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Nonparametric Statistical Methods, 3rd Edition by Eric Chicken, Douglas A. Wolfe, Myles Hollander

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Chapter 13

Wavelets

Introduction

In Chapters 9 and 11, methods were presented that estimated specific types of function: linear functions in Chapter 9 and survival curves in Chapter 11. Each of these methods made use of assumptions that specified the nature of the shape of the function to be estimated. This shape restriction is not discussed in this chapter. Here, the unknown function is not assumed to have any particular parametric shape or representation but rather the function belongs to a class of functions possessing more general characteristics, such as a certain level of smoothness. Using the observed data, one may estimate such a function by representing the function in another domain. One common way to approach this is to use an orthogonal series representation of the function. This shifts the estimation problem from directly trying to estimate the unknown function c13-math-0001, to estimating a set of scalar coefficients that represent c13-math-0002 in the orthogonal series domain. An efficient method for estimating such functions involves the use of wavelets. Wavelets are strong tool in such methods because they concentrate most of the information about the function in a much reduced set of data and have the ability to estimate both global and local features in the underlying function.

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