Linear Regression for Heavy Tails

Abstract

There exist several estimators of the regression line in the simple linear regression: Least Squares, Least Absolute Deviation, Right Median, Theil–Sen, Weighted Balance, and Least Trimmed Squares. Their performance for heavy tails is compared below on the basis of a quadratic loss function. The case where the explanatory variable is the inverse of a standard uniform variable and where the error has a Cauchy distribution plays a central role, but heavier and lighter tails are also considered. Tables list the empirical sd and bias for ten batches of one hundred thousand simulations when the explanatory variable has a Pareto distribution and the error has a symmetric Student distribution or a one-sided Pareto distribution for various tail indices. The results in the tables may be used as benchmarks. The sample size is n = 100 but results for n = ∞ are also presented. The error in the estimate of the slope tneed not be asymptotically normal. For symmetric errors, the symmetric generalized beta prime densities often give a good fit.

Keywords

• exponential generalized beta prime
• generalized beta prime
• hyperbolic balance
• least absolute deviation
• least trimmed squares
• Pareto distribution
• right median
• Theil–Sen
• weighted balance