Bernoulli

  • Bernoulli
  • Volume 15, Number 3 (2009), 659-686.

Local linear spatial quantile regression

Marc Hallin, Zudi Lu, and Keming Yu

Full-text: Open access

Abstract

Let {(Yi, Xi), i∈ℤN} be a stationary real-valued (d+1)-dimensional spatial processes. Denote by xqp(x), p∈(0, 1), x∈ℝd, the spatial quantile regression function of order p, characterized by P{Yiqp(x)|Xi=x}=p. Assume that the process has been observed over an N-dimensional rectangular domain of the form $\mathcal{I}_{\mathbf{n}}:=\{\mathbf{i}=(i_{1},\ldots,i_{N})\in\mathbb{Z}^{N}\vert1\leq i_{k}\leq n_{k},k=1,\ldots,N\}$, with n=(n1, …, nN)∈ℤN. We propose a local linear estimator of qp. That estimator extends to random fields with unspecified and possibly highly complex spatial dependence structure, the quantile regression methods considered in the context of independent samples or time series. Under mild regularity assumptions, we obtain a Bahadur representation for the estimators of qp and its first-order derivatives, from which we establish consistency and asymptotic normality. The spatial process is assumed to satisfy general mixing conditions, generalizing classical time series mixing concepts. The size of the rectangular domain $\mathcal{I}_{\mathbf{n}}$ is allowed to tend to infinity at different rates depending on the direction in ℤN (non-isotropic asymptotics). The method provides much richer information than the mean regression approach considered in most spatial modelling techniques.

Article information

Source
Bernoulli Volume 15, Number 3 (2009), 659-686.

Dates
First available in Project Euclid: 28 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.bj/1251463276

Digital Object Identifier
doi:10.3150/08-BEJ168

Mathematical Reviews number (MathSciNet)
MR2555194

Zentralblatt MATH identifier
05815950

Keywords
Bahadur representation local linear estimation random fields quantile regression

Citation

Hallin, Marc; Lu, Zudi; Yu, Keming. Local linear spatial quantile regression. Bernoulli 15 (2009), no. 3, 659--686. doi:10.3150/08-BEJ168. https://projecteuclid.org/euclid.bj/1251463276.


Export citation

References

  • [1] Altman, N. (2000). Krige, smooth, both or neither? Aust. N. Z. J. Stat. 42 441–461.
  • [2] Anselin, L. and Florax, R.J.G.M. (1995). New Directions in Spatial Econometrics. Berlin: Springer.
  • [3] Besag, J.E. (1974). Spatial interaction and the statistical analysis of lattice systems. J. Roy. Statist. Soc. Ser. B 36 192–225.
  • [4] Bhattacharya, P.K. and Gangopadhyay, A. (1990). Kernel and nearets neighbor estimation of a conditional quantile. Ann. Statist. 18 1400–1415.
  • [5] Bolthausen, E. (1982). On the central limit theorem for stationary random fields. Ann. Probab. 10 1047–1050.
  • [6] Bowman, A.W. and Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: The Kernel Approach With S-Plus Illustrations. Oxford: Oxford Univ. Press.
  • [7] Cai, Z.W. (2002). Regression quantiles for time series. Econometric Theory 18 169–192.
  • [8] Carbon, M., Hallin, M. and Tran, L.T. (1996). Kernel density estimation for random fields: The L1 theory. J. Nonparamet. Stat. 6 157–170.
  • [9] Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. 19 760–777.
  • [10] Cleveland, W.S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74 829–836.
  • [11] Cressie, N.A.C. (1993). Statistics for Spatial Data. New York: Wiley.
  • [12] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
  • [13] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The L1 View. New York: Wiley.
  • [14] Efron, B. (1991). Regression percentiles using asymmetric squared error loss. Statist. Sinica 1 93–125.
  • [15] Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 998–1004.
  • [16] Fan, J. and Gijbels, I. (1996). Local Polynomial Modeling and Its Applications. London: Chapman & Hall.
  • [17] Fan, J., Hu, T.C. and Truong, Y.K. (1994), Robust nonparametric function estimation. Scand. J. Statist. 21 433–446.
  • [18] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. New York: Springer.
  • [19] Gao, J., Lu, Z. and Tjøstheim, D. (2006). Estimation in semi-parametric spatial regression. Ann. Statist. 34 1395–1435.
  • [20] Guyon, X. (1987). Estimation d’un champ par pseudo-vraisemblable conditionelle: étude asymptotique et application au cas Markovien. In Spatial Processes and Spatial Time Series Analysis, Proceedings of the 6th Franco-Belgian Meeting of Statisticians (J.-J. Droesbeke et al., eds.). Brussels: FUSL.
  • [21] Guyon, X. (1995). Random Fields on a Network. New York: Springer Verlag.
  • [22] Hallin, M., Lu, Z. and Tran, L.T. (2001). Density estimation for spatial linear processes. Bernoulli 7 657–668.
  • [23] Hallin, M., Lu, Z. and Tran, L.T. (2004). Density estimation for spatial processes: The L1 theory. J. Multivariate Anal. 88 61–75.
  • [24] Hallin, M., Lu, Z. and Tran, L.T. (2004). Local linear spatial regression. Ann. Statist. 32 2469–2500.
  • [25] Hansen, B. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24 726–748.
  • [26] He, X. and Shao, Q.M. (1996). A general Bahadur representation of M-estimators and its application to linear regression with nonstochatic designs. Ann. Statist. 24 2608–2630.
  • [27] Honda T. (2000). Nonparametric estimation of a conditional quantile for α-mixing processes. Ann. Inst. Statist. Math. 52 459–470.
  • [28] Koenker, R. (2005). Quantile Regression. Cambridge, U.K.: Cambridge Univ. Press.
  • [29] Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33–50.
  • [30] Koenker, R. and Mizera, I. (2004). Penalized triograms: Total variation regularization for bivariate smoothing. J. Roy. Statist. Soc. Ser. B 66 145–164.
  • [31] Koenker, R. and Portnoy, S. (1987). L-estimation for linear models. J. Amer. Statist. Assoc. 82 851–857.
  • [32] Koenker, R. and Zhao, Q. (1996). Conditional quantile estimation and inference for ARCH models. Econometric Theory 12 793–813.
  • [33] Koul, H.L. and Mukherjee, K. (1994). Regression quantiles and related processes under long-range dependence. J. Multivariate Anal. 51 318–337.
  • [34] Loader, C. (1999). Local Regression and Likelihood. New York: Springer.
  • [35] Lu, Z. and Chen, X. (2002). Spatial nonparametric regression estimation: non-isotropic case. Acta Math. Appl. Sin. 18 641–656.
  • [36] Lu, Z. and Chen, X. (2004). Spatial kernel regression estimation: weak consistency. Statist. Probab. Lett. 68 125–136.
  • [37] Lu, Z., Hui, Y.V. and Zhao, Q. (1998). Local linear quantile regression under dependence: Bahadur representation and application. Discussion paper, Dept. Management Sciences, City Univ. Hong Kong.
  • [38] Lu, Z., Lundervold, A., Tjøstheim, D. and Yao, Q. (2007). Exploring spatial nonlinearity using additive approximation. Bernoulli 13 447–472.
  • [39] Nakhapetyan, B.S. (1980). The central limit theorem for random fields with mixing conditions. In Advances in Probability 6, Multicomponent Systems (R.L. Dobrushin and Ya.G. Sinai, eds.) 531–548. New York: Dekker.
  • [40] Neaderhouser, C.C. (1980). Convergence of blocks spins defined on random fields. J. Stat. Phys. 22 673–684.
  • [41] Portnoy, S.L. (1991). Asymptotic behavior of regression quantiles in non-stationary dependent cases. J. Multivariate Anal. 38 100–113.
  • [42] Possolo, A. (1991). Spatial Statistics and Imaging. Hayward: Institute of Mathematical Statistics.
  • [43] Ripley, B. (1981). Spatial Statistics. New York: Wiley.
  • [44] Ruppert, D. and Carroll, R.J. (1980). Trimmed least square estimation in the linear model. J. Amer. Statist. Assoc. 75 828–838.
  • [45] Ruppert, D. and Wand, M.P. (1994). Multivariate locally weighted least squares regression. Ann. Statist. 22 1346–1370.
  • [46] Stone, C.J. (1977). Consistent nonparametric regression. Ann. Statist. 5 595–620.
  • [47] Takahata, H. (1983). On the rates in the central limit theorem for weakly dependent random fields. Z. Wahrsch. Verw. Gebiete 64 445–456.
  • [48] Tran, L.T. (1990). Kernel density estimation on random fields. J. Multivariate Anal. 34 37–53.
  • [49] Tran, L.T. and Yakowitz, S. (1993). Nearest neighbor estimators for random fields. J. Multivariate Anal. 44 23–46.
  • [50] Welsh, A.H. (1996). Robust estimation of smooth regression and spread functions and their derivatives. Statist. Sinica 6 347–366.
  • [51] Whittle, P. (1954). On stationary processes in the plane. Biometrika 41 434–449.
  • [52] Whittle, P. (1963). Stochastic process in several dimensions. Bulletin of the International Statistical Institute 40 974–985.
  • [53] Yu, K. and Jones, M.C. (1997). A comparison of local constant and local linear regression quantile estimation. Comput. Statist. Data Anal. 25 159–166.
  • [54] Yu, K. and Jones, M.C. (1998). Local linear quantile regression. J. Amer. Statist. Assoc. 93 228–237.
  • [55] Yu, K. and Lu, Z. (2004). Local linear additive quantile regression. Scand. J. Statist. 31 333–346.
  • [56] Zhang, H. and Zimmerman, D.L. (2005). Towards reconciling two asymptotic frameworks in spatial statistics. Biometrika 92 921–936.