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Asymptotic normality of M-estimators in a semiparametric model with longitudinal data

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Abstract

Suppose that the longitudinal observations (Y ij , X ij , t ij ) for i = 1, . . . ,n; j = 1, . . . ,m i are modeled by the semiparamtric model \({Y_{ij}=X_{ij}^{T}\beta_{0}+g(t_{ij})+e_{ij},}\) where β 0 is a k × 1 vector of unknown parameters, g(·) is an unknown estimated function and e ij are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions. The M-estimator of the slope parameter β 0 is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(·) is also obtained by replacing β 0 in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of β 0 is derived. The asymptotic distributions of the local M-estimators of g(·) at both interior and boundary points are also established. Various applications of our main results are given.

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Authors and Affiliations

  1. School of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Tang Qingguo

  2. Institute of Sciences, PLA University of Science and Technology, Nanjing, 210007, People’s Republic of China

    Tang Qingguo

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  1. Tang Qingguo
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Correspondence to Tang Qingguo.

Additional information

The research is supported in part by National Natural Science Foundation of China (Grant No. 10671089).

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Qingguo, T. Asymptotic normality of M-estimators in a semiparametric model with longitudinal data. Metrika 69, 55–67 (2009). https://doi.org/10.1007/s00184-008-0175-6

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  • DOI: https://doi.org/10.1007/s00184-008-0175-6

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