Estimation And Application Of Partial Functional Linear Regression Model For Functional Response With Missing Data

Regression analysis is an important branch in statistics research,which is widely used.With the progress of data storage technology,various data can be measured and recorded continuously in a certain period of time in the era of information.When the time interval for recording is enough short,the data will take on the form of a function,then generating the concept of functional data.The essence of functional data analysis is to fit a smooth curve from dense discrete data,then analyze the characteristics of the curve,and establish a functional regression model for prediction and statistical decision.In statistical practice,some manmade or natural factors usually lead to the absence of the data when collecting sample data,thus generating the concept of missing data.If the samples with missing data are directly ignored,much data information will be lost,and then leading to a large prediction bias in the established regression model.Therefore,the problem of missing data should be solved first in statistical analysis.In recent years,the study of missing data has become one of the hot topics in the field of data analysis.In this aricle,the parameter estimation of partial functional linear regression model with functional response is studied when covariate is missing at random.In the model,the specific variable types are as follows:the response variable is functional data,the predictior is mixed data containing functional and scalar data.The scalar covariates are randomly missing.Firstly,we set the missing mechanism of covariates.Secondly,based on the method of functional principal component analysis,the estimated bivariate parameter function is obtained by using two sets of orthonormal principal component basis,and the asymptotic properties of the parameter are given under certain ragular assumptions.Thirdly,the empirical log-likelihood statistic of coefficient in the linear component is established.Under certain ragular assumptions,the empirical loglikelihood statistic is proved to be the chi-square distribution.Thus,the empirical likelihood confidence regions of the parameter is given.Finally,the performance of the proposed method under finite samples is verified by some numerical simulations and example simulations.