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Least square regression with indefinite kernels and coefficient regularization. (English) Zbl 1225.65015

Let ${\left({y}_{i},{x}_{i}\right)}_{i=1}^{m}$ be i.i.d. observations with ${y}_{i}\in ℝ$, ${x}_{i}\in X$, $X$ being some compact metric space. The authors consider estimates ${f}_{z}$ for the regression function ${f}_{\rho }\left(x\right)=E\left({y}_{i}|{x}_{i}\right)$, where ${f}_{z}={f}_{{\alpha }^{z}}$, ${f}_{\alpha }\left(x\right)={\sum }_{i=1}^{m}{\alpha }_{i}K\left(x,{x}_{i}\right)$,

${\alpha }^{z}=arg\underset{\alpha \in {R}^{m}}{min}\frac{1}{m}\phantom{\rule{0.166667em}{0ex}}\sum _{i=1}^{m}{\left({y}_{i}-{f}_{\alpha }\left({x}_{i}\right)\right)}^{2}+\lambda m\sum _{i=1}^{m}{\alpha }_{i}^{2},$

$K:X×X\to ℝ$ is a continuous bounded function (kernel), $\lambda$ is a regularization parameter.

Consistency of ${f}_{z}$ is demonstrated under the assumptions that $\lambda =\lambda \left(m\right)\to 0$, ${\lambda }^{3/2}\sqrt{m}\to \infty$ and the true regression function belongs to the closure of $\left\{{f}_{\alpha }\right\}$ in some suitable reproducing kernel Hilbert space.

To analyze the rates of convergence the authors make assumptions of the form $E\parallel {L}^{-r}{f}_{\rho }\left({x}_{i}\right){\parallel }^{2}<\infty$ for some $r>0$, where $Lf\left(x\right)=E\stackrel{˜}{K}\left(x,{x}_{i}\right)f\left({x}_{i}\right)$, $\stackrel{˜}{K}\left(x,t\right)={E}_{u}K\left(x,u\right)K\left(t,u\right)$. E.g. if $r>1$ then choosing $\lambda ={m}^{1/5}$ they get $\parallel {f}_{z}-{f}_{\rho }{\parallel }_{{L}^{2}}=O\left({m}^{-1/5}\right)$.

Results of simulations are presented for $X=\left[0,1\right]$ and the Gaussian kernel $K$.

##### MSC:
 65C60 Computational problems in statistics 62J02 General nonlinear regression 46E22 Hilbert spaces with reproducing kernels