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A regression type problem. (English) Zbl 0694.62018
Summary: Let \(X_ 1,...,X_ n\) be random vectors that take values in a compact set in \(R^ d\), \(d=1,2\). Let \(Y_ 1,...,Y_ n\) be random variables (the responses) which conditionally on \(X_ 1=x_ 1,...,X_ n=x_ n\) are independent with densities \(f(y| x_ i,\theta (x_ i))\), \(i=1,...,n\). Assuming that \(\theta\) lies in a sup-norm compact space \(\Theta\) of real-valued functions, an \(L_ 1\)-consistent estimator (of \(\vartheta)\) is constructed via empirical measures. The rate of convergence of the estimator to the true parameter \(\theta\) depends on Kolmogorov’s entropy of \(\Theta\).

62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions
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