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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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