Evaluating the Performance of Quantile Regression-Based Prediction Intervals
Keywords:
Quantile regression, Ordinary least squares, Prediction interval, Monte Carlo simulation, Interval scoreAbstract
This study compares the performance of prediction intervals from Ordinary Least Squares (OLS) and Quantile Regression (QR) in simple linear regression. Using Monte Carlo simulations, five error distributions were examined: the normal distribution (which satisfies the model assumptions) and four that violate the assumptions (skewed, heavy-tailed, extremely heavy-tailed, and heteroscedastic). Performance was evaluated using Coverage Probability (CP), Average Width (AW), and Average Interval Score (AIS).
The results showed that under the normal distribution, OLS achieved coverage close to the nominal level and provided lower AIS values across all conditions. In contrast, QR undercovered at small sample sizes but converged to the nominal level as the sample size increased. When assumptions were violated, QR outperformed OLS provided that the sample size was sufficiently large. Specifically, for skewed and extremely heavy-tailed errors, QR yielded a clearly lower AIS. Under heteroscedasticity, the coverage probability of OLS deviated significantly from the target level depending on the prediction point (x0), whereas QR adapted more effectively.
The findings suggest that selecting the appropriate method should take into account both the error distribution and the sample size. QR is a suitable alternative when OLS assumptions are violated, and the sample size is adequate.
References
Cade, B. S., & Noon, B. R. (2003). A gentle introduction to quantile
regression for ecologists. Frontiers in Ecology and the
Environment, 1(8), 412–420.
Casella, G., & Berger, R. L. (2002). Statistical Inference. 2nd ed., Duxbury.
Gneiting, T., & Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477), 359–378.
Greene, W. H. (2018). Econometric analysis. 8th ed. Pearson.
He, X. (1997). Quantile curves without crossing. The American Statistician, 51(2), 186–192.
Koenker, R. (2005). Quantile regression. Cambridge University Press.
Koenker, R., & Bassett, G. (1978). Regression quantiles. Econometrica, 46(1), 33–50.
Koenker, R., & Hallock, K. F. (2001). Quantile regression. Journal of Economic Perspectives, 15(4), 143–156.
Meinshausen, N. (2006). Quantile regression forests. Journal of Machine Learning Research, 7, 983–999.
Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression analysis. 5th ed., Wiley.
Olive, D. J. (2007). Prediction intervals for regression models. Computational Statistics & Data Analysis, 51(6), 3115–3122. https://doi.org/10.1016/j.csda.2006.02.006
R Core Team. (2025). R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. https://www.R-project.org/.
Schmoyer, R. L. (1992). Asymptotically valid prediction intervals for linear models. Technometrics, 34(4), 399–408.
Stine, R.A. (1985). Bootstrap prediction intervals for regression. Journal of the American Statistical Association, 80(392), 1026–1031.
Yang, M. C. K. & Mo, C. T. C. (1990). Coverage probability to the original data by prediction intervals in simple linear regression. Journal of Statistical Computation and Simulation, 36(4), 231–244.
Downloads
Published
Issue
Section
Categories
License
Copyright (c) 2026 Journal of Applied Science and Emerging Technology
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
