Bootstrap confidence sets for spectral projectors of sample covariance

Let $X_{1},\ldots,X_{n}$ be i.i.d. sample in $\mathbb{R}^{p}$ with zero mean and the covariance matrix $\mathbf{\Sigma}$. The problem of recovering the projector onto an eigenspace of $\mathbf{\Sigma}$ from these observations naturally arises in many applications. Recent technique from [Koltchinskii, Lounici, 2015] helps to study the asymptotic distribution of the distance in the Frobenius norm $\| \mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ between the true projector $\mathbf{P}_r$ on the subspace of the $r$-th eigenvalue and its empirical counterpart $\widehat{\mathbf{P}}_r$ in terms of the effective rank of $\mathbf{\Sigma}$. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector $\mathbf{P}_r$ from the given data. This procedure does not rely on the asymptotic distribution of $\| \mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ and its moments. It could be applied for small or moderate sample size $n$ and large dimension $p$. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.Comment: 39 pages, 3 figure

Regional differentiation of human potential indicators

The purpose of the study presented in this article is an interregional analysis of human potential. The quality of the population proper is investigated at the regional level far less than the quality of the population life. The article provides an extended characteristic of human potential in seven directions: economic activity, demographic processes, physical health, the cultural potential of population, social health, educational potential, the attitude of population to the environment. On the basis of official statistics for 2008–2012, there were selected 63 indicators characterizing human potential in all these directions. In the final result, the correlation analysis has led to the substantiation of the system of indicators for the level of human potential development, consisting of 10 indicators. The system included 3 economic indicators and 7 social indicators characterizing human potential. Upon these indicators the Russian regions were divided in two types of regions by means of hierarchic agglomerative (combining) methods of cluster analysis: the regions with economic indicators and without them. The performed calculations provided the typology of regions by the human potential indicators being stable over time and covering 74.4 % of the Russian population. A substantial interpretation of breaking down regions by groups, identification of both strong and weak aspects of each cluster were made, finding out specific features of the regions falling under the clusters. The obtained results can be used when working out measures for reducing the interregional inequality in the levels of human potential development. To find out what measures can be effective, it is possible to examine the strategic directions of regions’ development in the cluster that is the most successful with respect to the human potential characteristics under investigation

Corrosion of diffusion zinc coatings in sodium chloride solutions

Diffusion galvanizing is widely used in the pipe industry for coating the threaded surface of pipe couplings, protecting water and gas pipelines, and other metal products. Diffusion coatings have a number of advantages over other types of zinc coatings. In this work, electrochemical and gravimetric methods are used to study the corrosion behavior of diffusion zinc coatings in sodium chloride solutions. The corrosion rate depends non-linearly on the thickness of the coating. At the initial stages, the corrosion rate of coatings depends on the structure of the phases on the surface, and with an increase in the holding time, the corrosion rate depends to a greater extent on the properties of the products formed during the corrosion process. Films of corrosion products of diffusion zinc coatings consist of zinc oxide/hydroxide and basic zinc salts, while the composition of the film changes with increasing coating thickness

Composition and structure of tungsten antimony acid

Tungsten antimony acids (TAA) with the composition H(2)Sb(2)WXO6·nH2O (0 < x ≤ 1.45; 0 < n ≤ 2.0) have been synthesized by hydrolysis of antimony trichloride pre-oxidized with nitric acid in the presence of varying amounts of Na2WO4. To obtain TAA protonated forms, the samples were kept in a 96% solution of sulphuric acid, the precipitate was washed until reaction became neutral and dried in air. The amount of tungsten, antimony, and silver ions in TAA was determined using energy dispersive analysis. Changes in structural parameters upon doping of AA with tungsten ions were studied using a Bruker D8 ADVANCE X-ray diffractometer (CuKa1-radiation). The number of oxonium ions in TAA was determined by the substitution of these ions by silver ions in equivalent amounts (Ag+-TAA forms). All obtained TAA samples and Ag+ TAA forms had a pyrochlore-type structure, space group symmetry Fd3m. Refinement of the arrangement of atoms in the structure using the Rietveld method showed that tungsten ions replace antimony ions and are statistically located in 16c, oxygen anions in 48f, and oxonium ions and water molecules in 16d and 8b positions, respectively. When tungsten ions were introduced into samples, the structural parameters of the resulting phases changed. There was a decrease in the unit cell parameter and the distance between antimony ions and oxygen anions, while an increase in the distance between oxonium ions and oxygen anions located in 48f positions was observed. This allowed the removal of a proton from oxonium molecules and its transport via a system of hydrogen bonds formed by water molecule