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On the asymptotic properties of the shrinkage empirical likelihood estimators for longitudinal data

  • Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.03.003
  • Received Date: 20 November 2015
  • Accepted Date: 17 April 2016
  • Rev Recd Date: 17 April 2016
  • Publish Date: 30 March 2017
    Abstract

  • Abstract

    When there exist time-dependent covariates in some longitudinal study, it is well-known that the widely used generalized estimating equations approach would not preserve unbiasedness and robustness in an arbitrary working correlation structure. However, incorrect application of the working correlation structure could result in loss of efficiency and biased estimation. To deal with this problem, Leung et al. proposed a shrinkage empirical likelihood approach which combines the unbiased estimating equations and the extracted additional information from the estimating equations that excluded by the independence assumption. Although their simulations have shown the proposed estimators are efficient, the asymptotic properties of the proposed estimators are unknown. Here it is was shown that the proposed estimators are consistent and asymptotically normally distributed under some regular conditions.

    Abstract

    When there exist time-dependent covariates in some longitudinal study, it is well-known that the widely used generalized estimating equations approach would not preserve unbiasedness and robustness in an arbitrary working correlation structure. However, incorrect application of the working correlation structure could result in loss of efficiency and biased estimation. To deal with this problem, Leung et al. proposed a shrinkage empirical likelihood approach which combines the unbiased estimating equations and the extracted additional information from the estimating equations that excluded by the independence assumption. Although their simulations have shown the proposed estimators are efficient, the asymptotic properties of the proposed estimators are unknown. Here it is was shown that the proposed estimators are consistent and asymptotically normally distributed under some regular conditions.
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    • References(24)
  • [1]
    DIGGLE P J, HEAGERTY P J, LIANG K Y, et al. Analysis of Longitudinal Data[M]. 2nd ed. New York: Oxford University Press, 2002.
    [2]
    WU H, ZHANG J. Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches[M]. New York: Wiley, 2006.
    [3]
    LIANG K Y, ZEGER S L. Longitudinal data analysis using generalized linear models[J]. Biometrika, 1986, 73: 13-22.
    [4]
    ALEXANDER C S, MARKOWITZ R. Maternal employment and use of pediatric clinic services[J]. Medical Care, 1986, 24(2): 134-47.
    [5]
    PEPE M S, ANDERSON G L. A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data[J]. Communications in Statistics, Simulations and Computation, 1994, 23: 939-951.
    [6]
    FITZMAURICE G M. A caveat concerning independence estimating equations with multivariate binary data[J]. Biometrics, 1995, 51: 309-317.
    [7]
    LAI T L, SMALL D. Marginal regression analysis of longitudinal data with time-dependent covariates: A generalized method-of-moments approach[J]. Journal of the Royal Statistical Society B, 2007, 69: 79-99.
    [8]
    LEUNG D H, SMALL D S, QIN J, et al. Shrinkage empirical likelihood estimator in longitudinal analysis with time-dependent covariates: Application to modeling the health of Filipino children[J]. Biometrics, 2013, 69: 624-632.
    [9]
    NEWEY W K, SMITH R J. Higher order properties of GMM and generalized empirical likelihood estimators[J]. Econometrica, 2004, 72: 219-255.
    [10]
    QIN J, LAWLESS J. Empirical likelihood and general estimating equations[J]. Ann Statist B, 1994, 22: 300-325.
    [11]
    RAO C R. Linear Statistical Inference and Its Applications[M]. New York: Wiley, 1973.
    [12]
    ANDREWS D W K. Consistent moment selection procedures for generalized method of moments estimation[J]. Econometrica, 1999, 67: 543-564.
    [13]
    FITZMAURICE G M, DAVIDIAN M, VERBEKE G, et al. Longitudinal Data Analysis[M]. Boca Raton, Florida: Chapman & Hall/CRC, 2008.
    [14]
    HANSEN L P. Large sample properties of generalized method of moments estimators[J]. Econometrica, 1982, 50:1029-1054.
    [15]
    LAI T, SMALL D, LIU J. Statistical inference in dynamic panel data models[J]. Journal of Statistical Planning and Inference, 2008, 138: 2763-2776.
    [16]
    LIU X, ZHANG W. A moving average Cholesky factor model in joint mean-covariance modeling for longitudinal data[J]. Science in China A, 2013, 56(11): 2367-2379.
    [17]
    OKUI R. Instrumental variable estimation in the presence of many moment conditions[J]. Journal of Econometrics, 2011, 165: 70-86.
    [18]
    OWEN A B. Empirical likelihood ratio confidence intervals for a single functional[J]. Biometrika, 1988, 75: 237-249.
    [19]
    OWEN A B. Empirical Likelihood[M]. New York: Chapman and Hall/CRC, 2001.
    [20]
    PAN W, CONNETT J E. Selecting the working correlation structure in generalized estimating equations with application to the lung health study[J]. Statistica Sinica, 2002, 23: 475-490.
    [21]
    WANG L, QU A. Consistent model selection and data driven smooth tests for longitudinal data in the estimating equations approach[J]. Journal of the Royal Statistical Society B, 2009, 71: 177-190.
    [22]
    WANG S, QIAN L, CARROLL R J. Generalized empirical likelihood methods for analyzing longitudinal data[J]. Biometrika, 2010, 97(1): 79-93.
    [23]
    XUE L, ZHU L. Empirical likelihood for a varying coefficient model with longitudinal data[J]. Journal of the American Statistical Association, 2007, 102: 642-654.
    [24]
    YOU J, CHEN G, ZHOU Y. Block empirical likelihood for longitudinal partially linear regression models[J]. Can J Statist, 2006, 34: 79-96.)
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    [1]
    DIGGLE P J, HEAGERTY P J, LIANG K Y, et al. Analysis of Longitudinal Data[M]. 2nd ed. New York: Oxford University Press, 2002.
    [2]
    WU H, ZHANG J. Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches[M]. New York: Wiley, 2006.
    [3]
    LIANG K Y, ZEGER S L. Longitudinal data analysis using generalized linear models[J]. Biometrika, 1986, 73: 13-22.
    [4]
    ALEXANDER C S, MARKOWITZ R. Maternal employment and use of pediatric clinic services[J]. Medical Care, 1986, 24(2): 134-47.
    [5]
    PEPE M S, ANDERSON G L. A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data[J]. Communications in Statistics, Simulations and Computation, 1994, 23: 939-951.
    [6]
    FITZMAURICE G M. A caveat concerning independence estimating equations with multivariate binary data[J]. Biometrics, 1995, 51: 309-317.
    [7]
    LAI T L, SMALL D. Marginal regression analysis of longitudinal data with time-dependent covariates: A generalized method-of-moments approach[J]. Journal of the Royal Statistical Society B, 2007, 69: 79-99.
    [8]
    LEUNG D H, SMALL D S, QIN J, et al. Shrinkage empirical likelihood estimator in longitudinal analysis with time-dependent covariates: Application to modeling the health of Filipino children[J]. Biometrics, 2013, 69: 624-632.
    [9]
    NEWEY W K, SMITH R J. Higher order properties of GMM and generalized empirical likelihood estimators[J]. Econometrica, 2004, 72: 219-255.
    [10]
    QIN J, LAWLESS J. Empirical likelihood and general estimating equations[J]. Ann Statist B, 1994, 22: 300-325.
    [11]
    RAO C R. Linear Statistical Inference and Its Applications[M]. New York: Wiley, 1973.
    [12]
    ANDREWS D W K. Consistent moment selection procedures for generalized method of moments estimation[J]. Econometrica, 1999, 67: 543-564.
    [13]
    FITZMAURICE G M, DAVIDIAN M, VERBEKE G, et al. Longitudinal Data Analysis[M]. Boca Raton, Florida: Chapman & Hall/CRC, 2008.
    [14]
    HANSEN L P. Large sample properties of generalized method of moments estimators[J]. Econometrica, 1982, 50:1029-1054.
    [15]
    LAI T, SMALL D, LIU J. Statistical inference in dynamic panel data models[J]. Journal of Statistical Planning and Inference, 2008, 138: 2763-2776.
    [16]
    LIU X, ZHANG W. A moving average Cholesky factor model in joint mean-covariance modeling for longitudinal data[J]. Science in China A, 2013, 56(11): 2367-2379.
    [17]
    OKUI R. Instrumental variable estimation in the presence of many moment conditions[J]. Journal of Econometrics, 2011, 165: 70-86.
    [18]
    OWEN A B. Empirical likelihood ratio confidence intervals for a single functional[J]. Biometrika, 1988, 75: 237-249.
    [19]
    OWEN A B. Empirical Likelihood[M]. New York: Chapman and Hall/CRC, 2001.
    [20]
    PAN W, CONNETT J E. Selecting the working correlation structure in generalized estimating equations with application to the lung health study[J]. Statistica Sinica, 2002, 23: 475-490.
    [21]
    WANG L, QU A. Consistent model selection and data driven smooth tests for longitudinal data in the estimating equations approach[J]. Journal of the Royal Statistical Society B, 2009, 71: 177-190.
    [22]
    WANG S, QIAN L, CARROLL R J. Generalized empirical likelihood methods for analyzing longitudinal data[J]. Biometrika, 2010, 97(1): 79-93.
    [23]
    XUE L, ZHU L. Empirical likelihood for a varying coefficient model with longitudinal data[J]. Journal of the American Statistical Association, 2007, 102: 642-654.
    [24]
    YOU J, CHEN G, ZHOU Y. Block empirical likelihood for longitudinal partially linear regression models[J]. Can J Statist, 2006, 34: 79-96.)
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