Adaptive and minimax estimation of the cumulative distribution function given a functional covariate. (English) Zbl 1302.62082
Summary: We consider the nonparametric kernel estimation of the conditional cumulative distribution function given a functional covariate. Given the bias-variance trade-off of the risk, we first propose a totally data-driven bandwidth selection mechanism in the spirit of the recent Goldenshluger-Lepski method and of model selection tools. The resulting estimator is shown to be adaptive and minimax optimal: we establish nonasymptotic risk bounds and compute rates of convergence under various assumptions on the decay of the small ball probability of the functional variable. We also prove lower bounds. Both pointwise and integrated criteria are considered. Finally, the choice of the norm or semi-norm involved in the definition of the estimator is also discussed, as well as the projection of the data on finite dimensional subspaces. Numerical results illustrate the method.
MSC:
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62C20 | Minimax procedures in statistical decision theory |
Keywords:
adaptive kernel estimator; conditional cumulative distribution function; minimax estimation; functional random variable; small ball probabilitySoftware:
fda (R)References:
| [1] | Ait-Saïdi, A., Ferraty, F., Kassa, R. and Vieu, P. (2008). Cross-validated estimations in the single-functional index model., Statistics 42 475-494. · Zbl 1274.62519 · doi:10.1080/02331880801980377 |
| [2] | Akakpo, N. and Lacour, C. (2011). Inhomogeneous and anisotropic conditional density estimation from dependent data., Electron. J. Stat. 5 1618-1653. · Zbl 1271.62060 · doi:10.1214/11-EJS653 |
| [3] | Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities., Proc. Amer. Math. Soc. 6 170-176. · Zbl 0066.37402 · doi:10.2307/2032333 |
| [4] | Ash, R. B. and Gardner, M. F. (1975)., Topics in Stochastic Processes . Academic Press [Harcourt Brace Jovanovich Publishers], New York. Probability and Mathematical Statistics, Vol. 27. · Zbl 0317.60014 |
| [5] | Aspirot, L., Bertin, K. and Perera, G. (2009). Asymptotic normality of the Nadaraya-Watson estimator for nonstationary functional data and applications to telecommunications., J. Nonparametr. Stat. 21 535-551. · Zbl 1165.62029 · doi:10.1080/10485250902878655 |
| [6] | Azzedine, N., Laksaci, A. and Ould-Saïd, E. (2008). On robust nonparametric regression estimation for a functional regressor., Statist. Probab. Lett. 78 3216-3221. · Zbl 1489.62121 |
| [7] | Benhenni, K., Ferraty, F., Rachdi, M. and Vieu, P. (2007). Local smoothing regression with functional data., Comput. Statist. 22 353-369. · Zbl 1194.62042 · doi:10.1007/s00180-007-0045-0 |
| [8] | Bertin, K., Lacour, C. and Rivoirard, V. (2014). Adaptive estimation of conditional density function. Submitted, hal-00922555. · Zbl 1342.62090 |
| [9] | Birgé, L. and Massart, P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence., Bernoulli 4 329-375. · Zbl 0954.62033 · doi:10.2307/3318720 |
| [10] | Brunel, E., Comte, F. and Lacour, C. (2010). Minimax estimation of the conditional cumulative distribution function., Sankhya A 72 293-330. · Zbl 1213.62063 · doi:10.1007/s13171-010-0018-1 |
| [11] | Brunel, E., Mas, A. and Roche, A. (2013). Non-asymptotic Adaptive Prediction in Functional Linear Models. Submitted, hal-00763924. · Zbl 1328.62408 |
| [12] | Burba, F., Ferraty, F. and Vieu, P. (2009). \(k\)-nearest neighbour method in functional nonparametric regression., J. Nonparametr. Stat. 21 453-469. · Zbl 1161.62017 · doi:10.1080/10485250802668909 |
| [13] | Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression., Ann. Statist. 34 2159-2179. · Zbl 1106.62036 · doi:10.1214/009053606000000830 |
| [14] | Cardot, H., Ferraty, F. and Sarda, P. (1999). Functional linear model., Statist. Probab. Lett. 45 11-22. · Zbl 0962.62081 · doi:10.1016/S0167-7152(99)00036-X |
| [15] | Chagny, G. (2013a). Penalization versus Goldenshluger-Lepski strategies in warped bases regression., ESAIM Probab. Statist. 17 328-358 (electronic). · Zbl 06282476 · doi:10.1051/ps/2011165 |
| [16] | Chagny, G. (2013b). Estimation adaptative avec des données transformées ou incomplètes. Application à des modèles de survie PhD thesis, Univ. Paris, Descartes. |
| [17] | Chen, D., Hall, P. and Müller, H. G. (2011). Single and multiple index functional regression models with nonparametric link., Ann. Statist. 39 1720-1747. · Zbl 1220.62040 · doi:10.1214/11-AOS882 |
| [18] | Comte, F. and Genon-Catalot, V. (2012). Convolution power kernels for density estimation., J. Statist. Plann. Inference 142 1698-1715. · Zbl 1238.62041 · doi:10.1016/j.jspi.2012.02.038 |
| [19] | Comte, F. and Johannes, J. (2012). Adaptive functional linear regression., Ann. Statist. 40 2765-2797. · Zbl 1373.62350 · doi:10.1214/12-AOS1050 |
| [20] | Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Penalized contrast estimator for adaptive density deconvolution., Canad. J. Statist. 34 431-452. · Zbl 1104.62033 · doi:10.1002/cjs.5550340305 |
| [21] | Crambes, C., Delsol, L. and Laksaci, A. (2008). Robust nonparametric estimation for functional data., J. Nonparametr. Stat. 20 573-598. · Zbl 1147.62045 · doi:10.1080/10485250802331524 |
| [22] | Crambes, C., Kneip, A. and Sarda, P. (2009). Smoothing splines estimators for functional linear regression., Ann. Statist. 37 35-72. · Zbl 1169.62027 · doi:10.1214/07-AOS563 |
| [23] | Dabo-Niang, S., Kaid, Z. and Laksaci, A. (2012). On spatial conditional mode estimation for a functional regressor., Statist. Probab. Lett. 82 1413-1421. · Zbl 1246.62193 · doi:10.1016/j.spl.2012.03.029 |
| [24] | Dabo-Niang, S. and Rhomari, N. (2009). Kernel regression estimation in a Banach space., J. Statist. Plann. Inference 139 1421-1434. · Zbl 1153.62028 · doi:10.1016/j.jspi.2008.06.015 |
| [25] | Dabo-Niang, S. and Yao, A.-F. (2013). Kernel spatial density estimation in infinite dimension space., Metrika 76 19-52. · Zbl 1256.62016 · doi:10.1007/s00184-011-0374-4 |
| [26] | Delaigle, A. and Hall, P. (2010). Defining probability density for a distribution of random functions., Ann. Statist. 38 1171-1193. · Zbl 1183.62061 · doi:10.1214/09-AOS741 |
| [27] | Demongeot, J., Laksaci, A., Madani, F. and Rachdi, M. (2010). Estimation locale linéaire de la densité conditionnelle pour des données fonctionnelles., C. R. Math. Acad. Sci. Paris 348 931-934. · Zbl 1201.62048 · doi:10.1016/j.crma.2010.06.013 |
| [28] | Dunker, T., Lifshits, M. A. and Linde, W. (1998). Small deviation probabilities of sums of independent random variables. In, High Dimensional Probability (Oberwolfach, 1996) . Progr. Probab. 43 59-74. Birkhäuser, Basel. · Zbl 0902.60039 · doi:10.1007/978-3-0348-8829-5_4 |
| [29] | Ferraty, F., Laksaci, A. and Vieu, P. (2006). Estimating some characteristics of the conditional distribution in nonparametric functional models., Statistical Inference for Stochastic Processes. 9 47-76. · Zbl 1117.62030 · doi:10.1007/s11203-004-3561-3 |
| [30] | Ferraty, F., Mas, A. and Vieu, P. (2007). Nonparametric regression on functional data: inference and practical aspects., Aust. N. Z. J. Stat. 49 267-286. · Zbl 1136.62031 · doi:10.1111/j.1467-842X.2007.00480.x |
| [31] | Ferraty, F. and Romain, Y. (2011)., The Oxford Handbook of Functional Data Analysis . Oxford Handbooks in Mathematics . OUP Oxford. · Zbl 1284.62001 |
| [32] | Ferraty, F. and Vieu, P. (2002). The functional nonparametric model and application to spectrometric data., Comput. Statist. 17 545-564. · Zbl 1037.62032 · doi:10.1007/s001800200126 |
| [33] | Ferraty, F. and Vieu, P. (2006)., Nonparametric Functional Data Analysis . Springer Series in Statistics . Springer, New York. Theory and practice. · Zbl 1271.62085 |
| [34] | Ferraty, F., Laksaci, A., Tadj, A. and Vieu, P. (2010). Rate of uniform consistency for nonparametric estimates with functional variables., J. Stat. Plan. Inference 140 335-352. · Zbl 1177.62044 · doi:10.1016/j.jspi.2009.07.019 |
| [35] | Geenens, G. (2011). Curse of dimensionality and related issues in nonparametric functional regression., Stat. Surv. 5 30-43. · Zbl 1274.62283 · doi:10.1214/09-SS049 |
| [36] | Gheriballah, A., Laksaci, A. and Sekkal, S. (2013). Nonparametric \(M\)-regression for functional ergodic data., Statist. Probab. Lett. 83 902-908. · Zbl 1489.62127 |
| [37] | Gijbels, I., Omelka, M. and Veraverbeke, N. (2012). Multivariate and functional covariates and conditional copulas., Electron. J. Stat. 6 1273-1306. · Zbl 1295.62031 · doi:10.1214/12-EJS712 |
| [38] | Glaser, N. S., Barnett, P., McCaslin, I., Nelson, D., Trainor, J., Louie, J., Kaufman, F., Quayle, K., Roback, M., Malley, R. et al. (2001). Risk factors for cerebral edema in children with diabetic ketoacidosis., New Engl. J. Med. 344 264-269. |
| [39] | Glaser, N. S., Wootton-Gorges, S. L., Buonocore, M. H., Tancredi, D. J., Marcin, J. P., Caltagirone, R., Lee, Y., Murphy, C. and Kuppermann, N. (2013). Subclinical cerebral edema in children with diabetic ketoacidosis randomized to 2 different rehydration protocols., Pediatrics 131 e73-e80. |
| [40] | Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality., Ann. Statist. 39 1608-1632. · Zbl 1234.62035 · doi:10.1214/11-AOS883 |
| [41] | Goldenshluger, A. and Lepski, O. (2013). On adaptive minimax density estimation on Rd., Probab. Theory Related Fields · Zbl 1342.62053 · doi:10.1007/s00440-013-0512-1 |
| [42] | Hoffmann-Jørgensen, J., Shepp, L. A. and Dudley, R. M. (1979). On the lower tail of Gaussian seminorms., Ann. Probab. 7 319-342. · Zbl 0424.60041 · doi:10.1214/aop/1176995091 |
| [43] | Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising., Probab. Theory Related Fields 121 137-170. · Zbl 1010.62029 · doi:10.1007/s004400100148 |
| [44] | Klein, T. and Rio, E. (2005). Concentration around the mean for maxima of empirical processes., Ann. Probab. 33 1060-1077. · Zbl 1066.60023 · doi:10.1214/009117905000000044 |
| [45] | Lacour, C. (2006). Rates of convergence for nonparametric deconvolution., C. R. Math. Acad. Sci. Paris 342 877-882. · Zbl 1095.62056 · doi:10.1016/j.crma.2006.04.006 |
| [46] | Lacour, C. (2008). Adaptive estimation of the transition density of a particular hidden Markov chain., J. Multivariate Anal. 99 787-814. · Zbl 1286.62071 · doi:10.1016/j.jmva.2007.04.006 |
| [47] | Laib, N. and Louani, D. (2010). Nonparametric kernel regression estimation for functional stationary ergodic data: asymptotic properties., J. Multivariate Anal. 101 2266-2281. · Zbl 1198.62027 · doi:10.1016/j.jmva.2010.05.010 |
| [48] | Li, W. V. and Shao, Q. M. (2001). Gaussian processes: inequalities, small ball probabilities and applications. In, Stochastic Processes: Theory and Methods . Handbook of Statist. 19 533-597. North-Holland, Amsterdam. · Zbl 0987.60053 · doi:10.1016/S0169-7161(01)19019-X |
| [49] | Lifshits, M. A. (1997). On the lower tail probabilities of some random series., Ann. Probab. 25 424-442. · Zbl 0873.60012 · doi:10.1214/aop/1024404294 |
| [50] | Mas, A. (2012). Lower bound in regression for functional data by representation of small ball probabilities., Electron. J. Stat. 6 1745-1778. · Zbl 1295.62043 · doi:10.1214/12-EJS726 |
| [51] | Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality., Stochastic Process. Appl. 115 155-177. · Zbl 1101.62031 · doi:10.1016/j.spa.2004.07.006 |
| [52] | Plancade, S. (2013). Adaptive estimation of the conditional cumulative distribution function from current status data., J. Statist. Plann. Inference 143 1466-1485. · Zbl 1279.62086 · doi:10.1016/j.jspi.2013.04.005 |
| [53] | Rachdi, M. and Vieu, P. (2007). Nonparametric regression for functional data: automatic smoothing parameter selection., J. Statist. Plann. Inference 137 2784-2801. · Zbl 1331.62240 · doi:10.1016/j.jspi.2006.10.001 |
| [54] | Ramsay, J. O. and Silverman, B. W. (2005)., Functional Data Analysis , second ed. Springer Series in Statistics . Springer, New York. · Zbl 1079.62006 |
| [55] | Shang, H. L. (2013). Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density., Comput. Statist. Data Anal. 67 185-198. · Zbl 1471.62182 |
| [56] | Tsybakov, A. B. (2009)., Introduction to Nonparametric Estimation . Springer Series in Statistics . Springer, New York. Revised and extended from the 2004 French original, Translated by V. Zaiats. · Zbl 1176.62032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
