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M-estimators of structural parameters in pseudolinear models. (English) Zbl 1060.62029
The authors consider the consistency of M-estimators of the structural parameter $$\beta _0$$ in a pseudo-linear model, based on independent observations $$Y_1,\ldots ,Y_n,$$ with $$Y_i$$ distributed according to the distribution function $$F(y-\theta _i)$$, $$\theta _i=u(z_i^{\prime }\beta _0)$$, $$i=1,\ldots ,n.$$ The M-estimator is defined as $$\arg \min _{\beta \in B}\sum _{i=1}^n \rho (Y_i-\tau \circ u(z_i^{\prime }\beta ))$$ with given functions $$\tau$$ and $$u.$$ It explicitly depends on $$Y_n$$ and on the $$(p\times n)$$ matrix $$z_n.$$ Under a series of regularity conditions on $$\rho$$, $$F,$$ for monotone and continuous $$\tau$$ and $$u,$$ and bounded $$z$$ satisfying some concentration condition, the authors derive two identifiability conditions on $$M_n(\theta )=n^{-1} \sum _{i=1}^n\rho (Y_i-\tau (\theta ))$$ and on its expectation in a neighborhood of the true $$\theta _0,$$ sufficient for the consistency of the M-estimator of $$\beta _0.$$ Analogous sufficient conditions are derived for the consistency of M-estimators of $$\beta _0$$ in the model with random $$z_n.$$
The results are obtained with the aid of consistency considerations for the parent parameter $$\theta ,$$ that can be of interest of their own. The results apply also to regression quantiles and similar M-estimators generated by a not very smooth $$\rho$$. It is shown that the pseudolinear model is more general than the generalized linear model, with whom it coincides only in some special cases.
Reviewer: Jana Jurečkova

##### MSC:
 62F12 Asymptotic properties of parametric estimators 62J12 Generalized linear models (logistic models) 62F10 Point estimation 62F15 Bayesian inference 62F35 Robustness and adaptive procedures (parametric inference)
##### Keywords:
M-estimator; pseudo-linear models
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##### References:
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