Abstract
We consider the problem of nonparametric estimation of the conditional hazard function for spatial data. More precisely, given a strictly stationary random field \({Z_{\rm{i}}} = {({X_{\rm{i}}},\;{Y_{\rm{i}}})_{{\rm{i}} \in {{\mathbb{N}}^N}}}\), we investigate a kernel estimate of the conditional hazard function of univariate response variable Y i given the functional variable X i. The principal aims of this article are to give the mean squared convergence rate and to prove the asymptotic normality of the proposed estimator. Finally, a simulation study and an application on real data are carried out to illustrate, for finite samples, the behavior of our method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benhenni, K., F. Ferraty, M. Rachdi, and P. Vieu. 2007. Local smoothing regression with functional data. Comput. Stat., 22, 353–369.
Biau, G., and B. Cadre. 2004. Nonparametric spatial prediction. Stat. Inference Stochastic Proc., 7, 327–349.
Carbon, M., M. Hallin, and L. T. Tran. 1996. Kernel density estimation for rom fields. Stat. Prob. Lett., 36, 115–125.
Carbon, M., L. T. Tran, and B. Wu. 1997. Kernel density estimation for rom fields: The L 1 theory. JNonparametr. Stat., 6, 157–170.
Carbon, M., C. Francq, and L. T. Tran. 2007. Kernel regression estimation for rom fields. J. Stat. Plan. Inference, 137, 778–798.
Cressie, N. A. C. 1991. Statistics for spatial data. Wiley Series in Probability Mathematical Statistics. New York, NY: Wiley.
Cressie, N., and C. H. Wikle. 2011. Statistics for spatio-temporal data. Wiley Series in Probability and Statistics. New York, NY: Wiley.
Diggle, P., and P. J. Ribeiro. 2007. Model-based geostatistics. New York, NY: Springer.
Dabo-Niang, S., Z. Kaid, and A. Laksaci. 2012. On spatial conditional mode estimation for a functional regressor. Stat. Prob. Lett., 82, 1413–1421.
Dabo-Niang, S., M. Rachdi, and A.-F. Yao. 2011. Spatial kernel regression estimation prediction for functional rom fields. Far East J. Stat. 37, 77–113.
De Gooijer, J., and A. Gannoun. 2000. Nonparametric conditional predictive regions for time series. Comput. Stat. Data Anal., 33, 259–257.
Diggle, P., and P. J. Ribeiro. 2007. Model-based geostatistics. New York, NY: Springer.
Doukhan, P. 1994. Mixing: Properties examples. Lecture Notes in Statistics 85. New York, NY: Springer-Verlag.
Estévez-Pérez, G., H. Lorenzo-Cimadevila, and A. Quintela-del-Rìo. 2002. Nonparametric analysis of the time structure of seismicity in a geographic region. Ann. of Geophys., 45, 497–511.
Ferraty, F., A. Laksaci, A. Tadj, and P. Vieu. 2010. Rate of uniform consistency for nonparametric estimates with functional variables. J. Stat. Plan. Inference, 140, 335–352.
Ferraty, F., A. Mas, and P. Vieu. 2007. Advances in nonparametric regression for functional variables. Austr. NZ J. Stat., 49, 1–20.
Ferraty, F., A. Rabhi, and P. Vieu. 2008. Estimation non-parametrique de la fonction de hasard avec variable explicative fonctionnelle. Rev. Roumaine Math. Pures Appl., 53, 1–18.
Ferraty, F., and P. Vieu. 2006. Nonparametric functional data analysis. Springer Series in Statistics. New York, NY: Springer.
Guyon, X. 1987. Estimation d’un champ par pseudo-vraisemblance conditionnelle: Etude asymptotique et application au cas Markovien. Proc. Sixth Franco-Belgian Meeting of Statisticians, 15–62. Brussels: FUSL.
Hall, P., R. C. L. Wolef, and Q. Yao. 1999. Methods for estimating a conditional distribution function. Am. Stat. Assoc., 94, 154–163.
Hyndman, R. J., D. M. Bashtannyk, and G. K. Grunwald. 1996. Estimating and visualizing conditional densities. J. Comput. Graph. Stat., 5, 315–336.
Laksaci, A. 2007. Convergence en moyenne quadratique de l’estimateur à noyau de la densité conditionnelle avec variable explicative fonctionnelle. Pub. Inst. Stat. Univ. Paris, 3, 69–80.
Laksaci, A., F. Madani, and R. Mustapha. 2013. Kernel conditional density estimation when the regressor is valued in a semi-metric space. Commun. Stat. Theory Methods, 42, 3544–3570.
Laksaci, A., and B. Mechab. 2010. Estimation non-paramétrique de la fonction de hasard avec variable explicative fonctionnelle: cas des données spatiales. Rev. Roumaine Math. Pures Appl., 55, 35–51.
Lu, Z., and X. Chen. 2004. Spatial kernel regression estimation: Weak Concistency. Stat. Prob. Lett., 68, 125–136.
Li, J., and L. T. Tran. 2007 Hazard rate estimation on rom fields. J. Multivariate Anal., 98, 1337–1355.
Li, J., and L. T. Tran. 2009. Nonparametric estimation of conditional expectation. J. Stat. Plan. Inference, 139, 164–175.
Nakhapetyan, B. S. 1987. An approach to the proof of limit theorems for dependent rom variables. Theory Prob. Appl., 32, 535–539.
Quintela-del-Rìo, A. 2008. Hazard function given a functional variable: Non-parametric estimation under strong mixing conditions. J. Nonparametr. Stat., 20, 413–430.
Ramsay, J. 2008. FDA problems that I like to talk about. https://doi.org/www.psych.mcgill.ca/misc/fda
Ripley, B. 1981. Spatial statistics. New York, NY: Wiley.
Roussas, G. G. 1989. Hazard rate estimation under dependence conditions. J. Stat. Plan. Inference, 22, 81–93.
Tran, L. T. 1990. Kernel density estimation on rom fields. J. Multivariate Anal., 34, 37–53.
Tran, L. T., and S. Yakowitz. 1993. Nearest neighbor estimators for rom fields. J. Multivariate Anal., 44, 23–46.
Watson, G. S., and M. R. Leadbetter. 1964. Hazard analysis. I. Biometrika, 51, 175–184.
Author information
Authors and Affiliations
Corresponding author
Additional information
Color versions of one or more of the figures in the article can be found online at https://doi.org/www.tandfonline.com/ujsp.
Rights and permissions
About this article
Cite this article
Laksaci, A., Mechab, B. Conditional Hazard Estimate for Functional Random Fields. J Stat Theory Pract 8, 192–220 (2014). https://doi.org/10.1080/15598608.2014.847766
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/15598608.2014.847766