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Institute
- Fachbereich Medizintechnik und Technomathematik (1545) (remove)
Let X₁,…,Xₙ be independent and identically distributed random variables with distribution F. Assuming that there are measurable functions f:R²→R and g:R²→R characterizing a family F of distributions on the Borel sets of R in the way that the random variables f(X₁,X₂),g(X₁,X₂) are independent, if and only if F∈F, we propose to treat the testing problem H:F∈F,K:F∉F by applying a consistent nonparametric independence test to the bivariate sample variables (f(Xᵢ,Xⱼ),g(Xᵢ,Xⱼ)),1⩽i,j⩽n,i≠j. A parametric bootstrap procedure needed to get critical values is shown to work. The consistency of the test is discussed. The power performance of the procedure is compared with that of the classical tests of Kolmogorov–Smirnov and Cramér–von Mises in the special cases where F is the family of gamma distributions or the family of inverse Gaussian distributions.
The paper deals with an asymptotic relative efficiency concept for confidence regions of multidimensional parameters that is based on the expected volumes of the confidence regions. Under standard conditions the asymptotic relative efficiencies of confidence regions are seen to be certain powers of the ratio of the limits of the expected volumes. These limits are explicitly derived for confidence regions associated with certain plugin estimators, likelihood ratio tests and Wald tests. Under regularity conditions, the asymptotic relative efficiency of each of these procedures with respect to each one of its competitors is equal to 1. The results are applied to multivariate normal distributions and multinomial distributions in a fairly general setting.
Suppose we have k samples X₁,₁,…,X₁,ₙ₁,…,Xₖ,₁,…,Xₖ,ₙₖ with different sample sizes ₙ₁,…,ₙₖ and unknown underlying distribution functions F₁,…,Fₖ as observations plus k families of distribution functions {G₁(⋅,ϑ);ϑ∈Θ},…,{Gₖ(⋅,ϑ);ϑ∈Θ}, each indexed by elements ϑ from the same parameter set Θ, we consider the new goodness-of-fit problem whether or not (F₁,…,Fₖ) belongs to the parametric family {(G₁(⋅,ϑ),…,Gₖ(⋅,ϑ));ϑ∈Θ}. New test statistics are presented and a parametric bootstrap procedure for the approximation of the unknown null distributions is discussed. Under regularity assumptions, it is proved that the approximation works asymptotically, and the limiting distributions of the test statistics in the null hypothesis case are determined. Simulation studies investigate the quality of the new approach for small and moderate sample sizes. Applications to real-data sets illustrate how the idea can be used for verifying model assumptions.
The term ocular rigidity is widely used in clinical ophthalmology. Generally it is assumed as a resistance of the whole eyeball to mechanical deformation and relates to biomechanical properties of the eye and its tissues. Basic principles and formulas for clinical tonometry, tonography and pulsatile ocular blood flow measurements are based on the concept of ocular rigidity. There is evidence for altered ocular rigidity in aging, in several eye diseases and after eye surgery. Unfortunately, there is no consensual view on ocular rigidity: it used to make a quite different sense for different people but still the same name. Foremost there is no clear consent between biomechanical engineers and ophthalmologists on the concept. Moreover ocular rigidity is occasionally characterized using various parameters with their different physical dimensions. In contrast to engineering approach, clinical approach to ocular rigidity claims to characterize the total mechanical response of the eyeball to its deformation without any detailed considerations on eye morphology or material properties of its tissues. Further to the previous chapter this section aims to describe clinical approach to ocular rigidity from the perspective of an engineer in an attempt to straighten out this concept, to show its advantages, disadvantages and various applications.
Numerical methods for limit and shakedown analysis. Deterministic and probabilistic problems.
(2003)
Recently, we introduced and mathematically analysed a new method for grid deformation (Grajewski et al., 2009) [15] we call basic deformation method (BDM) here. It generalises the method proposed by Liao et al. (Bochev et al., 1996; Cai et al., 2004; Liao and Anderson, 1992) [4], [6], [20]. In this article, we employ the BDM as core of a new multilevel deformation method (MDM) which leads to vast improvements regarding robustness, accuracy and speed. We achieve this by splitting up the deformation process in a sequence of easier subproblems and by exploiting grid hierarchy. Being of optimal asymptotic complexity, we experience speed-ups up to a factor of 15 in our test cases compared to the BDM. This gives our MDM the potential for tackling large grids and time-dependent problems, where possibly the grid must be dynamically deformed once per time step according to the user's needs. Moreover, we elaborate on implementational aspects, in particular efficient grid searching, which is a key ingredient of the BDM.
A variety of transition metals, e.g., copper, zinc, cadmium, lead, etc. are widely used in industry as components for wires, coatings, alloys, batteries, paints and so on. The inevitable presence of transition metals in industrial processes implies the ambition of developing a proper analytical technique for their adequate monitoring. Most of these elements, especially lead and cadmium, are acutely toxic for biological organisms. Quantitative determination of these metals at low activity levels in different environmental and industrial samples is therefore a vital task. A promising approach to achieve an at-side or on-line monitoring on a miniaturized and cost efficient way is the combination of a common potentiometric sensor array with heavy metal-sensitive thin-film materials, like chalcogenide glasses and polymeric materials, respectively.