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On characteristic functions of kth record values from the generalized extreme value distribution and its characterization

M. A. W. Mahmoud, M. A. Atallah, M. Albassam (2011)

Applicationes Mathematicae

Recurrence relations for the marginal, joint and conditional characteristic functions of kth record values from the generalized extreme value distribution are established. These relations are utilized to obtain recurrence relations for single, product and conditional moments of kth record values. Moreover, by making use of the recurrence relations the generalized extreme value distribution is characterized.

On construction of confidence intervals for a mean of dependent data

Jan Ćwik, Jan Mielniczuk (2001)

Discussiones Mathematicae Probability and Statistics

In the report, the performance of several methods of constructing confidence intervals for a mean of stationary sequence is investigated using extensive simulation study. The studied approaches are sample reuse block methods which do not resort to bootstrap. It turns out that the performance of some known methods strongly depends on a model under consideration and on whether a two-sided or one-sided interval is used. Among the methods studied, the block method based on weak convergence result by...

On continuous convergence and epi-convergence of random functions. Part II: Sufficient conditions and applications

Silvia Vogel, Petr Lachout (2003)

Kybernetika

Part II of the paper aims at providing conditions which may serve as a bridge between existing stability assertions and asymptotic results in probability theory and statistics. Special emphasis is put on functions that are expectations with respect to random probability measures. Discontinuous integrands are also taken into account. The results are illustrated applying them to functions that represent probabilities.

On continuous convergence and epi-convergence of random functions. Part I: Theory and relations

Silvia Vogel, Petr Lachout (2003)

Kybernetika

Continuous convergence and epi-convergence of sequences of random functions are crucial assumptions if mathematical programming problems are approximated on the basis of estimates or via sampling. The paper investigates “almost surely” and “in probability” versions of these convergence notions in more detail. Part I of the paper presents definitions and theoretical results and Part II is focused on sufficient conditions which apply to many models for statistical estimation and stochastic optimization....

On cumulative process model and its statistical analysis

Petr Volf (2000)

Kybernetika

The notion of the counting process is recalled and the idea of the ‘cumulative’ process is presented. While the counting process describes the sequence of events, by the cumulative process we understand a stochastic process which cumulates random increments at random moments. It is described by an intensity of the random (counting) process of these moments and by a distribution of increments. We derive the martingale – compensator decomposition of the process and then we study the estimator of the...

On distributions of order statistics for absolutely continuous copulas with applications to reliability

Piotr Jaworski, Tomasz Rychlik (2008)

Kybernetika

Performance of coherent reliability systems is strongly connected with distributions of order statistics of failure times of components. A crucial assumption here is that the distributions of possibly mutually dependent lifetimes of components are exchangeable and jointly absolutely continuous. Assuming absolute continuity of marginals, we focus on properties of respective copulas and characterize the marginal distribution functions of order statistics that may correspond to absolute continuous...

On Dwass' method for deriving the distribution of rank order statistics

B. R. Handa, Sri Gopal Mohanty (1979)

Aplikace matematiky

This note presents a critical examination of Dwass' method for obtaining the distribution of rank order statistics defined on random samples obtained from the same continuous population. New situations are discussed for the usefulness of the method.

On EM algorithms and their proximal generalizations

Stéphane Chrétien, Alfred O. Hero (2008)

ESAIM: Probability and Statistics

In this paper, we analyze the celebrated EM algorithm from the point of view of proximal point algorithms. More precisely, we study a new type of generalization of the EM procedure introduced in [Chretien and Hero (1998)] and called Kullback-proximal algorithms. The proximal framework allows us to prove new results concerning the cluster points. An essential contribution is a detailed analysis of the case where some cluster points lie on the boundary of the parameter space.

On entropies for random partitions of the unit segment

Milena Bieniek, Dominik Szynal (2008)

Kybernetika

We prove the complete convergence of Shannon’s, paired, genetic and α-entropy for random partitions of the unit segment. We also derive exact expressions for expectations and variances of the above entropies using special functions.

On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane

Tomáš Mrkvička, Jan Rataj (2009)

Kybernetika

A method of estimation of intrinsic volume densities for stationary random closed sets in ℝ d based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the...

On Fourier coefficient estimators consistent in the mean-square sense

Waldemar Popiński (1994)

Applicationes Mathematicae

The properties of two recursive estimators of the Fourier coefficients of a regression function f ∈ L 2 [ a , b ] with respect to a complete orthonormal system of bounded functions (ek) , k=1,2,..., are considered in the case of the observation model y i = f ( x i ) + η i , i=1,...,n , where η i are independent random variables with zero mean and finite variance, x i ∈ [ a , b ] ⊂ R 1 , i=1,...,n, form a random sample from a distribution with density ϱ =1/(b-a) (uniform distribution) and are independent of the errors η i , i=1,...,n . Unbiasedness and mean-square...

On goodness-of-fit for the absence of memory model

Vilijandas Bagdonavičius, Rüta Levulienė (2001)

Kybernetika

Logrank-type and Kolmogorov-type goodness-of-fit tests for the absence of memory model are proposed when the accelerated experiments are done under step-stresses. The power of the test against the approaching alternatives is investigated. The theoretical results are illustrated with simulated data.

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