The almost sure limiting behavior and convergence rate of a kernel regression estimator are studied. As the domain of density function is compactly supported, it is known that the regressionestimator of Mack and Mu(5...
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The almost sure limiting behavior and convergence rate of a kernel regression estimator are studied. As the domain of density function is compactly supported, it is known that the regressionestimator of Mack and Mu(5)(5) ller [11] will also encounter the problem of the boundary effects, that is, it lacks the consistency property. In order to improve the limiting behavior and convergence rate as above, the idea of linear-fit method is used to construct a general weighted kernel regression estimator. In this paper, the almost sure limiting behavior, the convergence rate and some properties of the proposed estimator are given. Besides, the proposed estimator does not also need to adjust the boundary regions.
In linear and nonparametric regression models, the problem of testing for symmetry of the distribution of errors is considered. We propose a test statistic which utilizes the empirical characteristic function of the c...
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In linear and nonparametric regression models, the problem of testing for symmetry of the distribution of errors is considered. We propose a test statistic which utilizes the empirical characteristic function of the corresponding residuals. The asymptotic null distribution of the test statistic as well as its behavior under alternatives is investigated. A simulation study compares bootstrap versions of the proposed test to other more standard procedures.
Useful connections between radial basis function (RBF) nets and kernel regression estimators (KRE) are established. By using existing theoretical results obtained for KRE as tools, we obtain a number of interesting th...
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Useful connections between radial basis function (RBF) nets and kernel regression estimators (KRE) are established. By using existing theoretical results obtained for KRE as tools, we obtain a number of interesting theoretical results for RBF nets. Upper bounds are presented for convergence rates of the approximation error with respect to the number of hidden units. The existence of a consistent estimator for RBF nets is proven constructively. Upper bounds are also provided for the pointwise and L2 convergence rates of the best consistent estimator for RBF nets as the numbers of both the samples and the hidden units tend to infinity. Moreover, the problem of selecting the appropriate size of the receptive field of the radial basis function is theoretically investigated and the way this selection is influenced by various factors is elaborated. In addition, some results are also given for the convergence of the empirical error obtained by the least squares estimator for RBF nets.
Consistent procedures are constructed for testing the goodness-of-fit of the error distribution in nonparametric regression models. The test starts with a kernel-type regression fit and proceeds with the construction ...
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Consistent procedures are constructed for testing the goodness-of-fit of the error distribution in nonparametric regression models. The test starts with a kernel-type regression fit and proceeds with the construction of a test statistic in the form of an L (2) distance between a parametric and a nonparametric estimates of the residual characteristic function. The asymptotic null distribution and the behavior of the test statistic under alternatives are investigated. A simulation study compares bootstrap versions of the proposed test to corresponding procedures utilizing the empirical distribution function.
Let {(X-i, Y-i), i >= 1}be a strictly stationary sequence of associated random vectors distributed as (X, Y). This note deals with kernel estimation of the regression function r(x) E[Y]X = x] in the presence of ran...
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Let {(X-i, Y-i), i >= 1}be a strictly stationary sequence of associated random vectors distributed as (X, Y). This note deals with kernel estimation of the regression function r(x) E[Y]X = x] in the presence of randomly right censored data caused by another variable C. For this model we establish a strong uniform consistency rate of the proposed estimator, say r(n) (x) . Simulations are drawn to illustrate the results and to show how the estimator behaves for moderate sample sizes.
We first establish both the pointwise and the uniform almost complete consistencies with rate of a kernel type regression estimate in a doubly censoring setting. Then a simulation study illustrates the good behaviour ...
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We first establish both the pointwise and the uniform almost complete consistencies with rate of a kernel type regression estimate in a doubly censoring setting. Then a simulation study illustrates the good behaviour of the studied estimator. (C) 2012 Elsevier B.V. All rights reserved.
Large deviation results for the kernel density estimator and the kernel regression estimator have been given by Louani [Louani, D., 1998, Large deviations limit theorems for the kernel density estimator. Scandinavian ...
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Large deviation results for the kernel density estimator and the kernel regression estimator have been given by Louani [Louani, D., 1998, Large deviations limit theorems for the kernel density estimator. Scandinavian Journal of Statistics, 25, 243-253;Louani, D., 1999, Some large deviations limit theorems in conditional nonparametric statistics. Statistics, 33, 171-196]. We complete these works by establishing sharp large deviation results for the two estimators. This means that we study precisely the tail probabilities of the estimators. We distinguish two cases depending on the support of the kernel. To prove the results, we need an Edgeworth expansion obtained from a version of Cramer's condition.
Let nu be a vector field in a bounded open set G subset of R-d. Suppose that nu 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 t...
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Let nu be a vector field in a bounded open set G subset of R-d. Suppose that nu 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 = nu(x(t)), t >= 0, x(0) = x(0) epsilon G, starting at a given point x(0) = x0 epsilon G and to develop statistical tests for the hypothesis that the integral curve reaches a specified set Gamma subset of 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 Gamma subset of 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.
In this paper we propose and study a new kernel regression estimator in which the kernel is taken from a properly adapted location-scale family of the design distribution. We show that, while the original smoothing ma...
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In this paper we propose and study a new kernel regression estimator in which the kernel is taken from a properly adapted location-scale family of the design distribution. We show that, while the original smoothing may be performed with sub-optimal bandwidths, adaptation of proper scale parameters yields overall optimal estimators. Unlike traditional smoothing methodology, our approach does not aim at estimating pivotal higher order derivatives. (C) 2012 Elsevier B.V. All rights reserved.
Consistent procedures are constructed for testing independence between the regressor and the error in non-parametric regression models. The tests are based on the Fourier formulation of independence, and utilize the j...
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Consistent procedures are constructed for testing independence between the regressor and the error in non-parametric regression models. The tests are based on the Fourier formulation of independence, and utilize the joint and the marginal empirical characteristic functions of the regressor and of estimated residuals. The asymptotic null distribution as well as the behavior of the test statistic under alternatives is investigated. A simulation study compares bootstrap versions of the proposed tests to corresponding procedures utilizing the empirical distribution function. (C) 2011 Published by Elsevier Inc.
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