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On Parameters of Increasing Dimensions

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  • He, Xuming
  • Shao, Qi-Man
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    Abstract

    In statistical analyses the complexity of a chosen model is often related to the size of available data. One important question is whether the asymptotic distribution of the parameter estimates normally derived by taking the sample size to infinity for a fixed number of parameters would remain valid if the number of parameters in the model actually increases with the sample size. A number of authors have addressed this question for the linear models. The component-wise asymptotic normality of the parameter estimate remains valid if the dimension of the parameter space grows more slowly than some root of the sample size. In this paper, we consider M-estimators of general parametric models. Our results apply to not only linear regression but also other estimation problems such as multivariate location and generalized linear models. Examples are given to illustrate the applications in different settings.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 73 (2000)
    Issue (Month): 1 (April)
    Pages: 120-135

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    Handle: RePEc:eee:jmvana:v:73:y:2000:i:1:p:120-135

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    Related research

    Keywords: asymptotic approximation; exponential inequality; increasing dimension; linear regression; logistic regression; M-estimator; self-normalization; spatial median;

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    Cited by:
    1. Arun Chandrasekhar & Victor Chernozhukov & Francesca Molinari & Paul Schrimpf, 2012. "Inference for best linear approximations to set identified functions," CeMMAP working papers CWP43/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Victor Chernozhukov & Roberto Rigobon & Thomas M. Stoker, 2010. "Set identification and sensitivity analysis with Tobin regressors," Quantitative Economics, Econometric Society, vol. 1(2), pages 255-277, November.
    3. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2013. "Uniform post selection inference for LAD regression models," CeMMAP working papers CWP24/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2013. "Uniform post selection inference for LAD regression and other z-estimation problems," CeMMAP working papers CWP74/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Victor Chernozhukov & Ivan Fernandez-Val & Alfred Galichon, 2010. "Quantile and Probability Curves without Crossing," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
    6. Zheng, Qi & Gallagher, Colin & Kulasekera, K.B., 2013. "The growth rate of significant regressors for high dimensional data," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 1969-1972.
    7. Kato, Kengo & F. Galvao, Antonio & Montes-Rojas, Gabriel V., 2012. "Asymptotics for panel quantile regression models with individual effects," Journal of Econometrics, Elsevier, vol. 170(1), pages 76-91.
    8. V. Chernozhukov & I. Fernández-Val & A. Galichon, 2009. "Improving point and interval estimators of monotone functions by rearrangement," Biometrika, Biometrika Trust, vol. 96(3), pages 559-575.
    9. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP75/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    10. Du, Jiang & Sun, Zhimeng & Xie, Tianfa, 2013. "M-estimation for the partially linear regression model under monotonic constraints," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1353-1363.
    11. Leng, Chenlei & Li, Bo, 2010. "Least squares approximation with a diverging number of parameters," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 254-261, February.
    12. Victor Chernozhukov & Roberto Rigobon & Thomas Stoker, 2009. "Set identification with Tobin regressors," CeMMAP working papers CWP12/09, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    13. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2013. "Robust inference in high-dimensional approximately sparse quantile regression models," CeMMAP working papers CWP70/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    14. repec:spo:wpecon:info:hdl:2441/5rkqqmvrn4tl22s9mc4b6ga2g is not listed on IDEAS
    15. Lee, Sangin & Kim, Yongdai & Kwon, Sunghoon, 2012. "Quadratic approximation for nonconvex penalized estimations with a diverging number of parameters," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1710-1717.
    16. Zhijie Xiao & Roger Koenker, 2009. "Conditional Quantile Estimation for GARCH Models," Boston College Working Papers in Economics 725, Boston College Department of Economics.

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