Abstract
In this paper, we study the local asymptotic behavior of the regression spline estimator in the framework of marginal semiparametric model. Similarly to Zhu, Fung and He (2008), we give explicit expression for the asymptotic bias of regression spline estimator for nonparametric function f. Our results also show that the asymptotic bias of the regression spline estimator does not depend on the working covariance matrix, which distinguishes the regression splines from the smoothing splines and the seemingly unrelated kernel. To understand the local bias result of the regression spline estimator, we show that the regression spline estimator can be obtained iteratively by applying the standard weighted least squares regression spline estimator to pseudo-observations. At each iteration, the bias of the estimator is unchanged and only the variance is updated.
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References
Severini T A, Staniswalis J G. Quasi-likelihood estimation in semiparametric models. J Amer Statist Assoc, 89: 501–511 (1994)
Zeger S L, Diggle P J. Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. Biometrics, 50: 689–699 (1994)
Zhang D, Lin X, Raz J, et al. Semiparametric stochastic mixed models for longitudinal data. J Amer Statist Assoc, 93: 710–719 (1998)
Lin X, Carroll R J. Semiparametric regression for clustered data using generalized estimating equations. J Amer Statist Assoc, 96: 1045–1056 (2001)
He X, Zhu, Z Y, Fung W K. Estimation in a semiparametric model for longitudinal data with unspecified dependence structure. Biometrika, 89: 579–590 (2002)
Wang N, Carroll R J, Lin X. Efficient semiparametric marginal estimation for longitudinal/clustered data. J Amer Statist Assoc, 100: 147–157 (2005)
Chen K, Jin Z H. Local polynomial regression analysis of cluster data. Biometrika, 92: 59–74 (2005)
Huang J Z, Zhang L, Zhou L. Efficient estimation in marginal partially linear models for longitudinal/clustered data using spline. Scand J Statist, 34: 451–477 (2007)
Lin X, Wang N, Welsh A H, et al. Equivalent kernels of smoothing splines in nonparametric regression for clustered/longitudinal data. Biometrika, 91: 177–193 (2004)
Wang N. Marginal nonparametric kernel regression accounting for within-subject correlation. Biometrika, 90: 43–52 (2003)
Zhu Z Y, Fung W K, He X. On the asymptotics of marginal regression splines with longitudinal data. Biometrika, 95: 907–917 (2008)
Zhou S, Shen X, Wolfe D A. Local asymptotics for regression spline and confidence regions. Ann Statist, 26: 1760–1782 (1998)
Huang J Z. Local asymptotic properties for polynomial spline regression. Ann Statist, 31: 1600–1635 (2003)
Welsh A H, Lin X, Carroll R J. Marginal longitudinal nonparametric regression: locality and efficiency of spline and kernel methods. J Amer Statist Assoc, 97: 482–493 (2002)
Huang J Z, Wu C O, Zhou L. Polynomial spline estimation and inference for varying coefficient mdoels with longitudinal data. Statist Sinica, 14: 763–788 (2004)
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This work was supported by National Natural Science Foundation of China (Grant Nos. 10671038, 10801039), Youth Science Foundation of Fudan University (Grant No. 08FQ29) and Shanghai Leading Academic Discipline Project (Grant No. B118)
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Qin, G., Zhu, Z. Local asymptotic behavior of regression splines for marginal semiparametric models with longitudinal data. Sci. China Ser. A-Math. 52, 1982–1994 (2009). https://doi.org/10.1007/s11425-009-0115-6
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DOI: https://doi.org/10.1007/s11425-009-0115-6