#### Abstract

Estimation of the regression function has many applications in agriculture and industry. Usually, the regression function is assumed a known functional form which depends on unknown parameters. Nonparametric regression theory makes no such assumption and often uses some kernel functions to form the so-called Watson Nadaraya type estimators. Such estimators were extensively studied by Watson (1964), Nadaraya (1964, 1989) and Collomb (1981, 1985). When the data are independent, these estimators have nice asymptotic convergence properties. When the data are dependent, Gyorfi et al (1989) gave some large sample properties for the Watson-Nadaraya estimators. In this paper, the recently developed theory of wavelet will be used to estimate the regression function when the data are dependent. Large sample properties for the wavelet estimator will be proved, and the wavelet smoothing will be compared with the other well known nonparametric smoothing methods.

#### Keywords

Wavelet, Multiresolution Analysis, Mixing Conditions, Complete Convergence in Probability

#### Creative Commons License

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WAVELET NONPARAMETRIC REGRESSION WITH DEPENDENT DATA

Estimation of the regression function has many applications in agriculture and industry. Usually, the regression function is assumed a known functional form which depends on unknown parameters. Nonparametric regression theory makes no such assumption and often uses some kernel functions to form the so-called Watson Nadaraya type estimators. Such estimators were extensively studied by Watson (1964), Nadaraya (1964, 1989) and Collomb (1981, 1985). When the data are independent, these estimators have nice asymptotic convergence properties. When the data are dependent, Gyorfi et al (1989) gave some large sample properties for the Watson-Nadaraya estimators. In this paper, the recently developed theory of wavelet will be used to estimate the regression function when the data are dependent. Large sample properties for the wavelet estimator will be proved, and the wavelet smoothing will be compared with the other well known nonparametric smoothing methods.