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Convex analysis in the semiparametric model with Bernstein polynomials
Jianhua Ding,Zhongzhan Zhang 한국통계학회 2015 Journal of the Korean Statistical Society Vol.44 No.1
In this paper, we propose Bernstein polynomial estimation for the partially linear modelwhen the nonparametric component is subject to convex (or concave) constraint. Weemploy a nested sequence of Bernstein polynomials to approximate the convex (orconcave) nonparametric function. Bernstein polynomial estimation can be obtained as asolution of a constrained least squares method and hence we use a quadratic programmingalgorithm to compute efficiently the estimator. We show that the estimator of theparametric part is asymptotically normal. The rate of convergence of the nonparametricfunction estimator is established under very mild conditions. The small sample propertiesof our estimation are provided via simulation study and compared with regression splinesmethod. A real data analysis is conducted to illustrate the application of the proposedmethod.
Testing independence and goodness-of-fit jointly for functional linear models
Tingyu Lai,Zhongzhan Zhang,Wang Yafei 한국통계학회 2021 Journal of the Korean Statistical Society Vol.50 No.2
A conventional regression model for functional data involves expressing a response variable in terms of the predictor function. Two assumptions, that (i) the predictor function and the error are independent and (ii) the relationship between the response variable and the predictor function takes functional linear model, are usually added to the model. Checking the validation of these two assumptions is fundamental to statistic inference and practical applications. We develop a test procedure to check these assumptions simultaneously based on generalized distance covariance. We establish the asymptotic theory for the proposed test under null and alternative hypotheses, and provide a bootstrap procedure to obtain the critical value of the test. The proposed test is consistent against all alternatives provided that the semimetrics related to the generalized distance are strong negative, and can be readily generalized to other functional regression models. We explore the finite sample performance of the proposed test by using both simulations and real data examples. The results illustrate that the proposed method has favorable performance compared with the competing method.
Varying-coefficient partially functional linear quantile regression models
Ping Yu,Jiang Du,Zhongzhan Zhang 한국통계학회 2017 Journal of the Korean Statistical Society Vol.46 No.3
In this paper, we introduce a new varying-coefficient partially functional linear quantile regression model, which combines varying-coefficient quantile regression model with functional linear quantile regression model. The functional principal component basis and regression splines are employed to estimate the slope function and varying-coefficient functions, respectively, and the convergence rates of the estimators are obtained under some regularity conditions. Simulations and an illustrative real example are presented.
Yu Ping,Du Jiang,Zhang Zhongzhan 한국통계학회 2021 Journal of the Korean Statistical Society Vol.50 No.1
This paper investigates the hypothesis test of the parametric component in partial functional linear quantile regression model in which the dependent variable is related to both a vector of fnite length and a function-valued random variable as predictor variables. A quantile rank score test based on functional principal component analysis is developed. Under mild conditions, we establish the consistency of the proposed test statistic, and show that the proposed test can detect Pitman local alternatives converging to the null hypothesis at the usual parametric rate. A simulation study shows that the proposed test procedure has good size and power with fnite sample sizes. Finally, an illustrative example is given through ftting the Berkeley growth data and testing the efect of gender on the height of kids.
Longbing Wang,Ruiyuan Cao,Jiang Du,Zhongzhan Zhang 한국통계학회 2019 Journal of the Korean Statistical Society Vol.48 No.4
This paper considers the nonparametric inverse probability weighted estimation for functional data with missing response data at random. Under mild conditions, the asymptotic properties of the proposed estimation method are established. Based on the resampling method, the estimation of the asymptotic variance of the proposed estimator is obtained. Finally, the finite sample properties of the proposed estimation method are investigated via Monte Carlo simulation studies. A real data analysis is given to illustrate the use of the proposed method.