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Nonparametric smoothing methods for a class of non-standard curve estimation problems

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Abstract

This chapter discusses the nonparametric smoothing methods for a class of nonstandard curve estimation problems. A class of models is introduced where an unknown function is defined as solution of an integral equation. Intercept and kernel of the integral equation are unknown but they can be directly estimated by application of classical smoothing methods. Estimates of unknown function are given as solutions of the empirical integral equation. The key properties are the nature of the operator or family of operators that we define the integral equations. There are some results on the asymptotic properties of the estimated functions, which include point-wise normal distribution and uniform stochastic expansions. Simulations suggest that smooth backfitting works stable under weaker assumptions on the design and for quite larger number of additive components. The development of an asymptotic distribution theory for the estimate of unknown function is elaborated. A series of examples are given that motivate the class of models.

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... In nonparametric regression models, however, it often yields an additive model where classical smoothing methods can not be applied, as we illustrate on several cases in this section. Some of the models of this section were also discussed in the overview papers [31] and [44]. A general discussion of smooth least squares in a general class of nonparametric models can also be found in [39]. ...
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... Additive regression is an example of a nonparametric model where the nonparametric function is given as a solution of an integral equation. This has been outlined in Linton and Mammen [24] and Carrasco, Florens and Renault [6] where also other examples of statistical integral equations are given. Examples are additive models where the additive components are linked as in Linton and Mammen [25] and regression models with dependent errors where an optimal transformation leads to an additive model, see Linton and Mammen [26]. ...
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