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For the nonparametric estimation of regression functions with a one-dimensional design parameter, a new kernel estimate is defined and shown to be superior to the one introduced by Priestley and Chao (1972). The results are not restricted to positive kernels, but extend to classes of kernels satisfying certain moment conditions. An asymptotically valid solution for the boundary problem, arising for...
Recently Tukey has proposed several non-linear smoothers for time series, which have some properties that make them preferable in some ways to linear filters. We discuss these properties, and give some detailed results for one of these smoothers.
A class of robust smoother and interpolator algorithms is introduced. The motivation for these smoothers and interpolators is a theorem concerning approximate conditional-mean smoothers for vector Markov processes in additive non-Gaussian noise. This theorem is the smoothing analog of Masreliez’s approximate non-Gaussian filter theorem (IEEE-Auto. Control, AC-20, 1975). The theorem presented here...
We consider kernel estimates for the derivatives of a probability density which satisfies certain smoothness conditions. We derive the rate of convergence of the local and of the integrated mean square error (MSE and IMSE), by restricting us to kernels with compact support. Optimal kernel functions for estimating the first three derivatives are given. Adopting a technique developed by Farrel (1972)...
This paper describes the density-quantile function approach to statistical analysis of a sample as involving five phases requiring the study of various population raw and smoothed quantile and density-quantile functions. The phases can be succinctly described in terms of the notation for the functions studied: (1) Q, fQ, q, (ii) $$\tilde Q,\tilde q$$ , (iii) $$\tilde fQ$$ , (iv) $$\hat fQ,\hat...
A number of estimates of the probability density function (and regression function) have been introduced in the past few decades. The oldest are the kernel estimates and more recently nearest neighbor estimates have attracted attention. Most investigations have dealt with the local behavior of the estimates. There has, however, been some research and some heuristic comment on the utility of global...
In curve estimation, running M-estimates are a natural generalization of Kernel-type smoothers (moving averages). We find the rate of convergence that can be expected from these estimates and the leading bias and variance terms. We also explain the effect of twicing for Kernel-type smoothers and give some rationale for its use in robust curve estimation.
This paper aims to present some results on the asymptotic behaviour of a matrix associated with certain types of spline functions and shows how these results can be used to obtain a fast algorithm for choosing the smoothing parameter in the smoothing of noisy data by splines. First, we give a general theorem on the behaviour of the eigenvalues associated with a spline function. The spline functions...
We study the use of "thin plate" smoothing splines for smoothing noisy d dimensional data. The model is $$z_i = u(t_i ) + \varepsilon _i ,i = 1,2,...,n,$$ where u is a real valued function on a closed, bounded subset Ω of Euclidean d-space and the εi are random variables satisfying Eεi=0, Eεiεj=σ2, i=j, =0, i≠j, tiεΩ. The zi are observed. It is desired to estimate u, given zl, ..., zn...
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