Regression rank scores in nonlinear models
Jana Jurečková
Abstract
Consider the nonlinear regression model
Yi=g(xi, θ)+ei, i=1, …, n
with xi∈ℝk, θ=(θ0, θ1, …, θp)′∈Θ (compact in ℝp+1), where g(x, θ)=θ0+g̃(x, θ1, …, θp) is continuous, twice differentiable in θ and monotone in components of θ. Following Gutenbrunner and Jurečková (1992) and Jurečková and Procházka (1994), we introduce regression rank scores for model (1), and prove their asymptotic properties under some regularity conditions. As an application, we propose some tests in nonlinear regression models with nuisance parameters.
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Primary Subjects: 62G08
Secondary Subjects: 62J02
Keywords: nonlinear regression; regression quantile; regression rank scores
Full-text: Open access
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Permanent link to this document: http://projecteuclid.org/euclid.imsc/1207058272
Digital Object Identifier: doi:10.1214/193940307000000121
References
[1] Chen, L.-A. (1987). Regression quantiles and trimmed least squares estimator for structural equations and nonlinear models. Ph.D. thesis, Univ. Illinois at Urbana–Champaign.
[2] El-Attar, R., Vidyasagar, M. and Gupta, S. (1979). An algorithm for ℓ1 norm minimization with an application to nonlinear ℓ1 approximation. SIAM J. Numer. Anal. 16 70–86.
[3] Gutenbrunner, C. and Jurečková, J. (1992). Regression rank scores and regression quantiles. Ann. Statist. 20 305–330.
[4] Gutenbrunner, C. and Jurečková, J., Koenker, R. and Portnoy, S. (1993). Tests of linear hypotheses based on regression rank scores. J. Nonparametr. Statist. 2 307–331.
[5] Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. Academic Press, New York.
[6] Hallin, M. and Jurečková, J. (1999). Optimal tests for autoregressive models based on autoregression rank scores. Ann. Statist. 27 1385–1414.
[7] Hallin, M., J. Jurečková, J. Kalvová, J. Picek and Zahaf, T. (1997). Nonparametric tests in AR models with applications to climatic data. Environmetrics 8 651–660.
[8] Hallin, M., Jurečková, J. Picek, J. and Zahaf, T. (1999). Nonparametric of independence of two autoregressive time series based on autoregression rank scores. J. Statist. Plann. Inference 75 310–330.
[9] Jurečková, J. (1999). Regression rank scores tests against heavy-tailed alternatives. Bernoulli 5 659–676.
[10] Jurečková, J. and Procházka, B. (1994). Regression quantiles and trimmed least squares estimator in nonlinear regression model. J. Nonparametr. Statist. 3 201–222.
[11] Jurečková, J. and Sen, P. K. (1996). Robust Statistical Procedures: Asymptotics and Inter-Relations. Wiley, New York.
[12] Kalvová, J., Jurečková, J., Picek, J. and Nemešová, I. (2000). On the order of autoregressive (AR) model in temperature series. Meteor. Čas. 3 19–23.
[13] Koenker, R. (2005). Quantile Regression Cambridge Univ. Press.
Mathematical Reviews (MathSciNet):
MR2268657
[14] Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33–50.
Mathematical Reviews (MathSciNet):
MR474644
Digital Object Identifier: doi:10.2307/1913643
JSTOR: links.jstor.org
[15] Koenker, R. and Park, B. J. (1996). An interior point algorithm for nonlinear quantile regression. J. Econometrics 71 265–283.
Mathematical Reviews (MathSciNet):
MR1381085
Zentralblatt MATH:
0855.62030
Digital Object Identifier: doi:10.1016/0304-4076(96)84507-6
[16] Koul, H. L. (1996). Asymptotics of some estimators and sequential residual empiricals in nonlinear time series models. Ann. Statist. 24 380–404.
Mathematical Reviews (MathSciNet):
MR1389897
Zentralblatt MATH:
0865.62065
Digital Object Identifier: doi:10.1214/aos/1033066216
Project Euclid: euclid.aos/1033066216
[17] Koul, H. L. and Saleh, A. K. Md. E. (1996). Autoregression quantiles and related rank-scores processes. Ann. Statist. 23 670–689.
Mathematical Reviews (MathSciNet):
MR1332587
Zentralblatt MATH:
0848.62047
Digital Object Identifier: doi:10.1214/aos/1176324541
Project Euclid: euclid.aos/1176324541
[18] Mukherjee, K. (1999). Asymptotics of quantiles and rank scores in nonlinear time series. J. Time Ser. Anal. 20 173-192.
Mathematical Reviews (MathSciNet):
MR1701053
Zentralblatt MATH:
0938.62101
Digital Object Identifier: doi:10.1111/1467-9892.00132
[19] Schindler, M. (2006). Tests based on regression rank scores in nonlinear models and their computation. WDS’06 1 94–99. MATFYZPRESS Prague.
[20] Schindler, M. (2007). Inference based on regression rank scores. Ph.D. thesis, Charles Univ., Prague.
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