ﻻ يوجد ملخص باللغة العربية
Let $v$ be a vector field in a bounded open set $Gsubset {mathbb {R}}^d$. 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 [frac{dx(t)}{dt}=v(x(t)),qquad tgeq 0,x(0)=x_0in G,] starting at a given point $x(0)=x_0in G$ and to develop statistical tests for the hypothesis that the integral curve reaches a specified set $Gammasubset G.$ 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 $Gammasubset G$. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches $Gamma$. 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.
Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $int Phi(g(x)),dx$. Following a nonparametric
We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new p
We consider a stochastic individual-based model in continuous time to describe a size-structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We address here the problem of nonparametric estima
Kernel-based nonparametric hazard rate estimation is considered with a special class of infinite-order kernels that achieves favorable bias and mean square error properties. A fully automatic and adaptive implementation of a density and hazard rate e
In this paper we propose a new test for the hypothesis of a constant coefficient of variation in the common nonparametric regression model. The test is based on an estimate of the $L^2$-distance between the square of the regression function and varia