Non-Gaussian Component Analysis: New Ideas, New Proofs, New Applications

In this article, we present new ideas concerning Non-Gaussian Component Analysis (NGCA). We use the structural assumption that a high-dimensional random vector X can be represented as a sum of two components - a lowdimensional signal S and a noise component N. We show that this assumption enables us for a special representation for the density function of X. Similar facts are proven in original papers about NGCA ([1], [5], [13]), but our representation differs from the previous versions. The new form helps us to provide a strong theoretical support for the algorithm; moreover, it gives some ideas about new approaches in multidimensional statistical analysis. In this paper, we establish important results for the NGCA procedure using the new representation, and show benefits of our method.dimension reduction, non-Gaussian components, EDR subspace, classification problem, Value at Risk

Estimation of the signal subspace without estimation of the inverse covariance matrix

Let a high-dimensional random vector X can be represented as a sum of two components - a signal S, which belongs to some low-dimensional subspace S, and a noise component N. This paper presents a new approach for estimating the subspace S based on the ideas of the Non-Gaussian Component Analysis. Our approach avoids the technical difficulties that usually exist in similar methods - it doesn’t require neither the estimation of the inverse covariance matrix of X nor the estimation of the covariance matrix of N.dimension reduction, non-Gaussian components, NGCA

Statistical inference for generalized Ornstein-Uhlenbeck processes

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type $$X_{t} = e^{-\xi_{t}} \left( X_{0} + \int_{0}^{t} e^{\xi_{u-}} d u \right),$$ where $\xi_s$ is a L{\'e}vy process. Our primal goal is to estimate the characteristics of the L\'evy process $\xi$ from the low-frequency observations of the process $X$. We present a novel approach towards estimating the L{\'e}vy triplet of $\xi,$ which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.Comment: 32 pages. arXiv admin note: text overlap with arXiv:1312.473

Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems

The classical parametric and semiparametric Bernstein -- von Mises (BvM) results are reconsidered in a non-classical setup allowing finite samples and model misspecification. In the case of a finite dimensional nuisance parameter we obtain an upper bound on the error of Gaussian approximation of the posterior distribution for the target parameter which is explicit in the dimension of the nuisance and target parameters. This helps to identify the so called \emph{critical dimension} $p$ of the full parameter for which the BvM result is applicable. In the important i.i.d. case, we show that the condition "$p^{3} / n$ is small" is sufficient for BvM result to be valid under general assumptions on the model. We also provide an example of a model with the phase transition effect: the statement of the BvM theorem fails when the dimension $p$ approaches $n^{1/3}$. The results are extended to the case of infinite dimensional parameters with the nuisance parameter from a Sobolev class. In particular we show near normality of the posterior if the smoothness parameter $s$ exceeds 3/2

Non-gaussian component analysis: New ideas, new proofs, new applications

In this article, we present new ideas concerning Non-Gaussian Component Analysis (NGCA). We use the structural assumption that a high-dimensional random vector X can be represented as a sum of two components - a lowdimensional signal S and a noise component N. We show that this assumption enables us for a special representation for the density function of X. Similar facts are proven in original papers about NGCA ([1], [5], [13]), but our representation differs from the previous versions. The new form helps us to provide a strong theoretical support for the algorithm; moreover, it gives some ideas about new approaches in multidimensional statistical analysis. In this paper, we establish important results for the NGCA procedure using the new representation, and show benefits of our method