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A Computationally Eecient Oracle Estimator for Additive Nonparametric Regression with Bootstrap Conndence Intervals
| Content Provider | Semantic Scholar |
|---|---|
| Author | Linton, Oliver B. This, Niklaus W. Hengartner Abstract |
| Copyright Year | 1997 |
| Abstract | This paper makes three contributions. First, we introduce a computationally eecient esti-mator for the component functions in additive nonparametric regression exploiting a diierent motivation from the marginal integration estimator of Linton and Nielsen (1995). Our method provides a reduction in computation of order n; which is highly signiicant in practice. Second, we deene an eecient estimator of the additive components, by inserting the preliminary esti-mator into a backktting algorithm but taking one step only, and establish that it is equivalent in various sense to the oracle estimator based on knowing the other components. Our two-step estimator is minimax superior to that considered in Opsomer and Ruppert (1997), due to its better bias. Third, we deene a bootstrap algorithm for computing pointwise conndence intervals and show that it achieves the correct coverage. It is common practice to study the association between multivariate covariates and responses via regression analysis. While nonparametric models for the conditional mean m(x) = E(Y jX = x) are useful exploratory and diagnostic tools when X is one-dimensional, they suuer from the curse of dimensionality HHrdle (1990) and Wand and Jones (1995)] in that their best possible convergence rate is n ?q=(2q+d) ; where d is the dimension of X and m() is q-times continuously diierentiable. Additive regression models of the form 0 Woocheol Kim is doctoral candidate in the (1) with x = (x 1 ; : : :; x d) T 2 IR d ; ooer a compromise between the exibility of a full nonparametric regression and reasonable asymptotic behaviour. In particular, in such additive regression models, the functions m j () can be estimated with the one-dimensional rate of convergence ? e.g., n q=(2q+1) for q-times continuously diierentiable functions ? regardless of d; see Stone (1985,1986). The back-tting algorithm of Breiman and Friedman (1985), Buja, Hastie and Tibshirani (1989), and Hastie and Tibshirani (1990) is widely used to estimate the one dimensional components m j () and regression function m(). independently introduced the alternative method of marginal integration to estimate m ` (x `) see also earlier work by Auestad and Tjjstheim (1991)]. One advantage of the integration method is that its statistical properties are easier to describe; speciically, one can easily prove central limit theorems and give explicit expressions for the asymptotic bias and variance of the estimators. Hengartner (1997) provides the weakest set of conditions, showing that the curse of dimensionality does not … |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.econ.yale.edu/~linton/klh2.ps |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
