Abstract
We consider nonparametric estimation of conditional medians for time series data. The time series data are generated from two mutually independent linear processes. The linear processes may show long-range dependence. The estimator of the conditional medians is based on minimizing the locally weighted sum of absolute deviations for local linear regression. We present the asymptotic distribution of the estimator. The rate of convergence is independent of regressors in our setting. The result of a simulation study is also given.
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References
Beran J. (1994) Statistics for long-memory processes. Chapman & Hall, New York
Chaudhuri P. (1991) Nonparametric estimates of regression quantiles and their local Bahadur representation. The Annals of Statistics 19: 760–777
Chow Y.S., Teicher H. (1988) Probability theory, 2nd edn. Springer, New York
Csörgő S., Mielniczuk J. (2000) The smoothing dichotomy in random-design regression with long-memory errors based on moving averages. Statistica Sinica 10: 771–787
Doukhan, P. (1994). Mixing: properties and examples. In Lecture notes in statistics, vol. 85. New York: Springer.
Doukhan, P., Oppenheim, G., Taqqu M.S., (Eds). (2003) Theory and applications of long-range dependence. Birkhäser, Boston
Fan J., Gijbels I. (1996) Local polynomial modellingand its applications. Chapman & Hall, London
Fan J., Yao Q. (2003) Nonlinear time series. Springer, New York
Fan J., Hu T.-C., Truong Y.K. (1994) Robust nonparametric function estimation. Scandinavian Journal of Statistics 21: 867–885
Giraitis L., Koul H.L., Surgailis D. (1996) Asymptotic normality of regression estimators with long memory errors. Statistics & Probability Letters 29: 317–335
Guo H., Koul H.L. (2007) Nonparametric regression with heteroscedastic long memory errors. Journal of Statistical Planning and Inference 137: 379–404
Hall P., Peng L., Yao Q. (2002) Prediction and nonparametric estimation for time series with heavy tails. Journal of Time Series Analysis 23: 313–331
Härdle W., Müller M., Sperlich S., Werwatz A. (2004) Nonparametric and semiparametric models. Springer, Berlin
Hidalgo J. (1997) Non-parametric estimation with strongly dependent multivariate time series. Journal of Time Series Analysis 18: 95–122
Ho H.-C., Hsing T. (1996) On the asymptotic expansion of the empirical process of long-memory moving averages. The Annals of Statistics 24: 992–1024
Ho H.-C., Hsing T. (1997) Limit theorems for functionals of moving averages. The Annals of Probability 25: 1636–1669
Honda T. (2000) Nonparametric estimation of a conditional quantile for α-mixing processes. The Annals of the Institute of Statistical Mathematics 52: 459–470
Honda T. (2000) Nonparametric estimation of the conditional median function for long-range dependent processes. Journal of the Japan Statistical Society 30: 129–142
Honda, T. (2008). Nonparametric density estimation for linear processes with infinite variance. Forthcoming in The Annals of the Institute of Statistical Mathematics.
Koenker R. (2005) Quantile Regression. Cambridge University Press, New York
Koenker, R. (2008). Quantreg: Quantile Regression, R package version 4.17. http://www.r-project.org.
Koenker R., Basset G. (1978) Regression quantiles. Econometrica 46: 33–50
Koul, H. L. (2002). Weighted empirical processes in dynamic linear models, 2nd edn. In Lecture notes in statistics, vol. 85. New York: Springer.
Koul H.L., Mukherjee K. (1993) Asymptotics of R-, MD-, and LAD-estimators in linear regression models with long range dependent errors. Probability Theory and Related Fields 95: 535–553
Koul H.L., Surgailis D. (2001) Asymptotics of empirical processes of long memory moving averages with infinite variance. Stochastic Processes and their Application 91: 309–336
Koul H.L., Surgailis D. (2002) Asymptotic expansion of the empirical process of long memory moving averages. In: Dehling H., Mikosch T., Sørensen M.(eds) Empirical process techniques for dependent data. Birkhäuser, Boston, pp 213–239
Koul H.L., Baillie R.T., Surgailis D. (2004) Regression model fitting with a long memory covariate process. Econometric Theory 20: 485–512
Mielniczuk J., Wu W.B. (2004) On random design model with dependent errors. Statistica Sinica 14: 1105–1126
Nze P.A., Bühlman P., Doukhan P. (2002) Weak dependence beyond mixing and asymptotics for nonparametric regression. The Annals of Statistics 30: 397–430
Peng L., Yao Q. (2004) Nonparametric regression under dependent errors with infinite variance. The Annals of the Institute of Statistical Mathematics 56: 73–86
Pollard D. (1991) Asymptotics for least absolute deviation regression estimates. Econometric Theory 7: 186–98
R Development Core Team (2008). R: a Language and environment for statistical computing. Vienna: R roundation for statistical computing. http://www.R-project.org.
Robinson P.M. (1997) Large-sample inference for nonparametric regression with dependent errors. The Annals of Statistics 25: 2054–2083
Robinson, P.M. (eds) (2003) Time series with long memory. Oxford University Press, New York
Taniguchi M., Kakizawa Y. (2000) Asymptotic Theory of Statistical Inference for Time Series. Springer, New York
Truong Y.K., Stone C.J. (1992) Nonparametric function estimation involving time series. The Annals of Statistics 20: 77–97
Wu W.B. (2003) Empirical processes of long-memory sequences. Bernoulli 9: 809–831
Wu W.B., Mielniczuk J. (2002) Kernel density estimation for linear processes. The Annals of Statistics 30: 1441–1459
Zhao, Z., Wu, W. B. (2006). Kernel quantile regression for nonlinear stochastic models. In Technical report no.572, Department of Statistics, The University of Chicago.
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Honda, T. Nonparametric estimation of conditional medians for linear and related processes. Ann Inst Stat Math 62, 995–1021 (2010). https://doi.org/10.1007/s10463-008-0203-3
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DOI: https://doi.org/10.1007/s10463-008-0203-3