13 research outputs found
A new algorithm for estimating the effective dimension-reduction subspace
The statistical problem of estimating the effective dimension-reduction (EDR)
subspace in the multi-index regression model with deterministic design and
additive noise is considered. A new procedure for recovering the directions of
the EDR subspace is proposed. Under mild assumptions, -consistency of
the proposed procedure is proved (up to a logarithmic factor) in the case when
the structural dimension is not larger than 4. The empirical behavior of the
algorithm is studied through numerical simulations
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 \vX can be
represented as a sum of two components - a low-dimensional signal
\vS and a noise component \vN. We show that this assumption enables us for a special representation for the density function of \vX. Similar facts are proven in original papers about NGCA, 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
Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression
We consider the problem of testing a particular type of composite null
hypothesis under a nonparametric multivariate regression model. For a given
quadratic functional , the null hypothesis states that the regression
function satisfies the constraint , while the alternative
corresponds to the functions for which is bounded away from zero. On the
one hand, we provide minimax rates of testing and the exact separation
constants, along with a sharp-optimal testing procedure, for diagonal and
nonnegative quadratic functionals. We consider smoothness classes of
ellipsoidal form and check that our conditions are fulfilled in the particular
case of ellipsoids corresponding to anisotropic Sobolev classes. In this case,
we present a closed form of the minimax rate and the separation constant. On
the other hand, minimax rates for quadratic functionals which are neither
positive nor negative makes appear two different regimes: "regular" and
"irregular". In the "regular" case, the minimax rate is equal to
while in the "irregular" case, the rate depends on the smoothness class and is
slower than in the "regular" case. We apply this to the issue of testing the
equality of norms of two functions observed in noisy environments
Test function: A new approach for covering the central subspace
In this paper we offer a complete methodology for sufficient dimension
reduction called the test function (TF). TF provides a new family of methods
for the estimation of the central subspace (CS) based on the introduction of a
nonlinear transformation of the response. Theoretical background of TF is
developed under weaker conditions than the existing methods. By considering
order 1 and 2 conditional moments of the predictor given the response, we
divide TF in two classes. In each class we provide conditions that guarantee an
exhaustive estimation of the CS. Besides, the optimal members are calculated
via the minimization of the asymptotic mean squared error deriving from the
distance between the CS and its estimate. This leads us to two plug-in methods
which are evaluated with several simulations
Slice inverse regression with score functions
International audienceWe consider non-linear regression problems where we assume that the response depends non-linearly on a linear projection of the covariates. We propose score function extensions to sliced inverse regression problems, both for the first-order and second-order score functions. We show that they provably improve estimation in the population case over the non-sliced versions and we study finite sample estimators and their consistency given the exact score functions. We also propose to learn the score function as well, in two steps, i.e., first learning the score function and then learning the effective dimension reduction space, or directly, by solving a convex optimization problem regularized by the nuclear norm. We illustrate our results on a series of experiments