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Asymptotic normality of DHD estimators in a partially linear model

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Abstract

The paper studies a partially linear regression model given by

$$\begin{aligned} y_i=x_i^T\beta +f(t_i)+\varepsilon _i,i=1,2,\ldots ,n, \end{aligned}$$

where \(\{\varepsilon _i,i=1,2,\ldots , n\}\) are independent and identically distributed random errors with zero mean and finite variance \(\sigma ^2>0\). Using a difference based and the Huber–Dutter (DHD) approaches, the estimators of unknown parametric component \(\beta \) and root variance \(\sigma \) are given, and then the estimation of nonparametric component \(f(\cdot )\) is given by the wavelet method. The asymptotic normality of the DHD estimators of \(\beta \) and \(\sigma \) are investigated, and the weak convergence rate of the estimator of \(f(\cdot )\) is also investigated. In addition, for stationary \(m\)-dependent sequence of random variables, the central limit theorem is also obtained. At last, two examples are presented to illustrate the proposed method.

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References

  • Antoniads A, Gregoire G, Mckeague IW (1994) Wavelet methods for cure estimation. J Am Stat Assoc 89:1340–1353

    Article  MATH  Google Scholar 

  • Arashi M, Roozbeh M, Niroumand HA (2012) A note on Stein-type shrinkage estimator in partial linearmodels. Statistics 46(5):673–685

    Article  MathSciNet  MATH  Google Scholar 

  • Arashi M, Valizadeh T (2015) Performance of Kibrias methods in partial linear ridge regression model. Stat Pap 56:231–246

    Article  MathSciNet  MATH  Google Scholar 

  • Bianco A, Boente G (2004) Robust estimators in semiparametric partly linear regression models. J Stat Plan Inference 122:229–252

    Article  MathSciNet  MATH  Google Scholar 

  • Bianco A, Boente G, Martinez E (2006) Robust tests in semiparametric partly linear regression models. Scand J Stat 33(3):435–450

    Article  MathSciNet  MATH  Google Scholar 

  • Carroll RJ, Fan J, Gijbels I, Wand MP (1997) Generalized partially linear single-index models. J Am Stat Assoc 92:477–489

    Article  MathSciNet  MATH  Google Scholar 

  • Chang X, Qu L (2004) Wavelet estimation of partially linear models. Comput Stat Data Anal 47:31–48

    Article  MathSciNet  MATH  Google Scholar 

  • Chatterjee S, Hadi AS (2006) Regression analysis by example, 4th edn. Wiley, Hoboken

    Book  MATH  Google Scholar 

  • Chen H (1988) Convergence rates for parametric components in a partly linear model. Ann Stat 16(1):136–146

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng CL, Van Ness JW (1992) Generalized M-estimators for errors-in-variables regression. Ann Stat 20(1):385–397

    Article  MathSciNet  MATH  Google Scholar 

  • Engle FR, Granger WJ, Rice J, Weiss A (1986) Semiparametric estimates of the relation between weather and electricity sales. J Am Stat Assoc 80:310–319

    Article  Google Scholar 

  • Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman & Hall, London

    MATH  Google Scholar 

  • Fan J, Huang T (2005) Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11:1031–1057

    Article  MathSciNet  MATH  Google Scholar 

  • Fan J, Feng Y, Song R (2011) Nonparametric indendence screening in sparse ultra-high-dimensional additive models. J Am Stat Assoc 106(494):544–557

    Article  MathSciNet  MATH  Google Scholar 

  • Ferguson TS (1996) A course in large sample theory. Chapman & Hall, New York

    Book  MATH  Google Scholar 

  • Green PJ, Silverman BW (1995) Nonparametric regression and generalized linear models. Chapman and Hall, London

    MATH  Google Scholar 

  • Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Hampel FR, Ronchetti EM, Rousseeuw PJ et al (1986) Robust statistics. Wiley, New York

    MATH  Google Scholar 

  • Hu HC, Cui HJ, Li KC (2015) Asymptotic properties of wavelet estimators in partially linear Errors-in-Variables models with long-memory errors. Acta Math Appl Sin Ser B 31 (in press)

  • Hu HC (2005) Ridge estimation of a semiparametric regression model. J Comput Appl Math 176:215–222

    Article  MathSciNet  MATH  Google Scholar 

  • Hu HC, Wu L (2012) Convergence rates of wavelet estimators in semiparametric regression models under NA samples. Chin Ann Math 33(4):609–624

    Article  MathSciNet  MATH  Google Scholar 

  • Hu HC (2013) Asymptotic normality of Huber–Dutter in a linear model with AR(1) processes. J Stat Plan Inference 143:548–562

    Article  MathSciNet  MATH  Google Scholar 

  • Huber PJ, Ronchetti EM (2009) Robust statistics, 2nd edn. Wiley, New York

    Book  MATH  Google Scholar 

  • Lehmann EL (1999) Elements of large-sample theory. Springer, New York

    Book  MATH  Google Scholar 

  • Lenk PJ (1999) Bayesian inference for semiparametric regression using a Fourier representation. J R Stat Soc Ser B 61:863–879

    Article  MathSciNet  MATH  Google Scholar 

  • Li D, Stengos T (2002) The partially linear regressionmodel: Monte Carlo evidence fromthe projection pursuit regression approach. Econ Lett 75:11–16

    Article  MathSciNet  MATH  Google Scholar 

  • Silvapullé MJ (1985) Asymptotic behavior of robust estimators of regression and scale parameters with fixed carriers. Ann Stat 13(4):1490–1497

    Article  MathSciNet  MATH  Google Scholar 

  • Speckman P (1988) Kernel smoothing in partial linear models. J R Stat Soc Ser B 50:413–436

    MathSciNet  MATH  Google Scholar 

  • Tabakan G, Akdeniz F (2010) Difference-based ridge estimator of parameters in partial linear model. Stat Pap 51:357–368

    Article  MathSciNet  MATH  Google Scholar 

  • Tong XW, Cui HJ, Yu P (2008) Consistency and normality of Huber–Dutter estimators for partial linear model. Sci China Ser A 51(10):1831–1842

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Brown LD, Cai T (2011) A difference based approach to the semiparametric partial linear model. Electron J Stat 5:619–641

    Article  MathSciNet  MATH  Google Scholar 

  • Wu WB (2007) \(M\)-estimation of linear models with dependent errors. Ann Stat 35(2):495–521

    Article  MathSciNet  MATH  Google Scholar 

  • Yatchew A (1997) An elementary estimator for the partial linear model. Econ Lett 57:135–143

    Article  MathSciNet  MATH  Google Scholar 

  • Yatchew A (2003) Semiparametric regression for the applied econometrician. Cambridge Univerity Press, New York

    Book  MATH  Google Scholar 

  • Zhang T, Wang QH (2012) Semiparametric partially linear regression models for functional data. J Stat Plan Inference 142:2518–2529

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao H, You J (2011) Difference based estimation for partially linear regression models with measurement errors. J Multivar Anal 102:1321–1338

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou XC, Lin JG (2013) Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors. J Multivar Anal 122:251–270

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The first author’s work was supported by Natural Science Foundation of China (No. 11471105,11471223). The third author’s work was supported by Natural Science Foundation of China (No. 41374017). The authors thank the anonymous referees for their very valuable discussions and suggestions, which led to a great improvement of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China

    Hongchang Hu & Yu Zhang

  2. Faculty of Information Engineering, China University Geosciences, Wuhan, 430074, China

    Xiong Pan

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  1. Hongchang Hu
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  2. Yu Zhang
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  3. Xiong Pan
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Correspondence to Hongchang Hu.

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Hu, H., Zhang, Y. & Pan, X. Asymptotic normality of DHD estimators in a partially linear model. Stat Papers 57, 567–587 (2016). https://doi.org/10.1007/s00362-015-0666-2

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  • DOI: https://doi.org/10.1007/s00362-015-0666-2

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