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$$L_ 1$$-optimal estimates for a regression type function in $$R^ d$$. (English) Zbl 0744.62064
Summary: Let $$X_ 1,X_ 2,\dots,X_ n$$ be random vectors that take values in a compact set in $$R^ d$$, $$d\geq 1$$. Let $$Y_ 1,Y_ 2,\dots,Y_ n$$ be random variables (“the responses”) which conditionally on $$X_ 1=x_ 1,\dots,X_ n=x_ n$$ are independent with densities $$f(y\mid x_ i,\theta(x_ i))$$, $$i=1,\dots,n$$. Assuming that $$\theta$$ lives in a sup- norm compact space $$\Theta_{q,d}$$ of real valued functions, an optimal $$L_ 1$$-consistent estimator $$\hat\theta_ n$$ of $$\theta$$ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter $$\theta$$ depends on Kolmogorov’s entropy of $$\Theta_{q,d}$$.

##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions
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##### References:
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