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M-estimators of structural parameters in pseudolinear models

Authors:	Friedrich Liese Igor Vajda

Institution:	(1) Department of Mathematics, University of Rostock, Universitätsplatz 1, D-180 55 Rostock, Germany;(2) Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou ví 4, CZ-182 08 Praha, Czech Republic

Abstract:	Real valued M-estimators $\hat \theta _n : = \min \sum\limits_1^n {\varrho \left( {Y_i - \tau \left(\theta \right)} \right)}$ in a statistical model 1 with observations $Y_i \sim F_{{\theta }_0 }$ are replaced by $\mathbb{R}^p$ -valued M-estimators $\hat \beta _n : = \min \sum\limits_1^n {\varrho \left( {Y_i - \tau \left( {u\left( {z_i^T \beta } \right)} \right)} \right)}$ in a new model with observations $Y_i \sim F_{u\left( {z_i^t {\beta }}_{0} \right)}$ where $z_i \in \mathbb{R}^p$ are regressors, ${\beta }_{0} \in \mathbb{R}^p$ is a structural parameter and $u:\mathbb{R} \to \mathbb{R}$ 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 has a natural exponential density e^{x–b( x )} then our pseudolinear model with u = (g o )^–1 reduces to the well known generalized linear model, provided () = db()/d and g is the so-called link function of the generalized linear model. General results are illustrated for special pairs and leading to some classical M-estimators of mathematical statistics, as well as to a new class of generalized -quantile estimators.

Keywords:	M-estimator generalized linear models pseudolinear models
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