# $M$-estimators of structural parameters in pseudolinear models

Applications of Mathematics (1999)

- Volume: 44, Issue: 4, page 245-270
- ISSN: 0862-7940

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topLiese, 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 -

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