Daniela Jarušková

Testing appearance of linear trend

Abstract A test for the detection of a gradual change in simple linear regression is studied. Asymptotic distribution of the “maximum type” statistic is derived. Asymptotic critical values are compared with critical values obtained by simulations. The problem was motivated by the effort of meteorologists to discover a change in meteorological measurements.

On a Test Statistic for Linear Trend

Let {W(s)}s ≥ 0 be a standard Wiener process. The supremum of the squared Euclidian norm ⊬Y (t)⊬2, of the R2-valued process Y(t)=(√1/tW(t), √ {12/t3 int0ts dW (s)−√ {3/t} W(t)), t ∈ [α, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {supt ∈ [α, 1] ⊬Y(t)√2 > u as u → ∞, of this statistic, for a fixed α ∈ (0,1), and for a “moving” α=α (u) ↓ 0 at a suitable rate as u → ∞. The statistical interest of our results lie in their use as approximate test levels.

Off-Line Statistical Process Control

First part of this paper deals with tests on the stability of statistical models. The problem is formulated in terms of testing the null hypothesis H against the alternative hypothesis A. The null hypothesis H claims that the model remains the same during the whole observational period, usually it means that the parameters of the model do not change. The alternative hypothesis A claims that, at an unknown time point, the model changes, which means that some of the parameters of the model are subject to a change. In case we reject the null hypothesis H, i.e. when we decide that there is a change in the model, we concentrate on a number of questions that arise: when has the model changed; is there just one change or are there more changes; what is the total number of changes etc.

Testing Appearance of Polynomial Trend

For deciding whether a gradual polynomial trend appears in mean of a sequence of independent random variables a “trimmed maximum-type” statistic may be used. As the number of observations tends to infinity the suggested statistic converges in distribution to a maximum of a differentiable χ2-process. The method for asymptotic critical values approximation based on high level exceedence probability was developed.

Statistical analysis of trends in organic pollution and pollution by nutrients at selected Danube river stations.

The paper describes an application of a statistical analysis for estimating long-term trends in pollutant concentrations of selected pollutants in the Danube river. The results show the changes of concentrations of NH(4)(+)-N, NO(3)(-)-N, PO(4)(3-)-P, total P, BOD(5) and COD(Cr) in a ten year period with the aim to find how the concentrations vary in the whole stretch of this river. The study was based on the data collected in the frame of Transnational Monitoring Network of the ICPDR. To obtain plausible results we have chosen statistical methods, such as tests based on the Spearman correlation coefficient and median regression, which are not sensitive to departures from normality as high skewness or outliers.

Wang tiling aided statistical determination of the Representative Volume Element size of random heterogeneous materials

Abstract Wang tile based representation of a heterogeneous material facilitates fast synthesis of non-periodic microstructure realizations. In this paper, we apply the tiling approach in numerical homogenization to determine the Representative Volume Element size related to the user-defined significance level and the discrepancy between bounds on the apparent properties. First, the tiling concept is employed to efficiently generate arbitrarily large, statistically consistent realizations of investigated microstructures. Second, benefiting from the regular structure inherent to the tiling concept, the Partition theorem, and statistical sampling, we construct confidence intervals of the apparent properties related to the size of a microstructure specimen. Based on the interval width and the upper and lower bounds on the apparent properties, we adaptively generate additional microstructure realizations in order to arrive at an RVE satisfying the prescribed tolerance. The methodology is illustrated with the homogenization of thermo-mechanical properties of three two-dimensional microstructure models: a microstructure with mono-disperse elliptic inclusions, foam, and sandstone.