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JOURNALS // Teoriya Veroyatnostei i ee Primeneniya // Archive

Teor. Veroyatnost. i Primenen., 1997 Volume 42, Issue 3, Pages 531–552 (Mi tvp1951)

Asymptotics of statistical estimates constructed from censored samples of distributions with regularly varying tails

M. S. Tikhov

N. I. Lobachevski State University of Nizhni Novgorod

Abstract: We consider the asymptotic behavior of the Pitman estimators $\hat \theta_n$ for the density location parameter $f(x-\theta)=C(1+\alpha)(x-\theta)^{\alpha}L(x-\theta)$, $x\downarrow \theta $, $\alpha>-1$, $L(x)=1+D_1(1+\ell (1+\alpha)^{-1})x^{\ell }+o(x^{\ell })$, $\ell >0$, by observations over the first $k$ ordered statistics $(X_n^{(1)},\ldots,X_n^{(k)})$, when $k=k(n)\to \infty$, $k/n\to 0$ as $n\to \infty$. The limiting distributions of $\hat \theta_n$ are described for various values of $\alpha$. Our proofs use properties and asymptotic expansions of the hypergeometric functions in several variables. Simple asymptotically efficient estimators of $\theta$ are given as linear functionals of the ordered statistics.

Keywords: ensored samples, regularly varying density, location parameter, parameter estimation.

Received: 20.05.1996

DOI: 10.4213/tvp1951


 English version:
Theory of Probability and its Applications, 1998, 42:3, 495–512

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