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An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes

100%

Dudziński M.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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2017

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tom 71

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nr 1

EN

Let: $\mathbf{Y=}\left( \mathbf{Y}_{i}\right)$, where $\mathbf{Y}_{i}=\left( Y_{i,1},...,Y_{i,d}\right)$, $i=1,2,\dots $, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_{n}\left( \mathbf{x}\right) :=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}\left(Y_{i,1}\leq x_{1},\dots ,Y_{i,d}\leq x_{d}\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_{n}=\sqrt{n}\left( F_{n}-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

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Simultaneous testing procedures of many null hypotheses and their applications in the DNA microarrays analysis

63%

Dudziński M., Furmańczyk K.

Mathematica Applicanda

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2007

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tom 35

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nr 49/08

PL

W naszej pracy rozważamy różne podejścia do problematyki jednoczesnego testowania wielu hipotez zerowych. W tym kontekście omawiamy procedury testowania typu single-step, step-down i step-up. W szczególności, przedstawiamy własności i zastosowania takich miar błędów testowania, jak: FWER, k-FWER, FDP, FDR, pFDR. Wspomniane procedury testowania są intensywnie wykorzystywane w analizie mikromacierzy DNA, która to analiza umożliwia monitorowanie poziomów ekspresji wielu genów jednocześnie oraz znajduje ostatnio szerokie zastosowania w diagnostyce, leczeniu i badaniach medycznych.

EN

In our paper, we consider different approaches to the problem of simultaneous testing of many null hypotheses. In this context, we discuss the single-step, the step-down and the step-up procedures of multiple testing. In particular, we are concerned with their properties and applications in the control of the error rates, such as:FW ER, k-FWER, FDP, FDR, pFDR. The mentioned procedures are intensively used in the DNA microarrays analysis, which enables the monitoring of expression levels of many genes simultaneously and is widely applied in recent medical diagnostics, treatment and research.

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A note on control of the false discovery proportion

63%

Dudziński M., Furmańczyk K.

Applicationes Mathematicae

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2009

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tom 36

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nr 4

397-418

EN

We consider the problem of simultaneous testing of a finite number of null hypotheses $H_{i}$, i=1,...,s. Starting from the classical paper of Lehmann (1957), it has become a very popular subject of research. In many applications, particularly in molecular biology (see e.g. Dudoit et al. (2003), Pollard et al. (2005)), the number s, i.e. the number of tested hypotheses, is large and the popular procedures that control the familywise error rate (FWERM) have small power. Therefore, we are concerned with another error rate measure, called the false discovery proportion (FDP). We prove some theorems about control of the FDP measure. Our results differ from those obtained by Lehmann and Romano (2005).

4

Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima

63%

Dudziński M., Furmańczyk K.

Open Mathematics

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2017

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tom 15

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nr 1

1024-1034

EN

Our goal is to state and prove the almost sure central limit theorem for maxima (Mn) of X1, X2, ..., Xn, n ∈ ℕ, where (Xi) forms a stochastic process of identically distributed r.v.’s of the continuous type, such that, for any fixed n, the family of r.v.’s (X1, ...,Xn) has the Archimedean copula CΨ.

Ograniczanie wyników

1 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

1 Applicationes Mathematicae

1 Mathematica Applicanda

1 Open Mathematics

4 Dudziński M.

3 Furmańczyk K.

2 2017

1 2009

1 2007

An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes

Simultaneous testing procedures of many null hypotheses and their applications in the DNA microarrays analysis

A note on control of the false discovery proportion

Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima