Abstract
In this paper, we consider a nonparametric regression model with replicated observations based on the φ-mixing and the ρ-mixing error’s structures respectively, for exhibiting dependence among the units. The wavelet procedures are developed to estimate the regression function. Under suitable conditions, we obtain expansions for the bias and the variance of wavelet estimator, prove the moment consistency, the strong consistency, the strong convergence rate of it, and establish the asymptotic normality of wavelet estimator.
Highlights
Consider the following nonparametric regression model:Y (x) = g(x) + e(x), ( . )from a discrete set of observations of the process Y (·) at the points {xi, ≤ i ≤ n}, {e(·)} is a zero mean stochastic process, defined on a probability space (, A, P), and g(x) is an unknown function defined on a closed interval I = [, ].It is well known that regression model has a wide range of applications in filtering and prediction in communications and control systems, pattern recognition, classification and econometrics, and is an important tool of data analysis
We develop wavelet methods to estimate a regression function in the model ( . ) with the φ-mixing and ρ-mixing error’s structures respectively, that is, {e(j)(·), j ≥ } is a φ-mixing or ρ-mixing process
For the nonparametric regression model without repeated observations under weakly dependent processes, Lin and Zhang [ ] respectively adopted bootstrap wavelet and blockwise bootstrap wavelet to generate an independent blockwise sample from the original dependent data, defined the wavelet estimators of g(·), and took advantage of the independence of the blockwise sample to prove some asymptotic properties of the wavelet estimators
Summary
For a systematic discussion of wavelets and their applications in statistics, see the recent monographs by Härdle et al [ ] and Vidakovic [ ] Due to their ability to adapt to local features of unknown curves, many authors have applied wavelet procedures to estimate the general nonparametric model. For the nonparametric regression model without repeated observations under weakly dependent processes, Lin and Zhang [ ] respectively adopted bootstrap wavelet and blockwise bootstrap wavelet to generate an independent blockwise sample from the original dependent data, defined the wavelet estimators of g(·), and took advantage of the independence of the blockwise sample to prove some asymptotic properties of the wavelet estimators. We consider the nonparametric regression model with repeated observations under the specific ρ-mixing and φ-mixing dependent processes.
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