Integral Curves of Noisy Vector Fields and Statistical Problems in Diffusion Tensor Imaging: Nonparametric Kernel Estimation and Hypotheses Testing

dx(t) dt = v(x(t)), t≥ 0, x(0) = x0 ∈G, starting at a given point x(0) = x0 ∈ G and to develop statistical tests for the hypothesis that the integral curve reaches a specified set Γ ⊂ 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 Γ⊂G. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches Γ. 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