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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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