## Bernoulli

• Bernoulli
• Volume 20, Number 3 (2014), 1532-1559.

### Asymptotics of nonparametric L-1 regression models with dependent data

#### Abstract

We investigate asymptotic properties of least-absolute-deviation or median quantile estimates of the location and scale functions in nonparametric regression models with dependent data from multiple subjects. Under a general dependence structure that allows for longitudinal data and some spatially correlated data, we establish uniform Bahadur representations for the proposed median quantile estimates. The obtained Bahadur representations provide deep insights into the asymptotic behavior of the estimates. Our main theoretical development is based on studying the modulus of continuity of kernel weighted empirical process through a coupling argument. Progesterone data is used for an illustration.

#### Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1532-1559.

Dates
First available in Project Euclid: 11 June 2014

https://projecteuclid.org/euclid.bj/1402488949

Digital Object Identifier
doi:10.3150/13-BEJ532

Mathematical Reviews number (MathSciNet)
MR3217453

Zentralblatt MATH identifier
06327918

#### Citation

Zhao, Zhibiao; Wei, Ying; Lin, Dennis K.J. Asymptotics of nonparametric L-1 regression models with dependent data. Bernoulli 20 (2014), no. 3, 1532--1559. doi:10.3150/13-BEJ532. https://projecteuclid.org/euclid.bj/1402488949

#### References

• [1] Andrews, D.W.K. (1984). Nonstrong mixing autoregressive processes. J. Appl. Probab. 21 930–934.
• [2] Andrews, D.W.K. and Pollard, D. (1994). An introduction to functional central limit theorems for dependent stochastic processes. Int. Stat. Rev. 62 119–132.
• [3] Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33–45.
• [4] Bhattacharya, P.K. and Gangopadhyay, A.K. (1990). Kernel and nearest-neighbor estimation of a conditional quantile. Ann. Statist. 18 1400–1415.
• [5] Brumback, B.A. and Rice, J.A. (1998). Smoothing spline models for the analysis of nested and crossed samples of curves. J. Amer. Statist. Assoc. 93 961–994.
• [6] Cai, Z. (2002). Regression quantiles for time series. Econometric Theory 18 169–192.
• [7] Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. 19 760–777.
• [8] Dedecker, J. and Prieur, C. (2005). New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132 203–236.
• [9] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Series in Statistics. New York: Springer.
• [10] Fan, J. and Zhang, J.T. (2000). Two-step estimation of functional linear models with applications to longitudinal data. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 303–322.
• [11] Hallin, M., Lu, Z. and Yu, K. (2009). Local linear spatial quantile regression. Bernoulli 15 659–686.
• [12] He, X., Fu, B. and Fung, W.K. (2003). Median regression for longitudinal data. Stat. Med. 22 3655–3669.
• [13] He, X., Zhu, Z.Y. and Fung, W.K. (2002). Estimation in a semiparametric model for longitudinal data with unspecified dependence structure. Biometrika 89 579–590.
• [14] Ho, H.C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 992–1024.
• [15] Honda, T. (2000). Nonparametric estimation of a conditional quantile for $\alpha$-mixing processes. Ann. Inst. Statist. Math. 52 459–470.
• [16] Hoover, D.R., Rice, J.A., Wu, C.O. and Yang, L.P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85 809–822.
• [17] Koenker, R. (2004). Quantile regression for longitudinal data. J. Multivariate Anal. 91 74–89.
• [18] Koenker, R. (2005). Quantile Regression. Econometric Society Monographs 38. Cambridge: Cambridge Univ. Press.
• [19] Koenker, R. and Bassett, G. Jr. (1978). Regression quantiles. Econometrica 46 33–50.
• [20] Li, Q. and Racine, J.S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton, NJ: Princeton Univ. Press.
• [21] Rice, J.A. and Silverman, B.W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 233–243.
• [22] Romano, J.P. and Wolf, M. (2000). A more general central limit theorem for $m$-dependent random variables with unbounded $m$. Statist. Probab. Lett. 47 115–124.
• [23] Shao, Q.M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098–2127.
• [24] Shao, X. and Wu, W.B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773–1801.
• [25] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
• [26] Truong, Y.K. and Stone, C.J. (1992). Nonparametric function estimation involving time series. Ann. Statist. 20 77–97.
• [27] Walker, E. and Wright, S.P. (2002). Comparing curves using additive models. J. Qual. Technol. 34 118–129.
• [28] Wang, H.J. and Fygenson, M. (2009). Inference for censored quantile regression models in longitudinal studies. Ann. Statist. 37 756–781.
• [29] Wang, H.J., Zhu, Z. and Zhou, J. (2009). Quantile regression in partially linear varying coefficient models. Ann. Statist. 37 3841–3866.
• [30] Wei, Y., Zhao, Z. and Lin, D.K.J. (2012). Profile control charts based on nonparametric $L-1$ regression methods. Ann. Appl. Stat. 6 409–427.
• [31] Wu, H. and Zhang, J.T. (2002). Local polynomial mixed-effects models for longitudinal data. J. Amer. Statist. Assoc. 97 883–897.
• [32] Wu, W.B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154 (electronic).
• [33] Wu, W.B. (2008). Empirical processes of stationary sequences. Statist. Sinica 18 313–333.
• [34] Wu, W.B. and Zhao, Z. (2007). Inference of trends in time series. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 391–410.
• [35] Yao, F., Müller, H.G. and Wang, J.L. (2005). Functional linear regression analysis for longitudinal data. Ann. Statist. 33 2873–2903.
• [36] Yu, K. and Jones, M.C. (1998). Local linear quantile regression. J. Amer. Statist. Assoc. 93 228–237.
• [37] Yu, K., Lu, Z. and Stander, J. (2003). Quantile regression: Applications and current research areas. The Statistician 52 331–350.