Abstract
This paper focuses on semi-functional partially linear regression model, where a scalar response variable with missing at random is explained by a sum of an unknown linear combination of the components of multivariate random variables and an unknown transformation of a functional random variable which takes its value in a semi-metric abstract space \({\mathscr {H}}\) with a semi-metric \(d\left( \cdot , \cdot \right) \). The main purpose of this paper is to construct the estimators of unknown parameters and an unknown regression operator respectively. Then some asymptotic properties of the estimators such as almost sure convergence rates of the nonparametric component and asymptotic distribution of the parametric one are obtained under some mild conditions. Furthermore, a simulation study is carried out to evaluate the finite sample performances of the estimators. Finally, an application to real data analysis for food fat predictions shows the usefulness of the proposed methodology.
Similar content being viewed by others
References
Aneiros G, Bongiorno EG, Cao R, Vieu P (2017) Functional statistics and related fields. Contributions to statistics. Springer, Cham
Aneiros-Pérez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76:1102–1110
Aneiros-Pérez G, Vieu P (2008) Nonparametric time series prediction: a semi-functional partial linear modeling. J Multivar Anal 99:834–857
Aneiros-Pérez G, Vieu P (2011) Automatic estimation procedure in partial linear model with functional data. Stat Pap 52:751–771
Aneiros-Pérez G, Vieu P (2013) Testing linearity in semi-parametric functional data analysis. Comput Stat 28:413–434
Aneiros-Pérez G, Ling N, Vieu P (2015) Error variance estimation in semi-functional partially linear regression models. J Nonparametr Stat 27:316–330
Attouch M, Laksaci A, OuldSaid E (2009) Asymptotic distribution of robust estimator for functional nonparametric models. Commun Stat Theory Methods 38:1317–1335
Bongiorno EG, Salinelli E, Goia A, Vieu P (2014) Contributions in infinite-dimensional statistics and related topics, Societa Editrice Esculapio
Chaouch M, Khardani S (2015) Randomly censored quantile regression estimation using functional stationary ergodic data. J Nonparametr Stat 27:65–87
Cheng PE (1994) Nonparametric estimation of mean functionals with data missing at random. J Am Stat Assoc 89:81–87
Cuevas A (2014) A partial overview of the theory of ststistics with functional data. J Stat Plan Inference 147:1–23
Efromovich S (2011) Nonparametric regression with predictors missing at random. J Am Stat Assoc 106:306–319
Ferraty F, Laksaci A, Tadj A, Vieu P (2010) Rates of uniform consistency for nonparametric estimates with functional variables. J Stat Plan Infer 140:335–352
Ferraty F, Vieu P (2002) The functional nonparametric model and application to spectrometric data. Comput Stat 17:545–564
Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Theory and practice. Springer, New York
Ferraty F, Sued M, Vieu P (2013) Mean estimation with data missing at random for functional covariables. Statistics 47:688–706
Goia A, Vieu P (2016) An introduction to recent advances in high/infinite dimensional statistics. J Multivar Anal 146:1–6
Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer series in statistics. Springer, New York
Kraus D (2015) Components and completion of partially observed functional data. J R Stat Soc B 77:777–801
Lian H (2011) Functional partial linear model. J Nonparametr Stat 23:115–128
Liang H (1999) An application of Bernstein’s inequality. Econom Theory 15:151–160
Liang H, Wang S, Carroll R (2007) Partially linear models with missing response variables and error-prone covariates. Biometrika 94:185–198
Ling NX, Kan R (2016) Rate of uniform convergence of functional regression with missing responses at random. Working paper
Ling NX, Liang LL, Vieu P (2015) Nonparametric regression estimation for functional stationary ergodic data with missing at random. J Stat Plan Inference 162:75–87
Ling NX, Liu Y, Vieu P (2016) Conditional mode estimation for functional stationary ergodic data with responses missing at random. Statistics 50:1–23
Little R, Rubin D (2002) Statistical analysis with missing data, 2nd edn. Wiley, New York
Nittner T (2003) Missing at random (MAR) in nonparametric regression—a simulation experiment. Stat Methods Appl 12:195–210
Ramsay J, Silverman B (2005) Functional data analysis, 2nd edn. Springer series in statistics. Springer, New York
Shang H (2014) Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density. Comput Stat 29:829–848
Sued M, Yohai VJ (2013) Robust location estimation with missing data. Can J Stat 41(1):111–132
Tsiatis A (2006) Semiparametric theory and missing data. Springer, New York
Wang QH, Sun ZH (2007) Estimation in partially linear models with missing responses at random. J Multivar Anal 98:1470–1493
Wang QH, Linton O, Wolfgang H (2004) Semiparametric regression analysis with missing response at random. J Am Stat Assoc 99:334–345
Acknowledgements
The authors would like to appreciate the Editor in Chief and the two referees for their valuable comments and suggestions that are very helpful for them to improve the quality and presentation of the paper significantly. Ling’s work is supported by the National Social Science Funds of China (14ATJ005).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ling, N., Kan, R., Vieu, P. et al. Semi-functional partially linear regression model with responses missing at random. Metrika 82, 39–70 (2019). https://doi.org/10.1007/s00184-018-0688-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-018-0688-6