Milan Stehlík

Small Sample Robust Testing for Normality against Pareto Tails

The aim of this article is to introduce the general form (so called RT class) of the robust and classical Jarque–Bera (JB) test based on the location functional. We introduce the two-step procedure which is optimal for testing against the individual or contaminated Pareto alternative. As a reference for such a contamination we consider different Pareto distributions. We also give practical guidelines for robust testing for normality against short- and heavy-tailed alternatives. We concentrate mainly on simulation results for moderate and small samples. However, we also prove consistency and asymptotic distribution for introduced tests. We show that as the suitable measure of nominal level of Pareto tail parameter we may take the t-Hill estimator introduced in the article. To guarantee the consistency of the whole procedure, we also prove the consistency of t-Hill estimator. The introduced general class of robust tests of the normality is illustrated at the selected datasets of financial time series.

Homogeneity and scale testing of generalized gamma distribution

The aim of this paper is to derive the exact distributions of the likelihood ratio tests of homogeneity and scale hypothesis when the observations are generalized gamma distributed. The special cases of exponential, Rayleigh, Weibull or gamma distributed observations are discussed exclusively. The photoemulsion experiment analysis and scale test with missing time-to-failure observations are present to illustrate the applications of methods discussed.

Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass‐interaction process

Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.

Lattice-valued bornological systems

Motivated by the concept of lattice-valued topological system of J.T. Denniston, A. Melton, and S.E. Rodabaugh, which extends lattice-valued topological spaces, this paper introduces the notion of lattice-valued bornological system as a generalization of lattice-valued bornological spaces of M. Abel and A. Sostak. We aim at (and make the first steps towards) the theory, which will provide a common setting for both lattice-valued point-set and point-free bornology. In particular, we show the algebraic structure of the latter.

Optimal design for correlated processes with input-dependent noise

Optimal design for parameter estimation in Gaussian process regression models with input-dependent noise is examined. The motivation stems from the area of computer experiments, where computationally demanding simulators are approximated using Gaussian process emulators to act as statistical surrogates. In the case of stochastic simulators, which produce a random output for a given set of model inputs, repeated evaluations are useful, supporting the use of replicate observations in the experimental design. The findings are also applicable to the wider context of experimental design for Gaussian process regression and kriging. Designs are proposed with the aim of minimising the variance of the Gaussian process parameter estimates. A heteroscedastic Gaussian process model is presented which allows for an experimental design technique based on an extension of Fisher information to heteroscedastic models. It is empirically shown that the error of the approximation of the parameter variance by the inverse of the Fisher information is reduced as the number of replicated points is increased. Through a series of simulation experiments on both synthetic data and a systems biology stochastic simulator, optimal designs with replicate observations are shown to outperform space-filling designs both with and without replicate observations. Guidance is provided on best practice for optimal experimental design for stochastic response models.

Exact Likelihood Ratio Testing for Homogeneity of the Exponential Distribution

The aim of this article is to discuss homogeneity testing of the exponential distribution. We introduce the exact likelihood ratio test of homogeneity in the subpopulation model, ELR, and the exact likelihood ratio test of homogeneity against the two-components subpopulation alternative, ELR2. The ELR test is asymptotically optimal in the Bahadur sense when the alternative consists of sampling from a fixed number of components. Thus, in some setups the ELR is superior to frequently used tests for exponential homogeneity which are based on the EM algorithm (like the MLRT, ADDS, and D-tests). One important example of superiority of ELR and ELR2 tests is the case of lower contamination. We demonstrate this fact by both theoretical comparisons and simulations.

D-optimal Designs and Equidistant Designs for Stationary Processes

In this paper we discuss the structure of the information matrices of D-optimal experimental designs for the parameters in a stationary process when the parametrized correlation structure satisfies mild conditions. Such conditions are easily fulfilled by many correlation structures, e.g. structures from power exponential family and some members of the Matern class. We provide a lower bound for information on the mean parameter and prove it to be an increasing function of distances of design points. The design points can collapse under the presence of some covariance structures and a so called nugget effect can be employed in a natural way. We also show that the information of equidistant designs (designs with equally spaced design points) on the covariance parameter is increasing with the number of design points under our conditions on correlations. If only trend parameters are of interest, the designs covering the whole design space non-uniformly are rather efficient.

Decompositions of information divergences: Recent development, open problems and applications

What is the optimal statistical decision? And how it is related to the statistical information theory? By trying to answer these difficult questions, we will illustrate the necessity of understanding of structure of information divergences. This may be understand in particular through deconvolutions, leading to an optimal statistical inference. We will illustrate deconvolution of information divergence in the exponential family, which will gave us an optimal tests (optimal in the sense of Bahadur (see [3, 4]). We discuss about the results on the exact density of the I-divergence in the exponential family with gamma distributed observations (see [28]). Since the considered I-divergence is related to the likelihood ratio (LR) statistics, we deal with the exact distribution of the likelihood ratio tests and discuss the optimality of such exact tests. The both tests, the exact LR test of the homogeneity and the exact LR test of the scale parameter, are asymptotically optimal in the Bahadur sense when the obse...

Optimal designs for parameters of shifted Ornstein–Uhlenbeck sheets measured on monotonic sets

Measurement on sets with a specific geometric shape can be of interest for many important applications (e.g., measurement along the isotherms in structural engineering). The properties of optimal designs for estimating the parameters of shifted Ornstein–Uhlenbeck sheets are investigated when the processes are observed on monotonic sets. For Ornstein–Uhlenbeck sheets monotonic sets relate well to the notion of non-reversibility. Substantial differences are demonstrated between the cases when one is interested only in trend parameters and when the whole parameter set is of interest. The theoretical results are illustrated by simulated examples from the field of structure engineering. From the design point of view the most interesting finding of the paper is the possible loss of efficiency of the regular grid design compared to the optimal monotonic design.

On Small Samples Testing for Frailty Through Homogeneity Test

We derive a test in order to examine the need of modeling survival data using frailty models based on the likelihood ratio (LR) test for homogeneity. Test is developed for both complete and censored samples from a family of baseline distributions that satisfy a closure property. Approach motivated by I-divergence distance is used in order to determine “credible” regions for all parameters of baseline distribution for which homogeneity hypothesis is not rejected. Proposed test outperforms the usual asymptotic LR test both in very small samples with known frailty and for all small sample sizes under misspecified frailty.