On distinguishability of two nonparametric sets of hypothesis

Let we observe a signal S(t), t ∈ (0, 1) in Gaussian white noise ∈ dw(t). The problem is to test a hypothesis S ∈ Θ1 ⊂ L2 (0, 1) versus alternatives S ∈ Θ2 ⊂ L2(0, 1). The sets Θ1, Θ2 are closed and bounded. We show that there exists a statistical procedure allowing to make a true solution S ∈ Θ1 or S ∈ Θ2 with probability tending to one as ∈ → 0 ( i.e. to distinguish two nonparametric sets Θ1 and Θ2) iff there exists a finite-dimensional subspace H ⊂ L2 (0, 1) such that the projections Θ1 and Θ2 on H have no common points. A similar result is also obtained for the problems of testing hypotheses about density

Minimax and Bayes estimation in deconvolution problem

We consider the problem of estimation of solution of convolution equation on observations blurred a random noise. The noise is a product of Gaussian stationary process and a weight function $\epsilon h \in L_2(R1)$ with constant $\epsilon > 0$. The presence of weight function $h$ makes the power of noise finite on $R1$. This allows to suppose that the power of solution is also finite. For this model we find asymptotically minimax and Bayes estimators. The solution is supposed infinitely differentiable. The model with solutions having finite number of derivatives was studied in [5]

On asymptotic minimaxity of kernel based tests

In the problem of signal detection in Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L2-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that L2-norms of signal smoothed by the kernels exceed some constants ρε > 0. The constant ρε depends on the power ε of noise and ρε → 0 as ε → 0. Similar statements are proved also if an additional information on a signal smoothness is given. By theorems on asymptotic equivalence of statistical experiments these results are extended on the problems of testing nonparametric hypothesis on density and regression. The exact asymptotically minimax lower bounds of type II error probabilities are pointed out for all these settings. Similar results are also obtained in the problems of testing parametric hypothesis versus nonparametric sets of alternatives

On large and moderate large deviations of empirical bootstrap measure

We find the asymptotics for the large and moderate large deviation probabilities of common distribution of the empirical measure and the empirical bootstrap measure (empirical measure obtaining by the bootstrap method). For the most widespread statistical functionals depending on empirical measure we compare their asymptotics of moderate large deviation probabilities with similar asymptotics given by the bootstrap procedure

On moderate deviation probabilities of empirical bootstrap measure

We establish the moderate deviation principle for the common distribution of empirical measure and empirical bootstrap measure (empirical measure obtaining by the bootstrap procedure). For the most widespread statistical functionals depending on empirical measure (in particular differentiable and homogeneous functionals) we compare their asymptotic of moderate deviation probabilities with the asymptotic given by the bootstrap procedure

Лингвистический анализ юмора в английском языке

This article focuses on humour in form of linguistic analysis, describin

Macular pigment Raman detector for clinical applications

Journal ArticleClinical studies of carotenoid macular pigments (MP) have been limited by the lack of noninvasive, objective instruments. We introduce a novel noninvasive optical instrument, an MP Raman detector, for assessment of the carotenoid status of the human retina in vivo. The instrument uses resonant excitation of carotenoid molecules in the visible wavelength range, and quantitatively measures the highly specific Raman signals that originate from the single- and double-bond stretch vibrations of the pi-conjugated carotenoid molecule's carbon backbone. The instrument is a robust, compact device and suitable for routine measurements of MP concentrations in a clinical setting. We characterized and tested the instrument in clinical studies of human subjects to validate its function and to begin to establish its role as a possible screening test for macular pathologies. We also show that the MP Raman spectroscopy technology has potential as a novel, highly specific method for rapid screening of carotenoid antioxidant levels in large populations at risk for vision loss from age-related macular degeneration, the leading cause of blindness of the elderly in the developed world

Tonometric estimation of mechanical properties of a cornea and sclera

Purpose: To investigate degree of anisotropy and evaluate elastic constants of the eyeball tissues based on a shell theory. Methods: Simulated sclera was mechanically modeled as an orthotropic spherical shell. Scleral deformations were generated by simulated stresses in the intraocular pressure (IOP). Relations between the shape deflection mode in the shell and the ratio of tangential elasticity modules were calculated. In Maklakov’s method of IOP testing, the eyeball is modelled as two spherical segments with different mechanical properties. The baseline (prior to load) two-segment shell is assumed to be filled with uncompressible liquid under pressure. Young’s module of cornea is much lower than that of sclera. Hence, cornea is more compliant than rigid sclera and doesn’t resist to flexural deformations. A nonlinear shell theory was used to analyze deformations of both the cornea and the sclera. Results: Ratios of tangential elasticity modules of the orthotropic spherical shell in different directions appeared to have a great influence on the amplitude and shape of the deformation in the shell. Therefore, the degree of scleral anisotropy can be evaluated by ratio of anterior-posterior eye axis lengths to the equator diameter. Maklakov’s method of IOP measurement, with several different loads, can be also used to investigate the elasticity of the eye tunics and to study the range of application of physically linear theories of elasticity in mechanical models of pressure-related eyeball deformation. Conclusions: Pressure-related eyeball deformation varies significantly depending on the degree of anisotropy and heterogeneity of the eye tunics. Transversal shear modules have the greatest influence on the pressure-related deformation of the eyeball tunics. Physical nonlinear theories of elasticity are necessary tools to characterize details of pressure-related corneal and scleral deformations

The oxygen isotope effect on critical temperature in superconducting copper oxides

The isotope effect provided a crucial key to the development of the BCS (Bardeen-Cooper-Schrieffer) microscopic theory of superconductivity for conventional superconductors. In superconducting cooper oxides (cuprates) showing an unconventional type of superconductivity, the oxygen isotope effect is very peculiar: the exponential coefficient strongly depends on doping level. No consensus has been reached so far on the origin of the isotope effect in the cuprates. Here we show that the oxygen isotope effect in cuprates is in agreement with the bisoliton theory of superconductivity.Comment: 3 pages including 4 figures; version 2 is with minor correction