## 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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