Nonparametric recursive estimation of the derivative of the regression function with application to sea shores water quality

This paper is devoted to the nonparametric estimation of the derivative of the regression function in a nonparametric regression model. We implement a very efficient and easy to handle statistical procedure based on the derivative of the recursive Nadaraya–Watson estimator. We establish the almost sure convergence as well as the asymptotic normality for our estimates. We also illustrate our nonparametric estimation procedure on simulated data and real life data associated with sea shores water quality and valvometry.

[1]  B. Silverman,et al.  Density estimation in action , 1986 .

[2]  J. M. Vilar-Fernández,et al.  Recursive local polynomial regression under dependence conditions , 2000 .

[3]  T. Honjo,et al.  Detecting the shellfish killer Heterocapsa circularisquama (Dinophyceae) by measuring bivalve valve activity with a Hall element sensor , 2006 .

[4]  D. Mason,et al.  General Asymptotic Confidence Bands Based on Kernel-type Function Estimators , 2004 .

[5]  Gordon J Johnston,et al.  Probabilities of maximal deviations for nonparametric regression function estimates , 1982 .

[6]  Denis Efimov,et al.  Velocity estimation of valve movement in oysters for water quality surveillance , 2015 .

[7]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[8]  Jianqing Fan,et al.  Local polynomial estimation of regression functions for mixing processes , 1994, Proceedings of 1994 Workshop on Information Theory and Statistics.

[9]  J. M. Vilar-Fernández,et al.  Recursive Estimation of Regression Functions by Local Polynomial Fitting , 1998 .

[10]  J. R. García-March,et al.  Shell gaping behaviour of Pinna nobilis L., 1758: circadian and circalunar rhythms revealed by in situ monitoring , 2008 .

[11]  I. Ahmad,et al.  NONPARAMETRIC SEQUENTIAL ESTIMATION OF A MULTIPLE REGRESSION FUNCTION , 1976 .

[12]  L. Györfi,et al.  A Distribution-Free Theory of Nonparametric Regression (Springer Series in Statistics) , 2002 .

[13]  Mohamedou Sow,et al.  Water quality assessment by means of HFNI valvometry and high-frequency data modeling , 2011, Environmental monitoring and assessment.

[14]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[15]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[16]  David Blondin.  Rates of strong uniform consistency for local least squares kernel regression estimators , 2007 .

[17]  Alexandre B. Tsybakov,et al.  Introduction to Nonparametric Estimation , 2008, Springer series in statistics.

[18]  B. Bercu,et al.  A Robbins-Monro procedure for estimation in semiparametric regression models , 2011, 1101.0736.

[19]  B. Bercu,et al.  A nonparametric statistical procedure for the detection of marine pollution , 2019 .

[20]  Peter Hall,et al.  Data sharpening methods for bias reduction in nonparametric regression , 2000 .

[21]  M. H. Ngerng,et al.  Recursive Nonparametric Estimation of Local First Derivative Under Dependence Conditions , 2011 .

[22]  A uniform functional law of the logarithm for the local empirical process , 2004, math/0410143.

[23]  Theo Gasser,et al.  Finite-Sample Variance of Local Polynomials: Analysis and Solutions , 1996 .

[24]  L. Devroye,et al.  On the L1 convergence of kernel estimators of regression functions with applications in discrimination , 1980 .

[25]  Adam Krzyzak,et al.  A Distribution-Free Theory of Nonparametric Regression , 2002, Springer series in statistics.

[26]  A. Amiri,et al.  Recursive regression estimators with application to nonparametric prediction , 2012 .

[27]  Luc Devroye,et al.  Combinatorial methods in density estimation , 2001, Springer series in statistics.

[28]  Eugene F. Schuster,et al.  Joint Asymptotic Distribution of the Estimated Regression Function at a Finite Number of Distinct Points , 1972 .

[29]  H. U. Riisgård,et al.  VALVE-GAPE RESPONSE TIMES IN MUSSELS (MYTILUS EDULIS)—EFFECTS OF LABORATORY PRECEDING-FEEDING CONDITIONS AND IN SITU TIDALLY INDUCED VARIATION IN PHYTOPLANKTON BIOMASS , 2006 .

[30]  G. Durrieu,et al.  Influence of the parasite worm Polydora sp. on the behaviour of the oyster Crassostrea gigas: a study of the respiratory impact and associated oxidative stress , 2007 .

[31]  Xiaohong Chen,et al.  Recursive Nonparametric Estimation for Time Series , 2014, IEEE Transactions on Information Theory.

[32]  Kazuo Noda,et al.  Estimation of a regression function by the parzen kernel-type density estimators , 1976 .

[33]  Raphaël Coudret,et al.  Comparison of Kernel Density Estimators with Assumption on Number of Modes , 2012, Commun. Stat. Simul. Comput..

[34]  G. S. Watson,et al.  Smooth regression analysis , 1964 .

[35]  Damien Tran,et al.  Estimation of potential and limits of bivalve closure response to detect contaminants: Application to cadmium , 2003, Environmental toxicology and chemistry.

[36]  E. Nadaraya.  On Estimating Regression , 1964 .

[37]  I. Grama,et al.  Dynamic extreme values modeling and monitoring by means of sea shores water quality biomarkers and valvometry , 2016, Environmental Monitoring and Assessment.

[38]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[39]  小形 正男.  A. S. Davydov 著, E. S. Kryachko 訳: Solitons in Molecular Systems, D. Reidel, Dordrecht and Boston, 1985, xviii+320ページ, 23×16cm, 17,000円 (Mathematics and Its Applications, Soviet Series). , 1987 .

[40]  I. Grama,et al.  Nonparametric adaptive estimation of conditional probabilities of rare events and extreme quantiles , 2015 .

[41]  Donald S. Cherry,et al.  Valve closure responses of the Asiatic clam Corbicula fluminea exposed to cadmium and zinc , 1987, Hydrobiologia.