摘要:

In this article we study the method of nonparametric regression based on a weighted local linear regression. This method has advantages over other popular kernel methods. Moreover, such a regression procedure has the ability of design adaptation: It adapts to both random and fixed designs, to both highly clustered and nearly uniform designs, and even to both interior and boundary points. It is shown that the local linear regression smoothers have high asymptotic efficiency (i.e., can be 100% with a suitable choice of kernel and bandwidth) among all possible linear smoothers, including those produced by kernel, orthogonal series, and spline methods. The finite sample property of the local linear regression smoother is illustrated via simulation studies. Nonparametric regression is frequently used to explore the association between covariates and responses. There are many versions of kernel regression smoothers. Some estimators are not good for random designs, such as in observational studies, and others are not good for nonequispaced designs. Furthermore, most nonparametric regression smoothers have “boundary effects” and require modifications at boundary points. However, the local linear regression smoothers do not share these disadvantages. They adapt to almost all regression settings and do not require any modifications even at boundary. Besides, this method has higher efficiency than other traditional nonparametric regression methods.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476255

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

We propose a multivariate smoothing method based on products of localized orthogonal polynomial series estimators for a smooth regression surface in the fixed-design regression model. The estimation of partial derivatives is included. The proposed method provides for automatic and efficient boundary modifications near the edges of the surface, assuming that the boundary of the support of the regression function satisfies some regularity conditions. By allowing for a preaveraging step, the corresponding algorithms are speeded up considerably and are easy to implement. Computation of special boundary kernels, as required by the kernel method to avoid edge effects, is not necessary. It is shown that under sufficient smoothness assumptions, the global average mean squared error has the same optimal rate of convergence as the mean squared error at an interior point; that is, the boundary correction is asymptotically effective. The method depends on two smoothing parameters, one determining the amount of preaveraging and the other determining the amount of smoothing after preaveraging. Theoretical and practical bounds for the choice of these parameters are discussed. A Monte Carlo study based on a bivariate Gaussian surface indicates that increasing the preaveraging parameterδhas a negative effect on the average mean squared error, which is not unexpected. On the other hand, larger values ofδare computationally more economical. The effects of boundary correction as compared to noncorrected estimates are investigated for the example of a quadratic surface. The numerical complexity of the proposed method is discussed. The methods are demonstrated and compared to kriging for two data sets, one on nonuniformly measured groundwater levels in Arizona and the other on cover-clay thickness data from Iran measured on a regular mesh. The two data analyses include regular and irregular designs and supports; they seem to indicate that the method works well, particularly when compared to kriging.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476256

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

It is well known that kernel regression estimators are subject to so-called boundary or edge effects, a phenomenon in which the bias of an estimator increases near the endpoints of the estimation interval. When the regression curve is linear or nearly linear, the requisite amount of smoothing is so great that the boundary region is effectively the entire estimation interval. Special boundary kernels are proposed here to deal with such cases. It is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines; namely, as the estimator's smoothing parameter becomes large, the estimator tends to a straight line. The limiting straight line is essentially the least squares line when the design points are equally spaced. A simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths. Such estimates are an important component of a proposed test for the adequacy of a polynomial model. The test statistic is the bandwidth chosen to minimize an estimated risk function. An example illustrates the usefulness of the new boundary kernels.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476257

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Modern graphical tools have enhanced our ability to learn many things from data directly. With much user-friendly graphical software available, we are encouraged to plot a lot more often than before. The benefits from direct interaction with graphics have been enormous. But trailing behind these high-tech advances is the issue of appropriate guidance on what to plot. There are too many directions to project a high-dimensional data set and unguided plotting can be time-consuming and fruitless. In a recent article, Li set up a statistical framework for study on this issue, based on a notion of effective dimension reduction (edr) directions. They are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable. A methodology, sliced inverse regression, was introduced and shown to be useful in finding edr directions. This article introduces another method for finding edr directions. It begins with the observation that the eigenvectors for the Hessian matrices of the regression function are helpful in the study of the shape of the regression surface. A notation of principal Hessian directions (pHd's) is defined that locates the main axes along which the regression surface shows the largest curvatures in an aggregate sense. We show that pHd's can be used to find edr directions. We further use the celebrated Stein lemma for suggesting estimates. The sampling properties of the estimated pHd's are obtained. A significance test is derived for suggesting the genuineness of a view found by our method. Some versions for implementing this method are discussed, and simulation results and an application to real data are reported. The relationship of this method with exploratory projection pursuit is also discussed.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476258

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

A general nonlinear regression problem is considered with measurement error in the predictors. We assume that the response is related to an unknown linear combination of a multidimensional predictor through anunknownlink function. Instead of observing the predictor, we instead observe a surrogate with the property that its expectation is linearly related to the true predictor with constant variance. We identify an important transformation of the surrogate variable. Using this transformed variable, we show that if one proceeds with the usual analysis ignoring measurement error, then both ordinary least squares and sliced inverse regression yield estimates which consistently estimate the true regression parameter, up to a constant of proportionality. We derive the asymptotic distribution of the estimates. A simulation study is conducted applying sliced inverse regression in this context.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476259

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Recent developments of multivariate smoothing methods provide a rich collection of feasible models for nonparametric multivariate data analysis. Among the most interpretable are models with additive terms. Construction of various models and algorithms for computing the models have been the main concern of the existing literature in this area. Few results are available on the validation of computed fits, and many applications of nonparametric methods unfortunately end up interpreting the noise. This article proposes and illustrates some simple retrospective diagnostics to help data analysts in detecting possible aliasing effects in computed nonparametric fits and in building parsimonious models in an interactive fashion. It also discusses the concepts and rationale behind the proposal, including concurvity, diagnostics versus tests, and so forth. For their ready availability, interaction splines are used in the illustrations.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476260

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Toxicologists frequently conduct toxicity experiments in which different treatment conditions are applied to groups of animals and the resulting mortality in each group is measured at a number of discrete time points over the course of the experiment. In this article, we develop and extend a number of diagnostic tools for the detection of mean misspecification, or systematic departures of the mean-link specification, in cumulative multinomial generalized linear models fit to such data. Several real data sets are used to illustrate these diagnostics. These tools help the analyst to differentiate between two sources of lack of fit in such models: mean misspecification and extra-multinomial variation or overdispersion.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476261

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

Frequency domain analysis is used to examine estimates from linear smoothers operating on realizations of random fields over space and/or time. The estimates are expressed in terms of the Fourier transforms of the dependent variable and of the smoother weights. The latter is referred to as the equivalent transfer function. The data do not need to be evenly spaced to perform this analysis. The modulus of the equivalent transfer function characterizes the spectral content of an estimate and may reveal subtle sampling properties of the design. Frequency domain bias calculations are useful for comparing different smoothers and for assessing the resolution capabilities of a data set. These methods are used to compare six one-dimensional smoothers and analyze a complex three-dimensional example using data from a satellite altimeter.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476262

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

In the analysis of economic time series, a question often raised is whether a vector of variables causes another one in the sense of Granger. Most of the literature on this topic is concerned with bivariate relationships or uses finite-order autoregressive specifications. The purpose of this article is to develop a causality analysis in the sense of Granger for general vector autoregressive moving average (ARMA) models. We give a definition of Granger noncausality between vectors, which is a natural and simple extension of the notion of Granger noncausality between two variables. In our context, this definition is shown to be equivalent to a more complex definition proposed by Tjostheim. For the class of linear invertible processes, we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis. This result is then specialized to the class of stationary invertible ARMA processes. Further, relatively simple necessary and sufficient conditions are obtained for two important cases: (1) the case where the two vectors reduce to two variables inside a larger vector including other variables; and (2) the case where the two vectors embody all the variables considered. Test procedures for these necessary and sufficient conditions are discussed. Among other things, it is noted that the necessary and sufficient conditions for noncausality may involve singularities at which standard asymptotic regularity conditions do not hold. To deal with such situations, we propose a sequential approach that leads to bounds tests. Finally, the tests suggested are applied to Canadian money and income data. The tests are based on bivariate and trivariate models of changes in nominal income and two money stocks (M1 and M2). In contrast with the evidence based on bivariate models, we find from the trivariate model that money causes income unidirectionally.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476263

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

In this article we consider the case that each patient in a longitudinal study may experience two or more distinct failures. The corresponding failure times, which are possible censored, are recorded for each patient. The logarithm of each marginal failure time is assumed to be linearly related to its covariates; however, the distributional form of the error term in the model does not have to be specified in the analysis. Furthermore, no specific structure of dependence among the distinct failure times on each subject has to be imposed. Various linear regression methods for analyzing multivariate failure time observations are proposed. Our procedures do not involve the unstable nonparametric hazard function estimation. Extensive numerical studies are conducted to evaluate the new proposals. Recommendations are also made for their practical usage.

ISSN:0162-1459    

DOI:10.1080/01621459.1992.10476264

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor