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Direct use of regression quantiles to construct confidence sets in linear models. (English) Zbl 0853.62040

Summary: Direct use of the empirical quantile function provides a standard distribution-free approach to constructing confidence intervals and confidence bands for population quantiles. We apply this method to construct confidence intervals and confidence bands for regression quantiles and to construct prediction intervals based on sample regression quantiles. Comparison of the direct method with the studentization and the bootstrap methods are discussed. Simulation results show that the direct method has the advantage of robustness against departure from the normality assumption of the error terms.

MSC:

62G15 Nonparametric tolerance and confidence regions
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
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[1] BAHADUR, R. R. 1966. A note on quantiles in large samples. Ann. Math. Statist. 37 577 580. Z. · Zbl 0147.18805 · doi:10.1214/aoms/1177699450
[2] BASSETT, G. W. and KOENKER, R. W. 1982. An empirical quantile function for linear models with i.i.d. errors. J. Amer. Statist. Assoc. 77 407 415. Z. JSTOR: · Zbl 0493.62047 · doi:10.2307/2287261
[3] BASSETT, G. W. and KOENKER, R. W. 1986. Strong consistency of regression quantiles and related empirical processes. Econometric Theory 2 191 201. Z.
[4] BICKEL, P. J. 1975. One-step Huber estimates in the linear model. J. Amer. Statist. Assoc. 70 428 433. Z. JSTOR: · Zbl 0322.62038 · doi:10.2307/2285834
[5] BOFINGER, E. 1975. Estimation of a density function using order statistics. Austral. J. Statist. 17 1 7. Z. · Zbl 0346.62038 · doi:10.1111/j.1467-842X.1975.tb01366.x
[6] CSORGO, M. and REVESZ, P. 1984. Two approaches to constructing simultaneous confidence \" \' \' bounds for quantiles. Probab. Math. Statist. 4 221 236. Z. · Zbl 0591.62039
[7] EFRON, B. 1979. Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1 26. Z. · Zbl 0406.62024 · doi:10.1214/aos/1176344552
[8] FALK, M. and KAUFMANN, E. 1991. Coverage probabilities of bootstrap confidence intervals for quantiles. Ann. Statist. 19 485 495. Z. · Zbl 0725.62043 · doi:10.1214/aos/1176347995
[9] FELLER, W. 1966. An Introduction to Probability Theory and Its Applications. Wiley, New York. Z. · Zbl 0138.10207
[10] GUTENBRUNNER, C. and JURECKOVA, J. 1992. Regression rank-scores and regression quantiles. Ánn. Statist. 20 305 330. Z. · Zbl 0759.62015 · doi:10.1214/aos/1176348524
[11] GUTENBRUNNER, C., JURECKOVA, J., KOENKER, R. W. and PORTNOY, S. 1993. Tests of linear \' hy potheses based on regression rank scores. Journal of Nonparametric Statistics 2 307 331. Z. · Zbl 1360.62216 · doi:10.1080/10485259308832561
[12] HALL, P. 1992. The Bootstrap and Edgeworth Expansion. Springer, New York. Z. · Zbl 0744.62026
[13] HALL, P. and SHEATHER, S. J. 1988. On the distribution of a studentized quantile. J. Roy. Statist. Soc. Ser. B 50 381 391. Z. JSTOR: · Zbl 0674.62034
[14] JURECKOVA, J. 1977. Asy mptotic relations of M-estimates and R-estimates in linear regression ḿodel. Ann. Statist. 5 464 472. Z. · Zbl 0365.62034 · doi:10.1214/aos/1176343843
[15] KOENKER, R. W. 1994. Confidence intervals for regression quantiles. In Proceedings of the Fifth Z. Prague Sy mposium on Asy mptotic Statistics P. Mandl and M. Huskova, eds.. 349 359. physica, Heidelberg.
[16] KOENKER, R. W. and BASSETT, G. W. 1978. Regression quantiles. Econometrica 46 33 50. Z. JSTOR: · Zbl 0373.62038 · doi:10.2307/1913643
[17] KOENKER, R. W. and D’OREY, V. 1987. Computing regression quantiles. J. Roy. Statist. Soc. Ser. C 36 383 393. Z.
[18] KOENKER, R. W. and PORTNOY, S. 1987. L-estimation linear model. J. Amer. Statist. Assoc. 82 851 857. Z. JSTOR: · Zbl 0658.62078 · doi:10.2307/2288796
[19] PARZEN, M. I., WEI, L. J. and YING, Z. 1992. A resampling method based on pivotal estimating functions. Unpublished manuscript. Z. · Zbl 0807.62038
[20] PORTNOY, S. 1988. Asy mptotic behavior of regression quantiles in nonstationary, dependent cases. J. Multivariate Anal. 38 100 113. Z. · Zbl 0737.62078 · doi:10.1016/0047-259X(91)90034-Y
[21] PORTNOY, S. 1991. Asy mptotic number of regression quantile breakpoints. SIAM J. Sci. Statist. Comput. 12 867 883. Z. · Zbl 0736.62061 · doi:10.1137/0912047
[22] PORTNOY, S. and KOENKER, R. W. 1989. Adaptive L-estimation for linear models. Ann. Statist. 17 362 381. Z. · Zbl 0736.62060 · doi:10.1214/aos/1176347022
[23] RUPPERT, D. and CARROLL, R. J. 1980. Trimmed least squares estimation in the linear model. J. Amer. Statist. Assoc. 75 828 838. Z. · Zbl 0459.62055 · doi:10.2307/2287169
[24] SERFLING, R. 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York. Z. · Zbl 0538.62002
[25] ZHOU, K. Q. 1995. Quantile regression and survival analysis. Ph.D. dissertation, Univ. Illinois, Urbana Champaign.
[26] UNIVERSITY OF ILLINOIS, URBANA CHAMPAIGN 725 SOUTH WRIGHT STREET CHAMPAIGN, ILLINOIS 61820
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