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Adaptive \(L\)-estimation for linear models. (English) Zbl 0736.62060
Consider a linear regression model with intercept, \(y_ i=\alpha+x_ i'\gamma+u_ i\), \(i=1,\ldots,n\), where \(u_ i\) are independent random variables with common distribution function \(F\), and the explanatory variables \(x_ i\) are translated so that they sum to zero, \(\Sigma x_ i=0\). The authors suggest an asymptotically efficient adaptive \(L\)- estimator of the “slope” parameter \(\gamma\) under the least restrictive assumptions possible on \(F\).
In the first step of the construction of the estimator, regression quantiles \(\{\hat\alpha_ n(t),\hat\gamma_ n(t)\}\) are formed. The regression quantiles are defined by \[ \min_{\alpha,\gamma}\sum^ n_{i=1}\rho_ t(y_ i-\alpha-x_ i'\gamma) (1) \] for \(t\in(0,1)\), where \(\rho_ t(u)=u(t-I (u<0))\), and \(I\) is the indicator function. The adaptive estimator, \(T_ n\), of \(\gamma\) is a linear function of \(\hat\gamma_ n(t)\), \[ T_ n=\int^ 1_ 0\hat\gamma_ n(t)J(t)dt. \] The optimal choice for \(J(t)\) is \(J_ 0(t)=\psi'(F^{- 1}(t))\), where \(\psi(x)=-L'(x)\) and \(L(x)=\ln f(x)\). However, when \(F\) is unknown \(J_ 0(t)\) cannot be used. Instead a kernel estimator \(\hat J_ n(t)\) is used. This kernel estimator, in turn, is based on estimates of the conditional quantile and the conditional distribution functions. Denoting the set of solutions to (1) by \(\hat B_ n(t)\), the estimator of the conditional quantile estimator is defined by \[ \hat Q_ n(t\mid x)=\inf\{a+x_ i'g\mid (a,g)\in \hat B_ n(t)\}, \] and the estimator of the conditional distribution function is accordingly defined by \[ \hat F_ n(y\mid x)=\sup\{t\in(0,1)\mid \hat Q_ n(t\mid x)\leq y\}. \] At the mean of the design \(\hat F_ n(y)=\hat F_ n(y\mid\bar x)\) is a proper distribution function which can be used for estimating \(J_ 0(t)\).
Asymptotic efficiency for the adaptive estimator is proved under very mild regularity conditions. Small sample performances are illustrated in a small Monte Carlo experiment.
Reviewer: H.Nyquist (Umea)

MSC:
62J05 Linear regression; mixed models
62F35 Robustness and adaptive procedures (parametric inference)
62G35 Nonparametric robustness
62G07 Density estimation
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