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Nonparametric recursive estimation of the derivative of the regression function with application to sea shores water quality. (English) Zbl 1419.62498

The valvometry concept is generally used to determine water quality evolution on a specified area. The available techniques in this domain produce high-frequency data that enable the study of the behavior of bivalve molluscs, considering it is an indicator on water quality. The paper is devoted to the development of a nonparametric statistical procedure to evaluate the velocity of the valve open-closing activity. The relationship between the distance of two vales (\(Y_n\)) and the time of the measurement (\(X_n\)) is given by a nonparametric regression model of the form: \(Y_n = f(X_n) + \epsilon_n\) , where \(\epsilon_n\) are unknown random errors. The aim is to estimate the derivative of the unknown regression function \(f\) which is associated with the velocity of the valve opening-closing activities of the molluscs. The authors consider an estimator based on three recursive versions of the Nadaraya-Watson estimator and investigate the asymptotic behavior of the derivative of these three estimators. Under some special assumptions the almost sure convergence of the defined estimates and the pointwise asymptotic normality of defined estimates are proved. To evaluate the performance of the introduced estimates numerical examples as well as real data applications are shown.

MSC:

62P12 Applications of statistics to environmental and related topics
62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference

Software:

R; KernSmooth
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References:

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