Kernel estimators for multivariate regression

Staniswalis, Joan G.; Messer, Karen; Finston, David R.

doi:10.1080/10485259308832575

Cited by 18 publications

(11 citation statements)

References 17 publications

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“…In fact, they extend the minimum variance selection principle kernel used to select the optimal kernel in the interior region, as in Epanechnikov (1969) and Granovsky and Müller (1991). In the nonparametric regression context, the problem of boundary bias is developed by Gasser, Müller, and Mammitsch (1985), and Zhang, Karunamuni, and Jones (1999) for the univariate case, and Fan and Gijbels (1992), , Staniswalis, Messer, and Finston (1993), and Staniswalis and Messer (1997) for multivariate data. This paper proposes a nonparametric product kernel estimator for density functions of multivariate bounded data.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric density estimation for multivariate bounded data

Bouezmarni

Rombouts

2010

Journal of Statistical Planning and Inference

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We propose a new nonparametric estimator for the density function of multivariate bounded data. As frequently observed in practice, the variables may be partially bounded (e.g., nonnegative) or completely bounded (e.g., in the unit interval). In addition, the variables may have a point mass. We reduce the conditions on the underlying density to a minimum by proposing a nonparametric approach. By using a gamma, a beta, or a local linear kernel (also called boundary kernels), in a product kernel, the suggested estimator becomes simple in implementation and robust to the well known boundary bias problem. We investigate the mean integrated squared error properties, including the rate of convergence, uniform strong consistency and asymptotic normality. We establish consistency of the least squares cross-validation method to select optimal bandwidth parameters. A detailed simulation study investigates the performance of the estimators. Applications using lottery and corporate finance data are provided.

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Section: Introductionmentioning

confidence: 99%

Nonparametric density estimation for multivariate bounded data

Bouezmarni

Rombouts

2010

Journal of Statistical Planning and Inference

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“…Boundary problems are particularly severe in higher dimensions: if the support of the function to be estimated is a unit cube in d and the mean-squared optimal bandwidth b $ cn À1a4d is used for the situation of a twice continuously dierentiable function to be estimated with a non-negative kernel, then the volume of the boundary region is 2 d b ob $ 2 d n À1a4d , which is increasing in d. Thus the boundary region takes up an increasingly larger fraction of the support of the function to be estimated, as the dimension d increases. A correction for boundary eects arising for multivariate kernel estimates has been discussed previously by Staniswalis et al (1993), section 2.2, and Staniswalis and Messer (1996). This approach extends a method proposed by Rice (1984) for univariate boundary corrections to the multivariate case by linearly combining estimates with several dierent bandwidths.…”

Section: Introductionmentioning

confidence: 87%

Multivariate Boundary Kernels and a Continuous Least Squares Principle

Müller

Stadtmüller

1999

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss±Markov theorem.

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“…The former approach was explored, for example, by Cleveland and Devlin (1988), using locally weighted regression and by Nussbaum (1986), Georgiev (1987), Muller (1988, and Staniswalis, Messer, and Finston (1990), using kernel estimates. More restrictive additive models were considered by Hastie andTibshirani (1986, 1990) and others.…”

Section: Uj)es(n)mentioning

confidence: 99%

Preaveraged Localized Orthogonal Polynomial Estimators for Surface Smoothing and Partial Differentiation

Azari

Müller

1992

Journal of the American Statistical Association

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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 parameter (,has a negative effect on the average mean squared error, which is not unexpected. On the other hand, larger values of (, are computationally more economical. The effectsof 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.

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Kernel estimators for multivariate regression

Cited by 18 publications

References 17 publications

Nonparametric density estimation for multivariate bounded data

Nonparametric density estimation for multivariate bounded data

Multivariate Boundary Kernels and a Continuous Least Squares Principle

Preaveraged Localized Orthogonal Polynomial Estimators for Surface Smoothing and Partial Differentiation

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