Quantile regression using asymmetric Laplace distribution

비대칭 라플라스 분포를 이용한 분위수 회귀

  • Published : 2009.11.30


Quantile regression has become a more widely used technique to describe the distribution of a response variable given a set of explanatory variables. This paper proposes a novel modelfor quantile regression using doubly penalized kernel machine with support vector machine iteratively reweighted least squares (SVM-IRWLS). To make inference about the shape of a population distribution, the widely popularregression, would be inadequate, if the distribution is not approximately Gaussian. We present a likelihood-based approach to the estimation of the regression quantiles that uses the asymmetric Laplace density.


  1. Basset, G. and Koenker, R. (1982). An empirical quantile function for linear models with iid errors. Journal of the American Statistical Association, 77, 407-415. https://doi.org/10.2307/2287261
  2. He, X. (1997). Quantile curves without crossing. The American Statistician, 51, 186-192. https://doi.org/10.2307/2685417
  3. Heagerty, P. J. and Pepe, M. S. (1999). Semiparametric estimation of regression quantiles with application to standardizing weight for height and age in US children. Applied Statistics, 48, 533-551.
  4. Hwang, C. and Shim, J. (2005). A simple quantile regression via support vector machine. Lecture Notes in Computer Science, 3610, 512-520.
  5. Kimeldorf, G. S. and Wahba, G. (1971). Some results on Tchebycheffian spline functions. Journal of Mathematical Analysis and its Applications, 33, 82-95. https://doi.org/10.1016/0022-247X(71)90184-3
  6. Koenker, R. (2005). Quantile regression, Cambridge University Press, London.
  7. Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33-50. https://doi.org/10.2307/1913643
  8. Mercer, J. (1909). Functions of positive and negative and their connection with the theory of integral equations. Philosphical Transactions of the Royal Society, A, 415-446.
  9. Powell, J. L. (1986). Censored regression quantiles. Journal of Econometrics, 32, 143-155. https://doi.org/10.1016/0304-4076(86)90016-3
  10. Shim, J., Hwang, C. and Seok, K. (2009). Non-crossing quantile regression via doubly penalized kernel machine. Computational Statistics, 24, 83-94. https://doi.org/10.1007/s00180-008-0123-y
  11. Shim, J., Park, H. and Hwang, C. (2009). A kernel machine for estimation of mean and volatility functions. Journal of the Korean Data & Information Science Society, 20, 905-912.
  12. Shim, J., Park, H. and Seok, K. (2009). Variance function estimation with LS-SVM for replicated data. Journal of the Korean Data & Information Science Society, 20, 925-931.
  13. Smola, A. J. and Scholkopf, B. (1998). A tutorial on support vector regression. NeuroCOLT2 Technical Report, NeuroCOLT.
  14. Sohn, I., Kim, S., Hwang, C., Lee, J. W. and Shim, J. (2008). Support vector machine quantile regression for detecting differentially expressed genes in microarray analysis. Methods of Information in Medicine, 47, 459-467.
  15. Vapnik, V. N. (1998). Statistical learning theory, Springer.
  16. Weiss, A. (1991). Estimating nonlinear dynamic models using least absolute error estimation. Econometric Theory, 7, 46-68. https://doi.org/10.1017/S0266466600004230
  17. White, H. (1992). Nonparametric estimation of conditional quantile using neural networks, in H. White, eds., Artificial Neural Networks: Approximation and Learning Theory, Blackwell, Oxford, 191-205.
  18. Xiang, D. and Wahba, G. (1996). A generalized approximate cross validation for smoothing splines with non-Gaussian data. Statistica Sinica, 6, 675-692.
  19. Yuan, M. and Wahba, G. (2004). Doubly penalized likelihood estimator in heteroscedastic regression. Statistics and Probability Letter , 69, 11-20.