Abstract
In the paper, an exponential inequality for widely orthant dependent random variables is established without bounded condition. By using the inequality, we further investigate the strong linear representation for the M estimator of the regression parameter vector in linear regression models with widely orthant dependent random errors under some general conditions. In addition, we have conducted comprehensive simulation studies to demonstrate the validity of obtained theoretical results.
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References
Chen PY, Sung SH (2017) A Bernstein type inequality for NOD random variables and applications. J Math Inequal 11(2):455–467
Chen XR, Wu YH (1988) Strong consistency of \(M\)-estimates in linear models. J Multivar Anal 27:116–130
Chen XR, Zhao LC (1996) \(M\)-Methods in Linear Model. Scientific and Technical Publishers, Shanghai
Cheng CS, Li KC (1984) The strong consistency of \(M\)-estimators in linear models. J Multivar Anal 15:91–98
Chen W, Wang YB, Cheng DY (2016) An inequality of widely dependent random variables and its applications. Lith Math J 56(1):16–31
Cui H, He X, Ng KW (2004) \(M\)-estimation for linear models with spatially-correlated errors. Stat Probab Lett 66:383–393
Fan J (2012) Moderate deviations for \(M\)-estimators in linear models with \(\phi \)-mixing errors. Acta Math Sin Engl Ser 28:1275–1294
Fan J, Yan A, Xiu N (2014) Asymptotic properties for \(M\)-estimators in linear models with dependent random errors. J Stat Plann Inference 148:49–66
He W, Cheng DY, Wang YB (2013) Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables. Stat Probab Lett 83:331–338
He XM, Shao QM (1996) A general bahadur representation of \(M\) Estimator and its application to linear regression with nonstochastic designs. Ann Stat 24(6):2608–2630
Hu TZ (2000) Negatively superadditive dependence of random variables with applications. Chin J Appl Probab Stat 16:133–144
Huber PJ (1973) Robust regression: asymptotics, conjectures and Monte Carlo. Ann Stat 1(5):799–821
Joag-Dev K, Proschan F (1983) Negative association of random variables with applications. Ann Stat 11(1):286–295
Liu XJ, Gao QW, Wang YB (2012) A note on a dependent risk model with constant interest rate. Stat Probab Lett 82:707–712
Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York
Prakasa Rao BLS (1981) Asymptotic behavior of \(M\)-estimators for the linear model with dependent errors. Bull Inst Math Sin 9:367–375
Rao CR, Zhao LC (1992) Linear representation of \(M\)-estimates in linear models. Canad J Stat 20(4):359–368
Shen AT (2013) Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr Appl Anal 2013:862602
Shen AT, Yao M, Wang WJ, Volodin A (2016) Exponential probability inequalities for WNOD random variables and their applications. RACSAM 110(1):251–268
Wang XJ, Hu SH (2015) The consistency of the nearest neighbor estimator of the density function based on WOD samples. J Math Anal Appl 429(1):497–512
Wang XJ, Si ZY (2015) Complete consistency of the estimator of nonparametric regression model under ND sequence. Stat Pap 56(3):585–596
Wang K, Wang Y, Gao Q (2013) Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol Comput Appl Probab 15(1):109–124
Wang XJ, Xu C, Hu TC, Volodin A, Hu SH (2014) On complete convergence for widely orthant-dependent random variables and its applications in nonparametrics regression models. Test 23(3):607–629
Wu QY, Jiang YY (2011) The strong consistency of \(M\) estimator in linear model for negatively dependent random samples. Commun Stat Theory Methods 40:476–491
Wu WB (2007) \(M\)-estimation of linear models with dependent errors. Ann Stat 35:495–521
Yang WZ, Xu HY, Chen L, Hu SH (2016) Complete consistency of estimators for regression models based on extended negatively dependent errors. Stat Pap. https://doi.org/10.1007/s00362-016-0771-x
Yohai VJ, Maronna RA (1979) Asymptotic behavior of \(M\)-estimators for the linear model. Ann Stat 7:258–268
Zhao LC (2002) Strong consistency of \(M\)-estimates in linear model. Sci China Ser A 45:1420–1427
Zhao LC, Rao CR, Chen XR (1993) A note on the consistency of \(M\)-estimates in linear models. In: Cambanis S, Ghosh JK, Karandikar RL et al (eds) Stochastic process, a Festschrift in honour of Gopinath Kallianpur. Springer, New York, pp 359–367
Acknowledgements
The study was supported by the National Natural Science Foundation of China (11671012, 11501004, 11501005), the Natural Science Foundation of Anhui Province (1508085J06), the Key Projects for Academic Talent of Anhui Province (gxbjZD2016005), The Project on Reserve Candidates for Academic and Technical Leaders of Anhui Province (2017H123) and the Science Research Project of Anhui Colleges (KJ2017A027).
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Deng, X., Wang, X. An exponential inequality and its application to M estimators in multiple linear models. Stat Papers 61, 1607–1627 (2020). https://doi.org/10.1007/s00362-018-0994-0
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DOI: https://doi.org/10.1007/s00362-018-0994-0
Keywords
- Exponential inequality
- Strong linear representation
- M estimator
- Linear regression model
- Widely orthant dependent random variables