Marcin Dudziński

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

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.

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

Abstract 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.

The Frequentistic and Bayesian quantile estimation for the maxima of sea levels

Hydrological modelling has become an intensively studied subject in recent years. One of the most significant problems concerning this issue is to provide mathematical and statistical tools, which allow to forecast extreme hydrological events, such as severe sea or river floodings. In particular, the estimation of flood quantiles is often needed in hydrological design to construct reliable flood defences. In our paper, we establish procedures leading to the interval estimation of the quantiles of the maxima of sea levels. We mainly focus on the quantile estimation for the distributions of the 3-day maxima of sea levels, although some general patterns involving the quantile estimation for the distributions of them-day maxima of sea levels, wherem > 3, are also delivered. The introduced estimation scheme uses the Peaks over Threshold (POT) approach as one of theoretical frameworks. We propose the Frequentistic and Bayesian approaches in the constructions of our confidence intervals (CIs). The choice of optimal threshold in the presented method based on the POT approach is discussed as well. The coverage probabilities for the corresponding CIs are also computed.