fan EMPIRICAL+LIKELIHOOD+FOR+LONGITUDINAL+PARTIALLY+LINEAR+MODEL+WITH+α-MIXING+ERRORS

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FAN Guoliang School of Mathematics & Physics, Anhui Polytechnic University, Wuhu 241000, China. Email: guoliangfan@. LIANG Hanying (Corresponding author) Department of Mathematics, Tongji University, Shanghai 200092, China. Email: hyliang83@. ∗ This research is supported by the National Natural Science Foundation of China under Grant Nos. 11271286, 11271286, 71171003, and 11226218, Provincial Natural Science Research Project of Anhui Colleges under Grant No. KJ2011A032, Anhui Provincial Natural Science Foundation under Grant Nos. 1208085QA04 and 10040606Q03. This paper was recommended for publication by Editor ZOU Guohua.
J Syst Sci Complex (2013) 26: 232–248
EMPIRICAL LIKELIHOOD FOR LONGITUDINAL PARTIALLY LINEAR MODEL WITH α-MIXING ERRORS∗
FAN Guoliang · LIANG Hanying
DOI: 10.1007/s11424-013-0015-2 Received: 28 January 2010 / Revised: 17 September 2010 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2013 Abstract This paper considers large sample inference for the regression parameter in a partially linear regression model with longitudinal data and α-mixing errors. The authors introduce an estimated empirical likelihood for the regression parameter and show that its limiting distribution is a mixture of central chi-squared distributions. Also, the authors derive an adjusted empirical likelihood method which is shown to have a central chi-square limiting distribution. A simulation study is carried out to assess the performance of the empirical likelihood method. Key words Confidence region, empirical likelihood, longitudinal data, partially linear model, α-mixing sequence.
EMPIRICAL LIKELIHOOD FOR LONGITUDINAL PARTIALLY LINEAR
n
233
this paper we assume that n increases to push up the total sample size N = i=1 mi , while {mi } are the bounded sequences of positive integers. Obviously, model (1) includes many parametric, nonparametric and semiparametric regression models. For example, when mi ≡ 1, which corresponds to independent errors, model (1) reduces to the nonlongitudinal partially linear regression model. When g ≡ 0, i.e., the nonparametric component is removed, model (1) becomes the longitudinal linear regression model, which has been studied in [2], and the work referenced therein. On the other hand, when β = 0, model (1) reduces to the longitudinal nonparametric regression model, which has been investigated by Brumback and Rice[3] , Rice and Wu[4] , Fan and Zhang[5] , Wu and Zhang[6], among others. The empirical likelihood method has been applied to nonlongitudinal partially linear regression models; see Wang and Jing[7] , Shi and Lau[8] , Wang and Jing[9] , Fan and Liang[10] and Fan, et al.[11] . The empirical likelihood, proposed by Owen[12−13] , is a powerful tool for statistical inference. This method defines an empirical likelihood ratio function, and uses its maximum subject to a hypothesis that place restrictions on the parameter (or parametric vector) to construct confidence region. This method uses only the data to determine the shape and orientation of a confidence region and does not use the estimator of the asymptotic covariance. Therefore, the empirical likelihood is indeed appealing for the construction of confidence region of β . More references and techniques can be found in the recent monograph of Owen[14] . In addition, Jing, et al.[15] introduced a so-called jackknife empirical likelihood method which was shown to be very effective in handling one and two-sample U-statistics. Applying the empirical likelihood method to the model of longitudinal data is more difficult due to the correlation within groups. Recently, Li, et al.[16] , Xue and Zhu[17] investigated the empirical likelihood for longitudinal partially linear model. But these articles both assumed the errors εij from different subjects are independent. However, the independence assumption for the errors is not always valid in applications, especially for sequentially collected economic data, which often exhibit evident dependence in the errors. In the sequel, the error vectors {ε1 , ε2 , · · · } are assumed to form a stationary αl mixing sequence. We now introduce the α-mixing coefficients. Let Fk be the σ -algebra of events generated by {ζt , k ≤ t ≤ l} for l ≥ k . The α-mixing coefficient introduced by Rosenblatt[18] i j th measurement on the ith subject, the ith cluster has mi observations (i = d 1, 2, · · · , n), (xτ ij , tij ) ∈ R × R are known fixed design points, β is an unknown d-element vector of regression parameters, g (·) is an unknown regression function defined on the closed interval [0, 1], and εij are random errors with zero mean and positive definite covariance matrix Σ i , i.e., Var(εi ) = Σi where εi = (εi1 , εi2 , · · · , εimi )τ is random error vector of ith subject. Throughout