• Bernoulli
  • Volume 5, Number 4 (1999), 659-676.

Regression rank-scores tests against heavy-tailed alternatives

Jana Jureckova

Full-text: Open access


Statistical inference in the linear model based on the concept of regression rank scores is invariant to the nuisance regression; hence regression rank-scores tests need no estimation of the nuisance parameters. Such tests, already available in the literature, are manageable, asymptotically distribution-free and have many convenient properties, but they are either censored or applicable only to light-tailed distributions. To extend the universality of regression rank-scores tests, we propose modified tests applicable to heavy-tailed distributions including Cauchy. Depending on the alternative we want to treat by the test, we censor the score generating function but the censoring is asymptotically negligible. The proposed tests, being asymptotically distribution-free, are as efficient as the ordinary rank tests without nuisance parameters, for a broad class of density shapes.

Article information

Bernoulli, Volume 5, Number 4 (1999), 659-676.

First available in Project Euclid: 19 February 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

linear regression model regression quantile regression rank scores regression rank test


Jureckova, Jana. Regression rank-scores tests against heavy-tailed alternatives. Bernoulli 5 (1999), no. 4, 659--676.

Export citation


  • [1] Adichie, J.N. (1984) Rank tests in linear models. In P.R. Krishnaiah and P.K. Sen (eds), Handbook of Statistics, Vol. 4, pp. 229-257. New York: Elsevier Science.
  • [2] Gutenbrunner, C. (1986) Zur Asymptotik von Regressions-Quantil-Prozessen und daraus abgeleiten Statistiken. PhD thesis, Universität Freiburg.
  • [3] Gutenbrunner, C. (1994) Tests for heteroscedasticity based on regression quantiles and regression rank scores. In P. Mandl and M. Husková, (eds), Asymptotic Statistics, pp. 249-260. Heidelberg: Physica-Verlag.
  • [4] Gutenbrunner, C. and Jurecková, J. (1992) Regression rank scores and regression quantiles. Ann. Statist., 20, 305-330.
  • [5] Gutenbrunner, C., Jurecková, J., Koenker, R. and Portnoy, S. (1993) Tests of linear hypotheses based on regression rank scores. J. Nonparametr. Statist., 2, 307-331.
  • [6] Hájek, J. (1961) Some extensions of the Wald-Wolfowitz-Noether theorem. Ann. Math. Statist., 32, 506-523.
  • [7] Hájek, J. (1962) Asymptotically most powerful rank order tests. Ann. Math. Statist., 33, 1124-1147.
  • [8] Hájek, J. (1965) Extension of the Kolmogorov-Smirnov test to the regression alternatives. In L. Le Cam and J. Neyman (eds), Bernoulli (1713) - Bayes (1763) - Laplace (1813): Proceedings of an International Research Seminar, pp. 45-60. Berlin: Springer-Verlag.
  • [9] Hallin, M. and Jurecková, J. (1996) Optimal tests for autoregressive models based on autoregression rank scores. Submitted.
  • [10] Hallin, M., Zahaf, T., Jurecková, J., Kalvová, J. and Picek, J. (1997) Non-parametric tests in AR models with application to climatic data. Environmetrics, 8, 651-660.
  • [11] Hallin, M., Jurecková, J., Picek, J. and Zahaf, T. (1999) Nonparametric tests of independence of two autoregressive time series based on autoregression rank scores. J. Statist. Plann. Inference, 75, 319-330.
  • [12] Hettmansperger, T.P. (1984) Statistical Inference Based on Ranks. New York: Wiley.
  • [13] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 56, 13-30.
  • [14] Jurecková, J. (1992) Tests of Kolmogorov-Smirnov type based on regression rank scores. In J.A. Visek, (ed.), Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, pp. 41-49. Prague: Academia, and Amsterdam: Kluwer.
  • [15] Koenker, R. and Bassett, G. (1978) Regression quantiles. Econometrica, 46, 466-476.
  • [16] Koenker, R. and d'Orey, V. (1987) Algorithm AS 229: Computing regression quantiles. Appl. Statist., 36, 383-393.
  • [17] Koenker, R. and d'Orey, V. (1994) A remark on algorithm AS 229: Computing the dual regression quantiles and regression rank scores. Appl. Statist., 43, 410-414.
  • [18] Koul, H.L. and Saleh, A.K.Md.E. (1995) Autoregression quantiles and related rank scores processes. Ann. Statist., 23, 670-689.
  • [19] Osborne, M.R. (1992) An effective method for computing regression quantiles. IMA J. Numer. Anal., 12, 151-166.
  • [20] Pollard, D. (1991) Asymptotics for least absolute deviation regression estimation. Econom. Theory, 7, 186-200.
  • [21] Picek, J. (1997) Tests in linear models based on regression rank scores. PhD thesis (in Czech), Charles University, Prague.
  • [22] Puri, M.L. and Sen, P.K. (1985) Nonparametric Methods in General Linear Models. New York: Wiley.
  • [23] Steel, R.G.D. and Torrie, J.H. (1960) Principles and Procedures of Statistics. New York: McGraw-Hill.