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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

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)
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