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Bootstrap approximation of distributions of the \(L\)-statistics. (English. Russian original) Zbl 0995.62052

J. Math. Sci., New York 109, No. 6, 2088-2102 (2002); translation from Zap. Nauchn. Semin. POMI 260, 84-102 (1999).
Summary: We consider linear combinations \(L_n=\sum^n_{i=1} c_{n,i}X_{n,i}\) of order statistics \(X_{n,i}\) based on a sample of size \(n\) of independent, identically distributed random variables with coefficients \(c_{n,i}\) specified by a weight function \(J(u)\), \(u\in(0,1)\), as follows: (i) \(c_{n,i}= n^{-1}J(i(n+1)^{-1})\) or (ii) \(c_{n,i}= \int^{i/n}_{(i-1)/n}J(u)du\), \(i=1,\dots, n\). Consistency conditions of the bootstrap approximation of distributions of such linear combinations are studied. A difference in the asymptotic properties for these two types of weights, which are often used in statistical estimation problems, is established. Under the same natural moment assumptions, the continuity of the function \(J\) is sufficient for consistency (i.e., for convergence) in case (ii), while in case (i) a Hölder condition of order greater than 1/2 is needed. In case (i), an important role is played by the limit behavior of the ratio of sizes of the initial and artificial samples. This happens due to the bias of the bootstrap mathematical expectation with respect to \(L_n\). In case (ii), the value of this ratio does not affect the consistency.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G09 Nonparametric statistical resampling methods
62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
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