Hardle W. Applied nonparametric regression. Cambridge University Press, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....t=1 o#[BR(t)B ] whose minimization with respect to B yields an estimate of A 1 . In practice, however, the covariance matrices R(t) of X(t) are unknown. Therefore a sensible approach is to replace them by some non parametric estimator. We consider a kernel estimator for R(t) see for ex. [16], p. 25) i.e. R(t) X(#)X(#) This property can also be derived from the Hadamard inequality which states that det R det diagR with equality if and only if R is diagonal, see for ex. 13] where k( is a positive kernel function and M is a parameter controlling the window width. ....

Wolfgang Hardle., Applied nonparametric regression, Cambridge University press, 1990.

....of the observed dynamics. 2 Methods The two modeling approaches used here are only described briefly. For a deeper discussion of these methods, for proofs of convergence and numerical details, the reader is referred to the original works and reviews for the nonparametric regression method [7, 8, 9] and for the multiple shooting approach [10, 11, 12, 13] 2.1 Nonlinear regression and optimal transformations The task of searching for an appropriate functional form of the model can be performed best in a nonparametric way due to the high flexibility of nonparametrically given functions. Here ....

W. Hardle. Applied Nonparametric Regression. Cambridge University Press, Cambridge, 1990.

....is the corresponding hat matrix. In order to obtain a consistent estimator, in which the bias and the variance asymptotically vanish, the choice of the kernel and the scaling behavior respective the amount of data N is restricted. These conditions and some mathematical details are discussed in [10]. If the kernel has no finite support, Eq. 4) is an O(N ) computation time problem, which one usually wants to avoid. Therefore, kernel functions with finite support are commonly taken into account. Beside computation, the main disadvantage of kernel smoothers is the treatment of the boundary ....

Hardle W. Applied Nonparametric Regression. Cambridge University Press, Cambrige, 1990.

....initial resampling phase we fit a regression model to each of the found clusters. Although this general two stage schema can be applied to any regression method (and even to other supervised learning tasks) in this paper we concentrate our description on partial linear regression models (e.g. [12, 17]) Still, we also report some results using other regression techniques. Partial linear models belong to the class of semiparametric approaches that integrate parametric with non parametric techniques. In the case of partial linear models, a standard least squares linear polynomial (e.g. 8] is ....

....3 presents clustered partial linear models. In Section 4 we describe a series of experiments with these models. A further analysis of clustered partial linear regression is given in Section 5. Finally, we present the main conclusions of this work. 2 Partial Linear Models Partial linear regression [12, 17] is a semiparametric technique that integrates a linear polynomial with a kernel smoothing component. A prediction for a query case using these models is obtained by summing the value predicted by the linear model with the value resulting from smoothing over the residuals (errors) of the linear ....

Hardle, W. : Applied Nonparametric Regression. Econometric Society Monographs. Cambridge University Press, 1990.

....f and the set is the simplex i 0; i = 1; M and 1 i 1. Now let us present two specific application of functional aggregation which deal with dimensionality reduction in nonparametric regression estimation. The majority of the known estimates of multivariate regression functions, see [9, 16] and references therein, are aimed to restore smooth signals (f belongs to a Sobolev ball with known or unknown smoothness parameters) It is well known that in this case the rates of convergence degrade rather fast when the dimensionality d of f increases and become exceedingly slow when d ....

W. Hardle, W., Applied Nonparametric Regression, ES Monograph Series 19, Cambridge, U.K., Cambridge University Press, 1990.

....points t c[a, b] and f results from a super position of f at point ti with white noise i of variance r 2. fi f(ti) i i 0, 2 jmx 1 (1) There exists a variety of different approaches to solve this regression problem, such as kernel estimators with global bandwidth or spline estimators [4]. A major disadvantage of these standard smoothing techniques is the fact, that they do not resolve local structures well enough. This, however, is necessary when dealing with signals that contain structures of different scales and amplitudes such as neuro physiological signals (see Figure 1) In ....

Hardle W., Applied nonparametric regression, Econometric Society Mono- graphs, Cambridge University Press, 1990

.... one approach learning theory tools such as Vapnik Chervonenkis dimension, random covering numbers, and complexity regularization [3] were used to obtain universal convergence and rates of RBF nets [8, 9, 11] In another approach [16] a similarity between NRBF nets and kernel regression estimation [6, 13, 14] was pointed out and tools used in nonparametric regression estimation were applied to studying convergence and rates of NRBF nets. In [16] a bound on the empirical L 2 risk of NRBF net was expressed in terms of the kernel regression estimate and its performance on the training sequence was ....

.... i where K : IR IR is the normalized kernel ( K = 1) and h n is a scalar bandwidth ( 13] and references therein) Similar network called probabilistic neural network was proposed by Specht [15] Networks (2) with the same constrains as above are related to the kernel regression estimate [6, 14] R n (x) i=1 Y i K( i=1 K( which is the weighted average of Y i with nonlinear weights w i = adjusted with the sample size. This average is an estimate of the conditional mean of the output Y given input or equivalently regression function E(Y jX = x) This apparent connection ....

W. Hardle, Applied Nonparametric Regression. Cambridge University Press, Cambridge, 1990.

....3.04 1.09 100 0.694 0.313 1000 0.0923 0.0594 Table 1. Average estimation error of posterior output prediction for different sample sizes. The results for sample size n are based on 10 4 =n trials. estimate of w(a) can be obtained using some data smoothing algorithm such as kernel smoothing [3]. The resulting method henceforth is referred to as Smoothed Importance sampling. The usage of smoothing introduces bias (which can be made arbitrary small by decreasing the smoothing kernel width to zero as n 1) but in return can lead to a significant reduction of the variance, hence, a ....

W. Hardle. Applied nonparametric regression. Cambridge University Press, 1990.

....the value of a) It can be shown that if w(a) was known exactly, than (13) would have a lower variance than (11) A proof is given in [Zlochin and Baram, 2000] In practice, a good estimate of w(a) can be obtained using some data smoothing algorithm such as kernel smoothing, running means etc. [Hardle, 1990]. The resulting method is called, for obvious reasons, smoothed importance sampling. While introducing a small bias, this method can lead to a significant reduction of the variance, hence, a smaller estimation error than (11) as demonstrated by the following numerical experiments. 114.1.4 Two ....

....a sequence of three Metropolis updates with Gaussian proposal distributions centered in the current state and having covariances 0:05 2 I; 0:15 2 I; 0:5 2 I. In the Smoothed Importance sampling the estimate of the function w(a) was obtained using a kernel smoother with Epanechnikov kernel [Hardle, 1990] of width 1, i.e. w(a) was estimated to be a weighted average of the weights produced by Annealed Importance sampling: w(a) P a 0 w(a 0 )k(a; a 0 ) P a 0 w(a 0 ) 14) where k(a; a 0 ) 1 Gamma (a Gamma a 0 ) 2 , for ja Gamma a 0 j 1, and zero otherwise. 12 1.5 1 ....

[Article contains additional citation context not shown here]

Hardle, W. (1990). Applied nonparametric regression. Cambridge University Press.

....moving average being termed a kernel. By adopting the rather artificial device that observations are assumed to arrive more frequently (in a given time interval) as the sample size increases, it can be shown that a suitably designed kernel will estimate the trend consistently ; see, for example, Hardle (1990). The method assumes that the 16 0 10 20 30 40 50 60 70 80 90 100 5 10 (i) i) 10 8 6 4 2 0 2 4 6 8 10 .1 .2 .3 (ii) 10 8 6 4 2 0 2 4 6 8 10 .1 .2 .3 (iii) Figure 5: Data with break. i) simulated data, Gaussian signal (dotted) and t signal (solid) ii) weighting pattern for t signal at t ....

....8 Illustration Here we consider 133 observations of acceleration against time (measured in milliseconds) for a simulated motorcycle accident. This data set was originally analysed by Silverman (1985) and is often used as an example of nonparametric curve fitting techniques; see, for example, Hardle (1990) and Green and Silverman (1994) The observations are not equally spaced and at certain time points there are multiple observations; see figure 6. Nonparametric cubic spline and kernel smoothing techniques depend on some choice of a smoothness parameter. This is usually determined by cross ....

[Article contains additional citation context not shown here]

Hardle, W. (1990) Applied nonparametric regression. Cambridge: Cambridge University Press.

....as possible. This results in w t = # (t 1) # 1 t 1 n ## 1 2 # t # 1 t n ## 1 2 and the corresponding worst case is a single step change. Comparing the mean of time delayed moving windows with length m, which is a standard approach to detect systematic di#erences [68], has lower worst case discriminatory power since it corresponds to using a weighted sum with weights w 1 = 1 m, 1 m, 0, 0, 1 m, 1 m. The hypothesis of a constant mean should be rejected in favor of a monotone increasing (decreasing) mean if # n t=1 w t Y t is ....

Hardle, W. (1990), Applied nonparametric regression, Cambridge university press, Cambridge.

....problems of data analysis. The parametric models used in diverse areas have been too restrictive for many applications. The usual non parametric regression curves such as smoothing splines (Silverman, 1985; Wahba, 1990; Eubank, 1988) kernel estimators (Hardle and Gasser, 1984; Muller, 1988; Hardle, 1990; Chu and Marron, 1991) or locally weighted regression lowess (Cleveland, 1979) have the nice property to fit a vast class of smooth functions well. But they still may show many little wiggles which do not appear to be necessary for a good description of the data. A statistician using the ....

Hardle, W. (1990). Applied Nonparametric Regression, Cambridge U. Press, Cambridge, UK.

....kernel consists of the following fi les: kfunctio.src, kernel.src, In this documentation we discuss installation of the library and the functions that are included. This manual assumes that the reader is familiar with kernel estimation. The methods implemented in this library are based on Hardle (1990) and Silverman (1986) The library has three main procedures: ukernel, mkernel and nw. The procedure ukernel can be used to estimate the density of univariate data, the procedure mkernel estimates the density function of multivariate data. Finally, the procedure nw can be used to do ....

Hardle, W. (1990). Applied Nonparametric Regression. Cambridge: Cambridge University Press.

....spiked and the regression estimator becomes a step function. From the above mentioned observations, it is clear that a reliable selection procedure for the bandwidth parameter is of vital importance, and much work has been done in this area for the smooth kernel case (see Jones et al. 1996, or Hardle, 1990). The method usually involves minimizing, with respect to the bandwidth parameter, an estimate (or some variant) of Z [g(x) Gamma g(x) 2 f(x) dx; or in the density case, Z [ f(x) Gamma f(x) 2 dx; and the bandwidth chosen in this manner can have optimal properties. However, in the ....

Hardle, W. (1990), Applied Nonparametric Regression, Econometric Society Monographs 19, Cambridge University Press.

....A and the offset b were estimated with a standard SVD after the kernels had been pre allocated. Different functional forms were considered for the kernels: Gauss, Hardy multiquadrics, inverse of Hardy multiquadrics and Epanechnikov kernels, also in the Nadaraya Watson ( softmax ) normalized form [7, 21]. Backfitting with gradient descent was also tested in combination with the SVD for all the cited parameters [18] The normalized elliptical (i.e. with diagonal covariance matrix) Gaussian kernels were the most effective and the affine term Lx b was always included in the model. To illustrate ....

....to a model evaluated on the test set in terms of E and W . The expected monotonic behavior is observable: for each speaker, W decreases indeed with E, with a sigmoidal trend differently translated in the (E; W ) space. The solid line is an estimate of based on the Nadaraya Watson kernel smoother [21], computed for the available data irrespective of the speaker. The dashed lines indicate approximate 95 pointwise confidence intervals, built according to Theorem 4:2:1 in [21] on convergence in distribution of the Nadaraya Watson kernel estimator. A sigmoidal trend is in fact observable for the ....

[Article contains additional citation context not shown here]

W. Hardle. Applied nonparametric regression, volume 19 of Econom. Soc. Monographs. Cambridge Un. Press, New York, 1990.

....under H 1 , H 2 , and H 3 . The results based on the higher order kernel are similar but weaker. As one would expect, the rejection probabilities with the higher order kernel are always lower. This is consistent with the fact that higher order kernels tend to produce more variable estimates (e.g. Hardle 1990). Thus we focus on the normal kernel for the rest of the experiments. 21 20 We tried a cross validation method to determine constant c in h = cN 1 2k 1 . For the postwar monthly dataset, c = 1:04. 21 In terms of bias reduction, the higher order kernel is not eective in this nite sample ....

Hardle, W., 1990, Applied Nonparametric Regression, Cambridge University Press.

....h irrespective of the choice of k. If, for instance, k # (x) ck(cx) c 0, then k # is also a kernel, and the combination k # , ch, will lead to the same estimates as k, h. There has however been found little variation in performance due to the shape of k, provided that k is even and unimodal. Hardle (1990), page 138, ranks a number kernels in terms of their e#ciency, and found the Epanechnikov kernel k(u) 3 4 (1 u 2 )I( u # 1) cf. Epanechnikov (1969) to be the best polynomial kernel of degree 2 on [ 1, 1] in terms of the mean squared error. This kernel is not everywhere ....

....random denominator makes statistical treatment often rather cumbersome. 1.3.3 Other Nonparametric Estimates As noted earlier, there are many other ways to estimate densities or regression functions in a nonparametric fashion. Indeed, many standard texts [e.g. Prakasa Rao (1983) Silverman (1985) Hardle (1990)] cover a wide variety of such estimates as do survey articles, such as Buja, Hastie, and Tibshirani (1989) It would be well beyond the scope of this thesis to discuss every such estimate in detail, and we limit ourselves to give a short description of some of them. We moreover only examine the ....

[Article contains additional citation context not shown here]

Hardle, W. (1990), Applied Nonparametric Regression (Cambridge University Press, Cambridge MA).

....is common practice to study the association between multivariate covariates and responses via regression analysis. Although nonparametric models for the conditional mean m(x) E(YjX=x)are useful exploratory and diagnostic tools when X is onedimensional, they suffer from the curse of dimensionality (Hardle 1990; Wand and Jones 0q= 2q d) 1995) in that their best possible convergence rate is n ; where d is the dimension of X and m(1) is q times continuously differentiable. Additive regression models of the form m(x) c m(x) m(x) 111 m (x ) 1:1) 11 22 d d d T with x = x ; x ) 2IR ; offer a ....

....with the bootstrap method performs quite well, provided a good resampling bandwidth is chosen. We now evaluate a practical method for selecting the bootstrap bandwidth based on an elaboration of Silverman s rule of thumb idea. More sophisticated bandwidth selection rules, such as those reviewed by Hardle (1990) and Jones, Marron, and Sheather (1996) can also be applied here, but our purpose here is just to investigate one very simple method. The approach is to specify a parametric model for b the conditional distribution L(Y jX; for the purpose of selecting g: One estimates i i n o n b e by ....

Hardle, W. (1990), Applied Nonparametric Regression. Econometric Monograph Series 19, Cambridge: Cam- bridge University Press.

....density estimation also solves a problem involving the smoothed bootstrap, where it is sometimes desired to resample from a continuous distribution rather than the relatively rough empirical distribution. This problem is related to the wild bootstrap approach to resampling; see, for example, Hardle (1990, p. 247) and Mammen (1992, pp. 16 17) In such cases one would usually not wish the smoothing step to alter the main properties of the bootstrap; for example, properties of coverage or level accuracy of confidence or testing procedures. These properties are directly related to third and fourth ....

Hardle, W. (1990), Applied Nonparametric Regression, Cambridge: Cambridge University Press.

....real valued. Given data (X i ; Y i ) N i=1 and the model Y i = f(X i ) ffl i ; 1.1) where ffl is the noisy contribution, one wishes to estimate the unknown smooth function f . It is observed that well known regression methods such as kernel smoothing, nearest neighbor, spline smoothing (see Hardle, 1990 for details) may perform very badly in high dimensions because of the so called curse of dimensionality. The curse comes from the fact that when dealing with a finite amount of data, the high dimensional unit cube [0; 1] d is mostly empty, as discussed in the excellent paper of Friedman and ....

Hardle, W. (1990). Applied nonparametric regression. Cambridge, England: Cambridge University Press.

....Y is real valued. Given data (X i , Y i ) N i=1 and the model Y i = f(X i ) # i , 1.1) where # is the noisy contribution, one wishes to estimate the unknown smooth function f . It is observed that well known regression methods such as kernel smoothing, nearest neighbor, spline smoothing (see Hardle, 1990 for details) may perform very badly in high dimensions because of the so called curse of dimensionality. The curse comes from the fact that when dealing with a finite amount of data, the high dimensional unit cube [0, 1] d is mostly empty, as discussed in the excellent paper of Friedman and ....

Hardle, W. (1990). Applied nonparametric regression. Cambridge, England: Cambridge University Press.

.... distributed random variables with variance 1 and mean 0; g; oe are unknown smooth functions and ft i;n j i = 1; ng is a fixed design in the interval [0; 1] Much effort has been devoted to the problem of estimating the regression and variance function [see e.g. the recent monographs of Hardle (1990), Wand and Jones (1995) Fan and Gijbels (1996) and many of the developed estimation methods are meanwhile standard methods in applied regression analysis. 1 Although it is well known (and intuitively clear) that the asymptotic properties of the various nonparametric estimators depend ....

.... squared error of a nonparametric regression estimator For the introduction of the optimality criterion we need an asymptotic representation of the integrated mean squared error of a nonparametric estimate of the regression function g; which is nowadays standard in mathematical statistics [see Hardle (1990), Wand and Jones (1995) or Fan and Gijbels (1996) We assume that the regression function is k times continuously differentiable, i.e. g 2 C k [0; 1] and the variance function is Lipschitz continuous of order fl; i.e. oe 2 Lip fl [0; 1] for some fl 2 (0; 1] Following Gasser and Muller ....

Hardle, W. (1990). Applied nonparametric regression. Cambridge University Press, Boston.

....for estimating both probability densities and objects relating to such densities such as conditional expectations. Such approaches are explicitly density based in nature since they directly estimate the underlying probability structure. The interested reader is referred to Silverman (1986) Hardle (1990), Scott (1992) and Pagan Ullah (1999) for an overview of such approaches. Unlike parametric models, however, this flexibility typically comes at the cost of a slower rate of convergence which depends on the dimensionality of X . The benefit of employing nonparametric approaches lies in the ....

Hardle, W. (1990), Applied Nonparametric Regression, Cambridge, New Rochelle.

....and partial derivatives (response coefficients) of the form: E(X 1t jX 2t ; X pt ) X it ; i = 2; p (8) 9 See, for example, Bell (1972) Smith (1980) or Leith (1990) 10 Kernel estimators of the form used here have been discussed elsewhere in the literature. See for example, Hardle (1990), Silverman (1986) and Scott (1992) Some repetition is required here in order to explain our construction of asymptotic confidence intervals for the conditional mean and its derivatives at each data point. It should be noted that the important statistical properties of nonparametric estimators ....

Hardle, W. (1990), Applied Nonparametric Regression, Cambridge, New Rochelle.

....b(r; n) of the b(r; n) should be built in such a way that in the filtered frame Z(r; n) Z(r; n) Gamma b(r; n) the signal would be preserved, while the clutter would be removed almost completely. Kernel methods provide a powerful tool for such analysis due to both theoretic optimality (see [19, 23]) and computational transparency. In addition these methods are invariant to statistical properties and variations of clutter. Kernel estimators are weighted moving averages of observations b(r; n) 1 Q m i=1 N i X r1 ; r m Z(r; n)K i l 1 Gammar 1 N 1 ; l N Gammar 2 Nm ....

W. Hardle, Applied Nonparametric Regression. Cambridge: Cambridge University Press, 1990.

....different position, since in addition to the variance there is also a bias term present. Thus, we have to measure the accuracy of estimation in terms of the mean squared error, which incorporates both the bias and the variance. In the field of nonparametric regression (see [6] 16] 38] [13], 26] for monographs on this topic and the bibliography cited therein) one has to distinguish between two different statements of the problem: First, the so called fixed design case, when the choice of the input sequence is at the disposal of the experimenter and, second, the random design case, ....

Hardle W. Applied Nonparametric Regression. Cambridge University Press, Boston, 1990.

....window widths. There is no generally accepted method for choosing the window widths. Methods currently available include subjective choice and automatic methods such as the plug in , cross validation (CV) and penalizing function approaches (See Marron [3] for an excellent survey) Hardle [1] (page 173) compared various automatic methods and found that The best overall performance, though, showed GCV (generalized cross validation) One problem with the CV approach is that the CV function has to be repeatedly calculated over a range of window widths, which in turn requires repeated ....

....I consider two simulated examples to try to address these issues. Example 1. The first conditional mean function considered is y i = M (x i ) ffl i 4 = 2:0 sin(x i ) ffl i (6) where x i U ( Gamma10; 10) and ffl i N(0; 1) Example 2. The second function considered is one found in Hardle [1]. The model is y i = M (x i ) ffl i = 1:0 Gamma x i e Gamma200:0(x i Gamma0:5) 2 ffl i (7) where ffl i N(0; 0:25) and X i U(0; 1) To investigate the nature of this approach to window width selection, I consider varying the sample size n and the subsample size n s . The grid size ....

W. Hardle. Applied Nonparametric Regression. Cambridge, New Rochelle, 1990.

....(x i ) Clearly both estimators are kernel weighted sums, and the properties of the estimator follow from the properties of the weights W ( Delta) j . Given standard conditions regarding the kernel, bandwidth, and data generating process, these estimators are consistent, and one is referred to Hardle (1990, pg 29) and Fan (1992) for details. Table 1 presents the BIAS CORRECTED KERNEL REGRESSION 5 approximate bias and variance of each estimator for second order kernels. Note that both estimators are biased in finite samples. Both estimators suffers from curvature effects whereby bias increases ....

Hardle, W. (1990), Applied Nonparametric Regression, Cambridge, New Rochelle.

....set of individuals based on their initial probability of choosing either alternative being most sensitive to changes in the independent variables. The alternative would be to adopt a semi parametric estimation technique that would not assume any functional form for the distribution (Hardle 1990). If data gathering technology increased at the same speed as computing technology increased, this would be feasible. However, it is unlikely that the typical political science data set is up to the task of allowing for precise estimation this way. Now, by adding a parameter to the definition of ....

Hardle, Wolfgang. 1990. Applied Nonparametric Regression. Cambridge University Press.

....the over fitting problems should worsen. Ideally, we would like to use as much data as possible to construct our estimate while at the same time get the most reliable empirical estimate of the cost function for our estimate. The most common form of crossvalidation, Leave one out cross validation,(Hardle, 1990) provides a solution to this problem in the context of optimal smoothing parameter selection. Consider the following two examples. Define the Nadaraya Watson kernel regression function (Nadaraya, 1964) as f (x) P n i=1 K (x Gamma x i )y i P n i=1 K (x Gamma x i ) where K (x) is a ....

.... Z f 2 (x)dx: Note that the first term on the righthand side is a constant and can be ignored. The other terms can be calculated exactly for kernel regression and linear regression. The second term on the righthand side is known as the leave one out estimate of the expected value of f (x) (Hardle, 1990), 2 d E x [ f (x) 1 n n X k=1 f [k] x k ) This term is of interest to us since there is a striking similarity between this estimate and the BEM estimator from Section 2.2. Notice that the leave one out estimate is not a function of x; it is only a constant depending on the data. ....

[Article contains additional citation context not shown here]

Hardle, W. (1990). Applied Nonparametric Regression, volume 19 of Econometric Society Monographs. Cambridge University Press, New York.

....a random sample of N observations of m f , say, m 1 , m 2 , mN , calculated from a given q. Using kernel method, the probability density of m f at a given point y, denoted by f(y) is estimated by (6) h m y g 1 Nh = y f i N 1 = i ) S where h ( 0) is the bandwidth (Hardle (1990)) Silverman (1986) suggested that the choice of the kernel function has minimal effects on the density estimates. In this paper, we choose g( to be the density function of the standard normal random variable so that (7) g(z) 2p) 1 2 exp( z 2 2) and the lower partial moment is ....

Hardle, W. (1990): Applied Nonparametric Regression. Cambridge University Press: Cambridge.

....Hodson s theoretical analysis is in the spirit of the current model. 2.The Box Cox specification is flexible relative to the standard (inherently linear) specification of the wage equation. It is however considerably less flexible than a fully nonparametric specification, see Ullah (1988) and Hardle (1990). 3.The theoretical analysis is applicable at both the industry and firm levels. However, the current data contains only industry level variables. 4.For a more detailed theoretical analysis see Mason (forthcoming) 5.This data was graciously provided by Lawrence F. Katz, Ph. D. Harvard and ....

Hardle, W. (1990). Applied Nonparametric Regression, New York: Cambridge University Press.

....variables. There are many linear smoothers proposed to estimate nonparametric regression functions: kernel, spline, local polynomial, orthogonal series methods, and others. For the available methods and results on both theory and applications, we refer to the books by Eubank (1988) Muller (1988) Hardle (1990), Wahba (1990) Hastie and Tibshirani (1990) Green and Silverman (1994) Wand and Jones (1995) Simonoff (1996) Fan and Gijbels (1996) Bosq (1998) among others. Among the aforementioned linear smoothers, local polynomial method has become popular in recent years due to its attractive ....

Hardle, W. (1990). Applied Nonparametric Regression. Cambridge University Press, New York.

....of Y i Gamma X T i fi (the variation in Y i not accounted for by the linear component X T i fi) on T i : In the literature one can find various methods for estimating g(ffl) nonparametrically, e.g. kernel, nearest neighbor, orthogonal series, piecewise polynomial and smoothing splines. See Hardle (1990) for an extensive discussion of their statistical properties. All these estimators may be written as weighted local averages of the observed values of the dependent variable with the weights depending on the values of the explanatory variables. In our case, we can write (still assuming that fi is ....

Hardle, W. (1990). Applied Nonparametric Regression. Cambridge University Press, New York.

No context found.

Hardle W. Applied nonparametric regression. Cambridge University Press, 1993.

No context found.

W. Hardle, Applied nonparametric regression. Cambridge, University Press, Cambridge, 1990.

No context found.

W. Hardle, Applied Nonparametric Regression, Cambridge Univ. Press, Cambridge, 1990.

No context found.

W. Hardle, Applied nonparametric regression. Cambridge, University Press, Cambridge, 1990.

No context found.

W. Hardle. Applied Nonparametric Regression. Econometric Society Monographs. Cambridge University Press, 1989.

No context found.

Hardle, W. (1990). Applied Nonparametric Regression, Econometric Society Monographs, Cambridge University Press.

No context found.

Hardle, W. (1990), Applied nonparametric regression, Cambridge: Cambridge University Press.

No context found.

Wolfgang Hardle. Applied Nonparametric Regression. Cambridge University Press, Cambridge, 1990.

No context found.

Hardle, W. (1990): "Applied Nonparametric Regression," Cambridge: Cambridge Uni- versity Press.

No context found.

Hardle, W.: 1990, Applied Nonparametric Regression, Vol. 19 of Econometric Society Monographs, Cambridge University Press.

No context found.

Hardle, W., 1990, Applied nonparametric regression, Cambridge University Press, Cambridge, 1990.

No context found.

Hardle, W. Applied Nonparametric Regression. Cambridge: Cambridge University Press, 1989.

No context found.

Hardle, W. (1990), Applied Nonparametric Regression. London: Oxford University Press,.

No context found.

Hardle W., 1990 Applied Nonparametric Regression, Econometric Society Monograph, Cambridge University Press.

No context found.

Hardle, W., Applied Nonparametric Regression, University of Cambridge Press, New York, NY, 1990.

No context found.

Hardle, W. (1993), Applied nonparametric regression, Econometric Society Monographs, Cambridge University Press.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC