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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Asymptotically stable estimators of location and scale parameters II. Estimation of scale parameter

100%
Rychlik T.
Mathematica Applicanda
|
1987
|
tom 16
|
nr 30
PL
.
EN
Asymptotic robustness of estimators of scale parameter with respect to scale invariant bias oscillation function is studied for two types of disturbances. In the case of £-contamination, the most robust sequence of equivariant estimators for model distribution with a positive support and the most robust sequence of equivariant symmetric estimators for symmetric model distribution are constructed. In the case of Kolmogorov-Levy neighbourhoods, the solution is derived without any assumptions about the model distribution. As examples, the most bias-robust estimators for uniform, Pareto, Weibull, Laplace, normal, Cauchy and double-exponential distributions are presented.
2
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Asymptotically stable estimators of location and scale parameters I. Estimation of location parameter

100%
Rychlik T.
Mathematica Applicanda
|
1987
|
tom 16
|
nr 30
PL
.
EN
A sequence of equivariant estimators of a location parameter, which is asymptotically most robust with respect to bias oscillation function, is derived for two types of disturbances: e-contamination and Kolmogorov-Levy neighbourhoods. The sequence consists of properly chosen order statistics modified by adding a constant. As examples, the most bias-robust estimators for unimodal symmetric, Weibull, double-exponential and beta distributions are presented.
3
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A class of unbiased kernel estimates of a probability density function

100%
Rychlik T.
Applicationes Mathematicae
|
1993-1995
|
tom 22
|
nr 4
485-497
EN
We propose a class of unbiased and strongly consistent nonparametric kernel estimates of a probability density function, based on a random choice of the sample size and the kernel function. The expected sample size can be arbitrarily small and mild conditions on the local behavior of the density function are imposed.
4
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Evaluating improvements of records

100%
Rychlik T.
Applicationes Mathematicae
|
1996-1997
|
tom 24
|
nr 3
315-324
EN
We evaluate the extreme differences between the consecutive expected record values appearing in an arbitrary i.i.d. sample in the standard deviation units. We also discuss the relevant estimates for parent distributions coming from restricted families and other scale units.
5
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Optimal mean-variance bounds on order statistics from families determined by star ordering

100%
Rychlik T.
Applicationes Mathematicae
|
2002
|
tom 29
|
nr 1
15-32
EN
We present optimal upper bounds for expectations of order statistics from i.i.d. samples with a common distribution function belonging to the restricted family of probability measures that either precede or follow a given one in the star ordering. The bounds for families with monotone failure density and rate on the average are specified. The results are obtained by projecting functions onto convex cones of Hilbert spaces.
6
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The Ryszard Zielinski's works on nonparametric quantile estimators and their use in robust statistics

100%
Rychlik T.
Mathematica Applicanda
|
2012
|
tom 40
|
nr 2
PL
Celem tej przegladowej pracy jest opis wyników profesora RyszardaZielinskiego dotyczacych nieparametrycznych estymatorów kwantyli w skonczonychpróbach oraz ich zastosowania w odpornej estymacji parametru połozenia. Główneprzesłanie badan Zielinskiego było nastepujace: do estymacji kwantyli nalezy uzywacpojedynczych statystyk pozycyjnych, a juz ich liniowe kombinacje moga byc bardzoniedokładne w duzych modelach nieparametrycznych. Optymalny wybór statystykipozycyjnej zalezy od kryterium oceny błedu estymacji.
EN
This is a survey paper describing achievements of professor Ryszard Zieliński in the subject of nonparametric estimation of population quantiles based on samples of fixed size, and applications of the quantile estimators in the robust estimation of location parameter. Zielinski assumed that a finite sequence of independent identically distributed random variables X1, . . . ,Xn is observed, and their common distribution function F belongs to the family F of continuous and strictly increasing distribution functions. He considered the family T of randomized estimators XJ:n which are single order statistics based on X1, . . . ,Xn with a randomly determined number J. The random variable J is independent of the sample and has an arbitrary distribution on the numbers 1, . . . , n. It was proved that T is the maximal class of estimators which are functions of the complete and sufficient statistic (X1:n, . . . ,Xn:n), and are equivariant with respect to the strictly increasing transformations, i.e., satisfy T(φ(X1:n), . . . ,φ(Xn:n)) = φ(T(X1:n, . . . ,Xn:n)) for arbitrary strictly increasing φ. A number of examples showed that the estimators that do not belong to T are very inaccurate for some F€F.   For comparing estimators, there were used various accuracy criteria based on the difference F(T) - q, where 0 < q < 1 is the quantile order. They are invariant with respect to the strictly increasing transformations. Optimal estimators with respect to the mean absolute loss E|F(T)-q|, mean quadratic loss E(F(T)-q)2, expected LINEX loss E[exp(a[F(T)-q])-a[F(T)-q]-1], a≠0, and Pitman closeness measure were explicitly determined. Further, the best estimators in narrower classes of median-unbiased estimators U(q) = {T€T : med(T, F) = F-1(q)},  (where med(T, F) stands for the median of the distribution of estimator T when the parent distribution function is F), and F-unbiased estimators V(q) = {T € T : EF(T) = q} of quantiles F-1 (q), 0 < q < 1, are determined for some accuracy criteria. Also, random confidence intervals for F-1(q), F€F, of the form [XI:n,XJ:n] on a fixed confidence level 0 <  < 1, i.e. satisfying P(XI:n ≤F-1(q) ≤XJ:n)≥γ,  F € F, , and minimizing E(J - I), are described. Median-unbiased estimators of quantiles were applied by Zielinski in the robust estimation of location parameter. For the i.i.d. sample X1, . . . ,Xn from the location model Fμ(x) = F(x - μ), where μ€R and F is a known unimodal distribution function, and the ε-contamination of the model Z(μ) = {G = (1 -ε)Fμ +εH : H - arbitrary distribution function} for some fixed 0 <ε< 1/2 , the most robust translation equivariant estimator with respect to the median oscillation criterion bn(T, μ) = supG1,G2€Z(μ) |med(T,G1) - med(T,G2)| has the form XJ:n - F-1(q*), XJ:n  €U(q*). Number q*  is chosen so to minimize function (ε, 1 - ε)Э q→ F-1(q/(1-ε))-F-1((q-ε)/(1-ε)). If F is unimodal and symmetric, then q* = ½.. However, Zielinski also showed that a slight modification of the ε-contamination for symmetric unimodal F may imply that XJ:n - F-1(q*), XJ:n € U(q*), for some q*≠1/2 is the most robust estimator with respect to the median oscillation criterion. Celem tej przeglądowej pracy jest opis wyników profesora Ryszarda Zielińskiego dotyczącychnieparametrycznych estymatorów kwantyli w skończonych próbach oraz ich zastosowania w odpornej estymacjiparametru położenia. Główne przesłanie badań Zielińskiego było następujące:do estymacji kwantyli należy używać pojedynczych statystyk pozycyjnych, a już ich liniowekombinacje mogą być bardzo niedokładne w dużych modelach nieparametrycznych.Optymalny wybór statystyki pozycyjnej zależy od kryterium oceny błędu estymacji.
7
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Sharp bounds for expectations of spacings from decreasing density and failure rate families

64%
Danielak K., Rychlik T.
Applicationes Mathematicae
|
2004
|
tom 31
|
nr 4
369-395
EN
We apply the method of projecting functions onto convex cones in Hilbert spaces to derive sharp upper bounds for the expectations of spacings from i.i.d. samples coming from restricted families of distributions. Two families are considered: distributions with decreasing density and with decreasing failure rate. We also characterize the distributions for which the bounds are attained.
8
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Refined rates of bias convergence for generalized L-Statistics in the i.i.d. case

64%
Anastassiou G. A., Rychlik T.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 4
437-455
EN
Using tools of approximation theory, we evaluate rates of bias convergence for sequences of generalized L-statistics based on i.i.d. samples under mild smoothness conditions on the weight function and simple moment conditions on the score function. Apart from standard methods of weighting, we introduce and analyze L-statistics with possibly random coefficients defined by means of positive linear functionals acting on the weight function.
9
Content available remote

Evaluations of expected generalized order statistics in various scale units

51%
Cramer E., Kamps U., Rychlik T.
Applicationes Mathematicae
|
2002
|
tom 29
|
nr 3
285-295
EN
We present sharp upper bounds for the deviations of expected generalized order statistics from the population mean in various scale units generated by central absolute moments. No restrictions are imposed on the parameters of the generalized order statistics model. The results are derived by combining the unimodality property of the uniform generalized order statistics with the Moriguti and Hölder inequalities. They generalize evaluations for specific models of ordered observations.
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