Lyudmila Sakhanenko

Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: nonparametric kernel estimation and hypotheses testing

Let v be a vector field in a bounded open set G ⊂ Ρ d . Suppose that v is observed with a random noise at random points X i , i = 1,..., n, that are independent and uniformly distributed in G. The problem is to estimate the integral curve of the differential equation dx(t) dt =Q,x(0)=x 0 ∈G, starting at a given point x(0) = x 0 ∈ G and to develop statistical tests for the hypothesis that the integral curve reaches a specified set r C G. We develop an estimation procedure based on a Nadaraya-Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface r C G. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches r. The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter.

Testing for ellipsoidal symmetry: A comparison study

The focus of this paper is the methodology for testing ellipsoidal symmetry, which was recently proposed by Koltchinskii and Sakhanenko [Koltchinskii, V., Sakhanenko, L. 2000. Testing for ellipsoidal symmetry of a multivariate distribution. In: Gine, E., Mason, D., Wellner, J. (Eds.), High Dimensional Probability II. In: Progress in Probability, Birkhauser, Boston, pp. 493-510]. It is a class of omnibus bootstrap tests that are affine invariant and consistent against any fixed alternative. First, we study their behavior under a sequence of local alternatives. Secondly, a finite sample comparison study of this new class of tests with other popular methods given by Beran, Manzotti et al., and Huffer et al. is carried out. We find that the new tests outperform other methods in preserving the level and have superior power for the most of the chosen alternatives. We also suggest a tool for identifying periods of financial instability and crises when these tests are applied to the distribution of the return rates of stock market indices. These tests can be used in place of tests for normality of asset return distributions since ellipsoidally symmetric distributions are the natural extensions of multivariate normal distributions, so that the capital asset pricing model holds.

Lower Bounds for Accuracy of Estimation in Diffusion Tensor Imaging

A vector field is observed at random locations with additive noise. The corresponding integral curve is to be estimated based on the data. The focus of the current paper is to obtain lower bounds for the functions of deviations between true and estimated integral curves. In particular, we show that the estimation procedure in [Koltchinskii, Sakhanenko, and Cai, Ann. Statist., 35 (2007), pp. 1576–1607] yields estimates, which have the optimal rate of convergence in a minimax sense. Overall, this work is motivated by diffusion tensor imaging, which is a modern brain imaging technique. The integral curves are used to model axonal fibers in the brain. In medical research, it is important to estimate and map these fibers. The paper addresses statistical aspects pertinent to such an estimation problem.

Convergence in Distribution of Self-Normalized Sup-Norms of Kernel Density Estimators

Let fn denote a kernel density estimator of a density f on the real line, for a bounded, compactly supported probability kernel. Under relatively weak smoothness conditions on f and K it is proved, for every 0 < β < 1/2, that the sequence

Testing Group Symmetry of a Multivariate Distribution

{{\hat{A}}_{n}}\left( {\frac{{\sqrt {{n{{h}_{n}}}} }}{{\parallel K{{\parallel }_{2}}\parallel {{f}_{n}}\parallel _{\infty }^{{1/2 - \beta }}}}\mathop{{\sup }}\limits_{{t \in {{{\hat{D}}}_{{{{a}_{n}}}}}}} \frac{{|{{f}_{n}}(t) - f(t)|}}{{f_{n}^{\beta }(t)}} - {{{\hat{A}}}_{n}}} \right)

Asymptotics of Suprema of Weighted Gaussian Fields with Applications to Kernel Density Estimators

converges in distribution to the double exponential law. Here \({{\hat{A}}_{n}}\) is constructed from the sample, a n → ∞ as a power of n and \({{\hat{D}}_{{{{a}_{n}}}}} = \{ t:{{f}_{n}}(t) \geqslant a_{n}^{{ - 1}}\}\). Thus, this result provides distribution free asymptotic confidence bands for densities on the real line.

Kernel density estimators: convergence in distribution for weighted sup-norms

In their recent work Koltchinskii, Sakhanenko, and Cai [Ann. Statist., 35 (2007), pp. 1576–1607] proposed and studied estimators for integral curves based on noisy data of the corresponding gradient vector field. That estimation problem was motivated by diffusion tensor imaging, a popular brain imaging technique. Recently Sakhanenko [Theory Probab. Appl., 54 (2009), pp. 166–177] showed that those estimates have pointwise optimal convergence rate in a minimax sense. In this work we show that these estimators are convergence rate-optimal in the minimax sense with respect to the integral

Goodness-of-fit testing in regression: A finite sample comparison of bootstrap methodology and Khmaladze transformation

L_p