Jeffrey D. Hart

Kernel Regression Estimation Using Repeated Measurements Data

Abstract The estimation of growth curves has been studied extensively in parametric situations. Here we consider the nonparametric estimation of an average growth curve. Suppose that there are observations from several experimental units, each following the regression model y(xi)=f(xj)+e(j=1,…,n), where e1, …, e n are correlated zero mean errors and 0≤x1<…<xn≤1 are fixed constants. We study some of the properties of a kernel estimator of f(x). Asymptotic and finite-sample results concerning the mean squared error of the estimator are obtained. In particular, the influence of correlation on the bandwidth minimizing mean squared error is discussed. A data-based method for selecting the bandwidth is illustrated in a data analysis. Most previous research on kernel regression estimators has involved uncorrelated errors. We investigate how dependence of the errors changes the behavior of a kernel estimator. Our theorems concerning the asymptotic mean squared error show that the estimator cannot be consistent un...

Bootstrap test for difference between means in nonparametric regression

Abstract A bootstrap test is proposed for detecting a difference between two mean functions in the setting of nonparametric regression. Error distributions in the regression model are permitted to be arbitrary and unequal. The test enjoys power properties akin to those in a parametric setting, in the sense that it can distinguish between regression functions distant only n −1/2 apart, where n is the sample size. It has exceptional level accuracy, with level error of only n −2, and uses a very accurate estimate of the critical point of an exact test, being in error by only n −3/2 under the null hypothesis. The test admits several generalizations, for example to the case of testing for differences between several regression means. (This is a nonparametric regression analog of analysis of variance.) A simulation study using n as small as 15 corroborates the asymptotic result on level accuracy of the bootstrap test. Applications are illustrated with an example involving acid rain data.

Nonparametric regression with long-range dependence

The effect of dependent errors in fixed-design, nonparametric regression is investigated. It is shown that convergence rates for a regression mean estimator under the assumption of independent errors are maintained in the presence of stationary dependent errors, if and only if [Sigma] r(j)

Parametric modelling of growth curve data: An overview

In the past two decades a parametric multivariate regression modelling approach for analyzing growth curve data has achieved prominence. The approach, which has several advantages over classical analysis-of-variance and general multivariate approaches, consists of postulating, fitting, evaluating, and comparing parametric models for the datas mean structure and covariance structure. This article provides an overview of the approach, using unified terminology and notation. Well-established models and some developed more recently are described, with emphasis given to those models that allow for nonstationarity and for measurement times that differ across subjects and are unequally spaced. Graphical diagnostics that can assist with model postulation and evaluation are discussed, as are more formal methods for fitting and comparing models. Three examples serve to illustrate the methodology and to reveal the relative strengths and weaknesses of the various parametric models.

Testing the equality of two regression curves using linear smoothers

Suppose that data (y, z) are observed from two regression models, y = f(x) + [var epsilon] and z = g(x) + [eta]. Of interest is testing the hypothesis H: f [triple bond; length as m-dash] g without assuming that f or g is in a parametric family. A test based on the difference between linear, but nonparametric, estimates of f and g is proposed. The exact distribution of the test statistic is obtained on the assumption that the errors in the two regression models are normally distributed. Asymptotic distribution theory is outlined under more general conditions on the errors. It is shown by simulation that the test based on the assumption of normal errors is reasonably robust to departures from normality. A data analysis illustrates that, in addition to being attractive descriptive devices, nonparametric smoothers can be valuable inference tools.

Bandwidth Choice for Average Derivative Estimation

Abstract The average derivative is the expected value of the derivative of a regression function. Kernel methods have been proposed as a means of estimating this quantity. The problem of bandwidth selection for these kernel estimators is addressed here. Asymptotic representations are found for the variance and squared bias. These are compared with each other to find an insightful representation for a bandwidth optimizing terms of lower order than n –1. It is interesting that, for dimensions greater than 1, negative kernels have to be used to prevent domination of bias terms in the asymptotic expression of the mean squared error. The extent to which the theoretical conclusions apply in practice is investigated in an economical example related to the so-called “law of demand.”

Kernel Estimation of Densities with Discontinuities or Discontinuous Derivatives

In kernel density estimation, one usually assumes the density has two continous derivatives. In this paper we give precise experssions for the asymptotic mean integrated squared error in case the density has m-1 continuous derivatives and two more derivatives with simple discontinuities. We show that the convergence rate for the mean integrated squared error, with appropriate kernel, is proportional to n-v,ν =2m+1/2m+2. Furthermore the ratio of the (random) integrated squared error to the mean integrated squared error converges to 1 almost surely, if the bandwidth is chosen by cross-validation. In particular, it is possible to have n-5/6 with only one continuous derivative, if the kernel and bandwidth are appropriately chosen. When the density has known points of discontinuity, a symmetrization device suggested by SCHUSTER (1985) can be used to improve the convergence rate. The mean integrated squared error of the symmetrization estimator will converge as if the density had no discontinuities at those poi...

Testing the Fit of a Parametric Function

Abstract General methods for testing the fit of a parametric function are proposed. The idea underlying each method is to “accept” the prescribed parametric model if and only if it is chosen by a model selection criterion. Several different selection criteria are considered, including one based on a modified version of the Akaike information criterion and others based on various score statistics. The tests have a connection with nonparametric smoothing because they use orthogonal series estimators to detect departures from a parametric model. An important aspect of the tests is that they can be applied in a wide variety of settings, including generalized linear models, spectral analysis, the goodness-of-fit problem, and longitudinal data analysis. Implementation using standard statistical software is straightforward. Asymptotic distribution theory for several test statistics is described, and the tests are shown to be consistent against essentially any alternative hypothesis. Simulations and a data exampl...

Some automated methods of smoothing time-dependent data

Nonparametric function estimation based upon time-dependent data is a challenging problem to both the data analyst and the theoretician. This paper serves as an introduction to the problem and discusses some of the approaches that have been proposed for smoothing autocorrelated data. A principal theme will be accounting for correlation in the data driven choice of a function estimators smoothing parameter. Data-driven smoothing is considered in various settings including probability density estimation, repeated measures data, and time series trend estimation. Both applications and theoretical issues are addressed, and some open problems will be discussed.

Convergence rates in density estimation for data from infinite-order moving average processes

SummaryThe effect of long-range dependence in nonparametric probability density estimation is investigated under the assumption that the observed data are a sample from a stationary, infinite-order moving average process. It is shown that to first order, the mean integrated squared error (MISE) of a kernel estimator for moving average data may be expanded as the sum of MISE of the kernel estimator for a same-sizerandom sample, plus a term proportional to the variance of the moving average sample mean. The latter term does not depend on bandwidth, and so imposes a ceiling on the convergence rate of a kernel estimator regardless of how bandwidth is chosen. This ceiling can be quite significant in the case of long-range dependence. We show thatall density estimators have the convergence rate ceiling possessed by kernel estimators.