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$M$-estimator; generalized linear models; pseudolinear models
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.
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