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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 \(\Theta\).

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