Diagnostic Measures for Functional Linear Model with Nonignorable Missing Responses

Yujian Zhu, Puying Zhao

Communications in Mathematics and Statistics ›› 2023, Vol. 12 ›› Issue (3) : 543-562. DOI: 10.1007/s40304-022-00301-x
Article

Diagnostic Measures for Functional Linear Model with Nonignorable Missing Responses

Author information +
History +

Abstract

Assessing the influence of individual observations of the functional linear models is important and challenging, especially when the observations are subject to missingness. In this paper, we introduce three case-deletion diagnostic measures to identify influential observations in functional linear models when the covariate is functional and observations on the scalar response are subject to nonignorable missingness. The nonignorable missing data mechanism is modeled via an exponential tilting semiparametric functional model. A semiparametric imputation procedure is developed to mitigate the effects of missing data. Valid estimations of the functional coefficients are based on functional principal components analysis using the imputed dataset. A smoothed bootstrap sampling method is introduced to estimate the diagnostic probability for each proposed diagnostic measure, which is helpful to unveil which observations have the larger influence on estimation and prediction. Simulation studies and a real data example are conducted to illustrate the finite performance of the proposed methods.

Keywords

Case deletion / Diagnostic measure / Functional linear model / Nonignorable nonresponse / Semiparametric imputation

Cite this article

Download citation ▾
Yujian Zhu, Puying Zhao. Diagnostic Measures for Functional Linear Model with Nonignorable Missing Responses. Communications in Mathematics and Statistics, 2023, 12(3): 543‒562 https://doi.org/10.1007/s40304-022-00301-x

References

[1.]
Bugni FA. Specification test for missing functional data. Econom. Theory, 2011, 28(5): 959-1002,
CrossRef Google scholar
[2.]
Cai TT, Hall P. Prediction in functional linear regression. Ann. Stat., 2006, 34(5): 2159-2179,
CrossRef Google scholar
[3.]
Cai TT, Yuan M. Minimax and adaptive prediction for functional linear regression. J. Am. Stat. Assoc., 2012, 107(499): 1201-1216,
CrossRef Google scholar
[4.]
Cardot H, Ferraty F, Sarda P. Functional linear model. Stat. Probab. Lett., 1999, 45(1): 11-22,
CrossRef Google scholar
[5.]
Cardot H, Ferraty F, Sarda P. Spline estimators for the functional linear model. Stat. Sin., 2003, 13(3): 571-591
[6.]
Chiou JM, Müller HG. Diagnostics for functional regression via residual processes. Comput. Stat. Data Anal., 2007, 15(10): 4849-4863,
CrossRef Google scholar
[7.]
Cook RD. Detection of influential observations in linear regression. Technometrics, 1977, 19(1): 15-18,
CrossRef Google scholar
[8.]
Cook RD, Weisberg S. . Residuals and Influence in Regression, 1982 New York Chapman und Hall
[9.]
Crambes C, Kneip A, Sarda P. Smoothing splines estimators for functional linear regression. Ann. Stat., 2009, 37(1): 35-72,
CrossRef Google scholar
[10.]
Ferraty F, Sued F, Vieu P. Mean estimation with data missing at random for functional covariables. Statistics, 2013, 47(4): 688-706,
CrossRef Google scholar
[11.]
Febrero-Bande M, Galeano P, González-Manteiga W. Measures of influence for the functional linear model with scalar response. J. Multivar. Anal., 2010, 101(2): 327-339,
CrossRef Google scholar
[12.]
Febrero-Bande, M., Oviedo de la Fuente, M.: Statistical computing in functional data analysis: the R package fda.usc. J. Stat. Softw. 51(4), 1–28 (2012)
[13.]
Gao Q, Ahn M, Zhu H. Cook’s distance measures for varying coefficient models with functional responses. Technometrics, 2015, 57(2): 268-280,
CrossRef Google scholar
[14.]
Hall P, Hosseini-Nasab M. On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B, 2006, 68(1): 109-126,
CrossRef Google scholar
[15.]
Hall P, Horowitz JL. Methodology and convergence rates for functional linear regression. Ann. Stat., 2007, 35(1): 70-91,
CrossRef Google scholar
[16.]
Hsing, T., Eubank, R.: Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley & Sons, Chichester (2015)
[17.]
Jang JH, Manatunga AK, Chang C, Long Q. A Bayesian multiple imputation approach to bivariate functional data with missing components. Stat. Med., 2021, 40(22): 4772-4793,
CrossRef Google scholar
[18.]
Kim JK, Yu CL. A semiparametric estimation of mean functionals with nonignorable missing data. J. Am. Stat. Assoc., 2011, 106(493): 157-165,
CrossRef Google scholar
[19.]
Li T, Xie F, Feng X, Ibrahim JG, Zhu H. Functional linear regression models for nonignorable missing scalar responses. Stat. Sin., 2018, 28(4): 1867-1886
[20.]
Little, R.J.A., Rubin, D.B.: Statistical Analysis with Missing Data. Wiley, NJ (2002)
[21.]
Müller HG, Stadtmüller U. Generalized functional linear models. Ann. Stat., 2005, 33(2): 774-805,
CrossRef Google scholar
[22.]
Peña D. A new statistic for influence in linear regression. Technometrics, 2005, 47(1): 1-12,
CrossRef Google scholar
[23.]
Preda C, Saporta G. Régression pls sur un Processus Stochastique. Revue de Statistique Appliquée, 2002, 50(2): 27-45
[24.]
Ramsay JO, Silverman BW. . Functional Data Analysis, 2005 2 New York Springer,
CrossRef Google scholar
[25.]
Shen Q, Xu H. Diagnostics for linear models with functional responses. Technometrics, 2007, 49(1): 26-33,
CrossRef Google scholar
[26.]
Wang X, Song X, Zhu H. Bayesian latent factor on image regression with nonignorable missing data. Stat. Med., 2021, 40(1): 920-932,
CrossRef Google scholar
[27.]
Yao F, Müller HG, Wang JL. Functional linear regression analysis for longitudinal data. Ann. Stat., 2005, 33(6): 2873-2903,
CrossRef Google scholar
[28.]
Yao F, Müller HG. Functional quadratic regression. Biometrika, 2010, 97(1): 49-64,
CrossRef Google scholar
[29.]
Zhu H, Ibrahim JG, Cho H. Perturbation and scaled Cook’s distance. Ann. Stat., 2012, 40(2): 785-811,
CrossRef Google scholar
[30.]
Zhu H, Ibrahim JG, Lee S, Zhang H. Perturbation selection and influence measures in local influence analysis. Ann. Stat., 2007, 35(6): 2565-2588,
CrossRef Google scholar
[31.]
Zhu H, Ibrahim JG, Tang N, Zhang H. Diagnostic measures for empirical likelihood of general estimating equations. Biometrika, 2008, 95(2): 489-507,
CrossRef Google scholar
[32.]
Zhu H, Ibrahim JG, Tang N. Bayesian influence analysis: a geometric approach. Biometrika, 2011, 98(2): 307-323,
CrossRef Google scholar
[33.]
Zhu H, Lee S, Wei B, Zhou J. Case deletion measures for models with incomplete data. Biometrika, 2001, 88(3): 727-737,
CrossRef Google scholar
Funding
National Natural Science Foundation of China(12071416)

Accesses

Citations

Detail

Sections
Recommended

/