ASYMPTOTIC PROPERTIES OF ESTIMATORS IN PARTIALLY LINEAR SINGLE-INDEX MODEL FOR LONGITUDINAL DATA

doi:10.1016/S0252-9602(10)60069-0

Abstract

Abstract:

In this article, a partially linear single-index model for longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be estimated simultaneously by the proposed method while the feature of longitudinal data is considered. The existence, strong consistency and asymptotic normality of the estimators are proved under suitable conditions. A simulation study is conducted to investigate the finite sample performance of the proposed method. Our approach can also be used to study the pure single-index model for longitudinal data.

Key words: Longitudinal data, partially linear single-index model, penalized spline, strong consistency, asymptotic normality

CLC Number:

62G05

TIAN Ping, YANG Lin, XUE Liu-Gen. ASYMPTOTIC PROPERTIES OF ESTIMATORS IN PARTIALLY LINEAR SINGLE-INDEX MODEL FOR LONGITUDINAL DATA[J].Acta mathematica scientia,Series B, 2010, 30(3): 677-687.

Trendmd

References

[1] Stoker T M. Consistint estimation of scaled coefficients. Econometrica, 1986, 54(6): 1461--1481

[2] Härdle W, Stocker T M. Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association, 1989, 84(408): 986--995

[3] Ichimura H. Semiparametric least squares(SLS) and weighted SLS estimation of single-index models.
Journal of Econometrics, 1993, 58: 71--120

[4] Härdle W, Hall P, Ichimura H. Optimal smoothing in single-index models. The Annals of Statistics, 1993, 21(1): 157--178

[5] Yu Y, Ruppert D. Penalized spline estimation for partially linear single index models. Journal of the American Statistical Association, 2002, 97(460): 1042--1054

[6] Yu Y, Ruppert D. Root-n consistency of Penalized spline estimation for partially linear single index models under general euclidean space. Statistica Sinica, 2004, 14: 449--455

[7] Carroll R J, Fan J, Gijbels I, et al. Generalized partially linear single-index models. Journal of the American Statistical Association, 1997, 92(438): 477--489

[8] Delecroix M, H\"ardle W, Hristache M. Efficient estimation in conditional single-index regression. Journal of Multivariate Analysis, 2003, 86: 213--226

[9] Xue L G, Zhu L X. Empirical likelihood confidence regions of the parameters in a partially linear single-index model. Science in China Ser.A(in Chinese), 2005, 48(10): 1333--1348

[10] Diggle P J, Heagerty P J, Liang K Y. Analysis of Longitudinal Data. New York: Oxford University Press Inc., 2002. 1--1

[11] Lin Z Y, Lu C R, Su Z G. Foundations of the Theory of Probalility Limits (in Chinese). Beijing: Higher Education Press, 1999: 95--95

[12] Chen M H. Consistency of a class of estimators in semiparametric regression model under fixed design.
Applied Mathematics A Journal of Chinese Universities(in Chinese), 1998, 13(3): 301--310

[13] Jennrich R I. Asymptotic properties of nonlinear least squares estimators. The Annals of Mathematical Statistics, 1969, 40(2): 633--643

[14] Wu C F. Asymptotic theory of nonlinear least squares estimation. The Annals of Statistics, 1981, 9(3): 501--513