Some asymptotic results of the estimators for conditional mode for functional data in the single index model missing data at random

Souad Mekkaoui; Nadia Kadiri; Abbes Rabhi

doi:https://doi.org/10.59139/ps.2023.02.2

Souad Mekkaoui Djillali LIABES University of Sidi Bel Abbes, Laboratory of Mathematics, Algeria ORCID:https://orcid.org/0009-0005-8611-3333 , Nadia Kadiri Djillali LIABES University of Sidi Bel Abbes, Higher School of Economics of Oran 22000, Sidi Bel Abbes, Algeria ORCID:https://orcid.org/0000-0002-3405-7414 , Abbes Rabhi Djillali LIABES University of Sidi Bel Abbes, Laboratory of Mathematics, Algeria ORCID:https://orcid.org/0000-0001-6740-0226 Przegląd Statystyczny. Statistical Review, vol. 70, 2023, 2, pages: 20-45 Published online: 30 November 2023 DOI https://doi.org/10.59139/ps.2023.02.2 Citation: Mekkaoui, S., Kadiri, N., & Rabhi, A. (2023). Some asymptotic results of the estimators for conditional mode for functional data in the single index model missing data at random. Przegląd Statystyczny. Statistical Review, 70(2), 20–45. https://doi.org/10.59139/ps.2023.02.2.

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ARTICLE

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ABSTRACT

In this work, we consider the problem of non-parametric estimation of a regression function, namely the conditional density and the conditional mode in a single functional index model (SFIM) with randomly missing data. The main result of this work is the establishment of the asymptotic properties of the estimator, such as almost complete convergence rates. Moreover, the asymptotic normality of the constructs is obtained under certain mild conditions. We finally discuss how to apply our result to construct confidence intervals.

KEYWORDS

functional data analysis, functional single-index process, kernel estimator, missing at random, non-parametric estimation, small ball probability

JEL

C10, C13, C14, C19, C24

REFERENCES

Ait-Saidi, A., Ferraty, F., Kassa, R., & Vieu, P. (2008). Cross-validated estimation in the singlefunctional index model. Statistics. A Journal of Theoretical and Applied Statistics, 42(6), 475–494. https://doi.org/10.1080/02331880801980377.

Attaoui, S.(2014). On the non-parametric conditional density and mode estimates in the single functional index model with strongly mixing data. Sankhya. The Indian Journal of Statistics, 76(2), 356–378. https://doi.org/10.1007/s13171-014-0051-6.

Attaoui, S., & Ling, N. (2016). Asymptotic results of a non-parametric conditional cumulative distribution estimator in the single functional index modeling for time series data with applications. Metrika. International Journal for Theoretical and Applied Statistics, 79(5), 485–511. https://doi.org/10.1007/s00184-015-0564-6.

Chaudhuri, P., Doksum, K., & Samarov, A. (1997). On average derivative quantile regression. The Annals of Statistics, 25(2), 715–744. https://doi.org/10.1214/aos/1031833670.

Cheng, P. E. (1994). Non-parametric Estimation of Mean Functional with Data Missing at Random. Journal of the American Statistical Association, 89(425), 81–87. https://doi.org/10.2307/2291203.

Ezzahrioui, M., & Ould-Said, M. (2008). Asymptotic normality of a non-parametric estimator of the conditional mode function for functional data. Journal of Non-parametric Statistics, 20(1), 3–18. https://doi.org/10.1080/10485250701541454.

Ferraty, F., Rabhi, A., & Vieu, P. (2005). Conditional Quantiles for Dependent Functional Data with Application to the Climatic El Nino Phenomenon. Sankhya. The Indian Journal of Statistics, 67(2), 378–398.

Ferraty, F., Sued, F., & Vieu, P. (2013). Mean estimation with data missing at random for functional covariables. Statistics. A Journal of Theoretical and Applied Statistics, 47(4), 688–706.

Ferraty, F., & Vieu, P. (2003). Functional Non-parametric Statistics. A Double Infinite Dimensional Framework. In M. G. Akritas & D. N. Politis (Eds.), Recent Advances and Trends in Non-parametric Statistics (pp. 61–78). Elsevier.

Ferraty, F., & Vieu, P. (2006). Non-parametric Functional Data Analysis. Theory and Practice. Springer. https://doi.org/10.1007/0-387-36620-2.

Hamri, M. M., Mekki, S. D., Rabhi, A., & Kadiri, N. (2022). Single functional index quantile regression for independent functional data under right-censoring. Econometrics. Ekonometria, 26(1), 31–62. https://doi.org/10.15611/eada.2022.1.03.

Kadiri, N., Rabhi, A., & Bouchentouf, A. A. (2018). Strong uniform consistency rates of conditional quantile estimation in the single functional index model under random censorship. Dependence Modeling, 6(1), 197–227. https://doi.org/10.1515/demo-2018-0013.

Laib, N., & Louani, D. (2010). Non-parametric Kernel Regression Estimation for Functional Stationary Ergodic Data: Asymptotic Properties. Journal of Multivariate Analysis, 101(10), 2266–2281. https://doi.org/10.1016/j.jmva.2010.05.010.

Lemdani, M., Ould-Said, E., & Poulin, N. (2009). Asymptotic properties of a conditional quantile estimator with randomly truncated data. Journalof Multivariate Analysis, 100(3), 546–559. https://doi.org/10.1016/j.jmva.2008.06.004.

Liang, H.-Y., & de Una-Álvarez, J. (2010). Asymptotic normality for estimator of conditional mode under left-truncated and dependent observations. Metrika. International Journal for Theoretical and Applied Statistics, 72(1), 1–19. https://doi.org/10.1007/s00184-009-0237-4.

Ling, N., Liang, L., & Vieu, P. (2015). Non-parametric regression estimation for functional stationary ergodic data with missing at random. Journal of Statistical Planning and Inference, 162, 75–87. https://doi.org/10.1016/j.jspi.2015.02.001.

Ling, N., Liu, Y., & Vieu, P. (2016). Conditional mode estimation for functional stationary ergodic data with responses missing at random. Statistics. A Journal of Theoretical and Applied Statistics, 50(5), 991–1013. https://doi.org/10.1080/02331888.2015.1122012.

Mekki, S. D., Kadiri, N., & Rabhi, A. (2021). Asymptotic Properties of the Semi-Parametric Estimators of the Conditional Density for Functional Data in the Single Index Model with Missing Data at Random. Statistica, 81(4), 399–422. https://doi.org/10.6092/issn.1973-2201/10472.

Ould-Said, E., & Cai, Z. (2005). Strong uniform consistency of non-parametric estimation of the censored conditional mode function. Journal of Non-parametric Statistics, 17(7), 797–806. https://doi.org/10.1080/10485250500038561.

Ould-Said, E., & Tatachak, A. (2011). A non-parametric conditional mode estimate under RLT model and strong mixing condition. International Journal of Statistics&Economics, 6(S11), 76–92.

Ould-Said, E., & Yahia, D. (2011). Asymptotic normality of a kernel conditional quantile estimator under strong mixing hypothesis and left-truncation. Communications in Statistics – Theory and Methods, 40(14), 2605–2627. https://doi.org/10.1080/03610926.2010.489171.

Rabhi, A., Kadiri, N., & Akkal, F. (2021). On the Central Limit Theorem for Conditional Density Estimator In the Single Functional Index Model. Applications and Applied Mathematics: An International Journal, 16(2), 844–866. https://digitalcommons.pvamu.edu/aam/vol16/iss2/4.

Roussas, G. G. (1969). Non-parametric estimation of the transition distribution function of a Markov Process. The Annals of Mathematical Statistics, 40(4), 1386–1400. https://doi.org/10.1214/aoms/1177697510.

Samanta, M. (1989). Non-parametric estimation of conditional quantiles. Statistics & Probability Letters, 7(5), 407–412. https://doi.org/10.1016/0167-7152(89)90095-3.