-\tau (u(z_i^T\,\beta )))$ in a new model with observations $Y_i\sim F_{u(z_i^t\beta _0)}$, where $z_i\in \mathbb {R}^p$ are regressors, $\beta _0\in \mathbb {R}^p$ is a structural parameter and $u:\mathbb {R}\rightarrow \mathbb {R}$ a structural function of the new model. Sufficient conditions for the consistency of $\hat{\beta }_n$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” $\hat{\theta }_n$. The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If $F_\theta $ has a natural exponential density $\mathrm {e}^{\theta x-b(x)}$ then our pseudolinear model with $u=(g\circ \mu )^{-1}$ reduces to the well known generalized linear model, provided $\mu (\theta )= {\mathrm {d}}b(\theta )/{\mathrm {d}}\theta $ and $g$ is the so-called link function of the generalized linear model. General results are illustrated for special pairs $\rho $ and $\tau $ leading to some classical $M$-estimators of mathematical statistics, as well as to a new class of generalized $\alpha $-quantile estimators.
LA - eng
KW - $M$-estimator; generalized linear models; pseudolinear models; M-estimator; pseudo-linear models
UR - http://eudml.org/doc/33033
ER -