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

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.