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1

Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units

Kozyra P. M., Rychlik T.

Probability and Mathematical Statistics

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2018

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Vol. 38, Fasc. 1

39--59

EN

We describe a method of calculating sharp lower and upper bounds on the expectations of linear combinations of kth records Expressem in the Gini mean difference units of the original i.i.d. observations. In particular, we provide sharp lower and upper bounds on the expectations of kth records and their differences. We also present the families of distributions which attain the bounds in the limit.

2

Prace Ryszarda Zielińskiego o nieparametrycznych estymatorach kwantyli i ich zastosowaniu w statystyce odpornej

Rychlik T.

Mathematica Applicanda

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2012

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Vol. 40, No. 2

65–-82

PL

Celem tej przegladowej pracy jest opis wyników profesora Ryszarda Zielinskiego dotyczacych nieparametrycznych estymatorów kwantyli w skonczonych próbach oraz ich zastosowania w odpornej estymacji parametru połozenia. Główne przesłanie badan Zielinskiego było nastepujace: do estymacji kwantyli nalezy uzywac pojedynczych statystyk pozycyjnych, a juz ich liniowe kombinacje moga byc bardzo niedokładne w duzych modelach nieparametrycznych. Optymalny wybór statystyki pozycyjnej 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.

3

On characterization of distributions by expectations of transformed generalized order statistics

Pawlas P., Rychlik T., Szynal D.

Demonstratio Mathematica

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2003

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Vol. 36, nr 1

239--245

EN

We generalize results of Swanepoel [10] on the characterization of distributions by expectations of transformed order statistics using lower generalized order statistics. In particular we characterize continuous distributions via expectations of kth lower record values.

4

Upper bounds for the expected Jefferson rounding under mean-variance-skewness conditions

Rychlik T.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 1

1--18

EN

A for the class of nonnegative random variables with given mean, variance, and skewness and support bound, we present a sharp upper bound for the expectation of rounding due to the Jefferson rule. The result gives an estimate for average extra gains due to rounding down payments. Arguments of four-dimensional geometric moment theory implemented in the proof provide tools for refined evaluations of rates of convergence of probability distributions and positive linear operators.