Pointwise Estimation of Density of Heteroscedastistic Response in Regression

Hyun, Ji-Hoon;Kim, Si-Won;Lee, Sung-Dong;Byun, Wook-Jae;Son, Mi-Kyoung;Kim, Choong-Rak

  • Received : 2011.11.24
  • Accepted : 2012.01.16
  • Published : 2012.02.29


In fitting a regression model, we often encounter data sets which do not follow Gaussian distribution and/or do not have equal variance. In this case estimation of the conditional density of a response variable at a given design point is hardly solved by a standard least squares method. To solve this problem, we propose a simple method to estimate the distribution of the fitted vales under heteroscedasticity using the idea of quantile regression and the histogram techniques. Application of this method to a real data sets is given.


Conditional distribution function;heteroscedasticity;histogram;quantile regression


  1. Cole, T. J. and Green, P. J. (1992). Smoothing reference centile curves: The LMS method and penalized likelihood, Statistics in Medicine, 11, 1305-1319.
  2. Copas, J. B. (1995). Local likelihood based on kernel censoring, Journal of the Royal Statistical Society, Series B, 57, 221-235.
  3. Hall, P. and Presnell, B. (1997). Intentionally Biased Bootstrap Methods, unpublished manuscript.
  4. Hall, P., Wolff, R. C. and Yao, Q. (1999). Methods for estimating a conditional distribution, Journal of the American Statistical Association, 94, 154-163.
  5. 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, Journal of the Royal Statistical Society (Applied Statistics), 48, 533-551.
  6. Hendricks, W. and Koenker, R. (1992). Hierarchical spline models for conditional quantiles and the demand for electricity, Journal of the American Statistical Association, 87, 58-68.
  7. Hjort, N. L. and Jones, M. C. (1996). Locally parametric nonparametric density estimation, Annals of Statistics, 24, 1619-1647.
  8. Koenker, R. (2005). Quantile Regression, Cambridge, U.K., Cambridge University Press.
  9. Koenker, R. and Bassett, G. (1978). Asymptotic theory of least absolute error regression, Journal of the American Statistical Association, 73, 618-622.
  10. Koenker, R. and Hallock, K. F. (2001). Quantile regression, The Journal of Economic Perspectives, 15, 143-156.
  11. Loader, C. R. (1996). Local likelihood density estimation, Annals of Statistics, 24, 1602-1618.
  12. Portnoy, S. and Koenker, R. (1997). The Gaussian hare and the Laplacian tortoise: Computability of squared-error versus absolute-error estimators (with discussion), Statistical Science, 12, 279-300.
  13. R Development Core Team (2011). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL
  14. Ruppert, D., Sheather, S. J. and Wand, M. P. (1995). An effective bandwidth selector for local least squares regression, Journal of the American Statistical Association, 90, 1257-1270.
  15. Yu, K. and Jones, M. C. (1998). Local linear regression quantile estimation, Journal of the American Statistical Association, 93, 228-238.
  16. Yu, K., Lu, Z. and Stander, J. (2003). Quantile regression: Applications and current research areas, Journal of the Royal Statistical Society, 52, 331-350.


Supported by : National Research Foundation of Korea(NRF)