Nonparametric quantile regression estimation for functional data with responses missing at random
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Nonparametric quantile regression estimation for functional data with responses missing at random Dengke Xu1 · Jiang Du2 Received: 25 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents the nonparametric quantile regression estimation for the regression function operator when the functional data with the responses missing at random are considered. Then, the large sample properties of the proposed estimator are established under some mild conditions. Finally, a simulation study is conducted to investigate the finite sample properties of the proposed method. Keywords Quantile regression · Functional data analysis · Missing at random · Inverse probability weighting estimator · Asymptotic normality Mathematics Subject Classification 62G08 · 62G20
1 Introduction Functional data analysis is a powerful tool for studying about the analysis of high dimensional data, which has been applied in many fields such as medicine, economics, environmetrics, chemometrics and others. In recent years, statistical inference for functional data has received much attention in statistic literatures, and relevant results are summarized very nicely in Horváth and Kokoszka (2012). The functional linear model (FLM) may be regarded as one of the most useful and popular models in functional data analysis, which is used to model the relationship between the real value response variable and the functional predictive variable. Ever since the work by Ramsay and Dalzell (1991), many developments concerning building theories and methods for FLM have been achieved. As is well known, the study of statistical models for FLM has received an increasing interest in recent years, such as
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Jiang Du [email protected]
1
Department of Statistics, Zhejiang Agriculture and Forestry University, Hangzhou 311300, China
2
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
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D. Xu, J. Du
Ramsay and Silverman (2005), Cai and Hall (2006), Lu et al. (2014), Yu et al. (2017) for various estimation and test methods. However, all the results mentioned above are focused on the situation that the samples are observed completely. Unfortunately, in many practical works, such as sampling survey, pharmaceutical tracing test, reliability test and so on, some pairs of observations may be incomplete, which is often called the case of missing data. Many examples of missing data and its statistical inferences for regression model can be found in statistical literatures when the regressive variables are of finite dimensionality. For more details, see Cheng (1994), Little and Rubin (2002), Tang et al. (2014) and references therein. When the regressive variables are of infinite dimensionality or of functional formation, only very few literatures were reported on investigating the statistical properties of functional nonparametric regression model with missing data. Recently, Ling et al. (2015) investigated the asymptotic properties of the estimator for the regression function operator when
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