Non-parametric Multivariate Kernel Regression Estimation to Describe Cognitive Processes and Mental Representations

  • Sahar Slama Université de Sousse, Laboratoire de Mathématiques Modélisation Déterministe et Aléatoire (LAMMDA), Hammam Sousse, Tunisie
  • Yousri Slaoui Université de Poitiers, Laboratoire des Mathématiques et Applications (LMA), Futuroscope Chasseneuil, France
  • Gwendoline Le Du UMR-S INSERM 1237- Physiopathology & Imaging of Neurological Disorders (PhIND), Université de Caen Normandie, France
  • Cyril Perret Université de Poitiers, Centre de Recherches sur la Cognition et l'Apprentissage (CeRCa), Poitou-Charentes, France
Keywords: regression function estimation, stochastic approximation algorithm, classification and cluster analysis, plug-in method, handwritten and cognitive psychology, propensity score matching.


In this research paper, we set forward a non-parametric multivariate recursive kernel regression estimator under missing data using the propensity score approach in order to describe writing word production. Our main objective is to explore cognitive processes and mental representations mobilized when a human being prepares to write a word according to the idea developed in Perret and Olive (2019). We investigate the asymptotic properties of the proposed recursive estimator and compare them to the well known Nadaraya-Watson's regression estimator. We calculate the bias and the variance of the proposed estimator which depend on the choice of some parameters such as the stepsize and the bandwidth. We examine some data-driven procedures to select these parameters. Thus, we demonstrate that, under some optimal choices of these parameters, the MSE (Mean Squared Error) of the proposed estimator can be smaller than the one obtained by using Nadaraya Watson's regression estimator. The elaborated estimator is then applied to the behavioral data to classify some participants in groups. This classication may stand for a departure point to tackle written behavior variations.


Almanjahie, I. M, Aissiri, K. A., Laksaci, A. and Zouaoui Chikr Elmezouar (2020). The k nearest neighbors smoothing of the relative-error regression with functional regressor Comm. Statist. Theory Methods. doi: 10.1080/03610926.2020.1811870

Altman, N. and Leger, C. (1995). Bandwidth Selection for Kernel Distribution Function Estimation. J. Statist. Plann. Inference. 46, 195-214.

Bonin, P., Chalard, M., Meot, A. and Fayol, M. (2002). The determinants of spoken and written picture naming latencies. British Journal of Psychology, 93, 89-114.

Cousineau, D. and Chartier, S. (2010). Outliers detection and treatment: a review. Journal of Psychological Research 3, 58-67. International Journal of Psychological Research 3, 58-67.

Boukabour, S. and Masmoudi, A. (2020). Semiparametric Bayesian networks for continuous data. Comm. Statist. Theory Methods doi:10.1080/03610926.2020.1738486.

Delaigle, A and Gijbels, I. (2004), Practical bandwidth selection in deconvolution kernel density estimation, Comput. Statist. Data Anal. 45, 249-267.

Galambos, J. and Seneta E. (1973). Regularly Varying Sequences. Proc. Amer. Math. Soc. 41, 110-116. Hardle, W. and Marron, J.S.(1985).Optimal Bandwidth Selection in Nonparametric Regression Function Estimation. The Ann. Statist. 13, 1465-1481.

Jmaei, A. Slaoui, Y. and Dellagi, W. (2017). Recursive distribution estimators defined by stochastic approximation method using Bernstein polynomials. J. Nonparametr. Stat. 29, 792-805.

Kara, L.Z., Laksaci, A., Rachdi, M. and Vieu, P. (2017) Data-driven kNN estimation in nonparametric functional data analysis, J. Multivariate Anal. 153, 176-188.

Kiefer, J. and Wolfowitz, J. (1952). Stochastic Estimation of the Maximum of a Regression Function. Ann. Math. Statist. 23, 462-466.

Luce, R.D. (1986). Response times: their role in inferring elementary mental organization. Handbook of Mathematical PsychologyNew York: Oxford.

McCormack, P.F. and Wright, N.M. (1964). The positive skew observed in reaction time distributions. Canadian Journal of Psychology 18, 43-51.

McGill, W.J. (1963). Stochastic latency mechanisms. Handbook of Mathematical Psychology 1, 309-360. New York: John Wiley and Sons, Inc.

Mokkadem, A. Pelletier, M. and Slaoui, Y. (2009a). The stochastic approximation method for the estimation of a multivariate probability density. J. Statist. Plann. Inference. 139, 2459-2478.

Mokkadem, A. Pelletier, M. and Slaoui, Y. (2009b). Revisiting Revesz's Stochastic Approximation Method for the Estimation of a Regression Function. ALEA Lat. Amer.J. Probab.Math. Statist. 6, 63-114.

Mokkadem, A. and Pelletier, M. (2016). The Multivariate Revesz's Online Estimator of a Regression Function and Its Averaging.. M. Math. Meth. Stat. 25, 151-167.

Nadaraya, E.A. (1964). On Estimating Regression. Theory Probab. Appl. 10, 186-190.

Perret, C. and Bonin, P. (2019). Which variables should be controlled for to investigate picture naming in adults A Bayesian meta-analysis. Behavior Research Methods 51, 2533-2545.

Perret, C. and Laganaro, M. (2013). Why are written naming latencies (not) longer than spoken naming ?Reading and Writing An Interdisciplinary Journal 26, 225-239.

Perret, C. and Olive, T. (2019). Spelling and Writing Words: Theoretical and Methodological Advances. Brills Edition.

Perret, C., Bonin, P., and Laganaro, M. (2014). Exploring the multiple-level hypothesis of AoA effects in spoken and written picture naming using a topographic ERP analysis. Brain and Language 135, 20-31.

Révész, P. (1977). How to Apply the Method of Stochastic Approximation in the Non parametric Estimation of a Regression Function. Math. Operationsforsch. Statist. Ser. Statistics. 8, 119-126.

Schmiedek, F., Oberauer, K., Wilhelm, O., Süβ , H.-M. and Wittmann, W.W. (2007). Individual differences in components of reaction time distributions and their relations to working memory and intelligence. Journal of Experimental Psychology: General, 136, 414-429.

Slaoui, Y. (2014a). Bandwidth selection for recursive kernel density estimators defined by stochastic approximation method. Journal of Probability and Statistics, ID 739640, doi:10.1155/2014/739640.

Slaoui, Y. (2014b). The stochastic approximation method for the estimation of a distribution function. Math. Methods Statist. 23, 306-325.

Slaoui, Y. (2016). Optimal bandwidth selection for semi-recursive kernel regression estimators. Statistics and Its Interface. 9, 375-388.

Slaoui, Y. (2017). Recursive kernel density estimators under missing data. Methods. 18, 9101-9125. Comm. Statist.Theory Methods. 18, 9101-9125.

Slaoui, Y. (2018). Bias reduction in kernel density estimation. J. Nonparametr. Stat. 30, 505-522.

Slaoui, Y. (2019). Wild Bootstrap Bandwidth Selection of Recursive Nonparametric Relative Regression for Independent Functional Data, J. Multivariate Anal., 173, 494-511.

Slaoui, Y. (2020). Recursive non-parametric regression estimation for independent functional data, Statist. Sinica, 30, 417-437.

Tsybakov, A.B. (1990). Recurrent Estimation of the Mode of a Multidimensional Distribution. Probl. Inf. Transm. 8, 119-126.

Watson, G.S. (1964). Smooth regression analysis. Sankhya Ser. A. 26, 359-372.

Unsworth, N., Redick, T.S., Lakey, C.E., and Young, D.L. (2010). Lapses in sustained attention and their relation to executive control and uid abilities: An individual dierences investigation. Intelligence, 38, 111-122.

How to Cite
Slama, S., Slaoui, Y., Le Du, G., & Perret, C. (2022). Non-parametric Multivariate Kernel Regression Estimation to Describe Cognitive Processes and Mental Representations. Statistics, Optimization & Information Computing, 10(4), 1021-1043.
Research Articles