Asymptotic Integrated Mean Square Error Using Least Squares and Bias Minimizing Splines

Agarwal, Gaurav; Studden, W. J.

doi:10.1214/aos/1176345203

Cited by 124 publications

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“…There are a large amount of papers on regression splines. Among these, Agarwal and Studden [20] and Huang and Studden [21] considered the rates of convergence and the connection between splines and kernels in the univariate case. More recently, Zhou, Shen and Wolfe [22] studied the asymptotic distribution of the regression spline.…”

Section: Spline Approximation 21 Asymptotic Properties Of Sir Matrixmentioning

confidence: 99%

On spline approximation of sliced inverse regression

Zhu

2007

SCI CHINA SER A

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The dimension reduction is helpful and often necessary in exploring the nonparametric regression structure. In this area, Sliced inverse regression (SIR) is a promising tool to estimate the central dimension reduction (CDR) space. To estimate the kernel matrix of the SIR, we herein suggest the spline approximation using the least squares regression. The heteroscedasticity can be incorporated well by introducing an appropriate weight function. The root-n asymptotic normality can be achieved for a wide range choice of knots. This is essentially analogous to the kernel estimation. Moreover, we also propose a modified Bayes information criterion (BIC) based on the eigenvalues of the SIR matrix. This modified BIC can be applied to any form of the SIR and other related methods. The methodology and some of the practical issues are illustrated through the horse mussel data. Empirical studies evidence the performance of our proposed spline approximation by comparison of the existing estimators.

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Section: Spline Approximation 21 Asymptotic Properties Of Sir Matrixmentioning

confidence: 99%

On spline approximation of sliced inverse regression

Zhu

2007

SCI CHINA SER A

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“…Agarwal and Studden (1980) derived asymptotic bias and variance for regression splines which are linear in the observations of the response variable, including in particular the least squares estimator and a bias minimizing estimator. Least squares spline in nonparametric regression was also considered in .…”

Section: Introductionmentioning

confidence: 99%

Bivariate Tensor-Product B-Splines in a Partly Linear Model

Shi

1996

Journal of Multivariate Analysis

136

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In some applications, the mean or median response is linearly related to some variables but the relation to additional variables are not easily parameterized. Partly linear models arise naturally in such circumstances. Suppose that a random samplewhere Y i is a real-valued response, X i # R p and T i ranges over a unit square, and g 0 is an unknown function with a certain degree of smoothness. We make use of bivariate tensor-product B-splines as an approximation of the function g 0 and consider M-type regression splines by minimization of;& g n (T i )) for some convex function \. Mean, median and quantile regressions are included in this class. We show under appropriate conditions that the parameter estimate of ; achieves its information bound asymptotically and the function estimate of g 0 attains the optimal rate of convergence in mean squared error. Our asymptotic results generalize directly to higher dimensions (for the variable T) provided that the function g 0 is sufficiently smooth. Such smoothness conditions have often been assumed in the literature, but they impose practical limitations for the application of multivariate tensor product splines in function estimation. We also discuss the implementation of B-spline approximations based on commonly used knot selection criteria together with a simulation study of both mean and median regressions of partly linear models.

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“…As shown in the proof of Lemma 6.10 of Agarwal and Studden (1980), it hold that max 1≤k≤K r ς k = o(h m r +1 r ). Note that B rk (t) ≤ 1 for any t ∈ [T 1 , T 2 ] and 1 ≤ k ≤ K r .…”

Section: Proofsmentioning

confidence: 85%

Componentwise B-spline estimation for varying coefficient models with longitudinal data

Tang

Cheng

2011

Stat Papers

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A componentwise B-spline method is proposed for estimating the unknown functions in the varying-coefficient models with longitudinal data. Different amounts of smoothing are used for different individual coefficient functions and the estimators of different coefficient functions are obtained by different minimization operations. The local asymptotic bias and variance of the estimators are derived. It is shown that our estimators achieve the local and global optimal convergence rates even if the coefficient functions belong to different smoothness families. The asymptotic distributions of the estimators are also established and are used to construct approximate pointwise confidence intervals for coefficient functions. Finite sample properties of our procedures are studied through Monte Carlo simulations.

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Asymptotic Integrated Mean Square Error Using Least Squares and Bias Minimizing Splines

Cited by 124 publications

References 0 publications

On spline approximation of sliced inverse regression

On spline approximation of sliced inverse regression

Bivariate Tensor-Product B-Splines in a Partly Linear Model

Componentwise B-spline estimation for varying coefficient models with longitudinal data

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