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Article

Linear Regression for Heavy Tails

1
Department of Mathematics, Universiteit van Amsterdam, 1098xh Amsterdam, The Netherlands
2
RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
*
Author to whom correspondence should be addressed.
Received: 29 June 2018 / Revised: 18 August 2018 / Accepted: 21 August 2018 / Published: 10 September 2018
(This article belongs to the Special Issue Heavy Tailed Distributions in Economics)
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. View Full-Text
Keywords: exponential generalized beta prime; generalized beta prime; hyperbolic balance; least absolute deviation; least trimmed squares; Pareto distribution; right median; Theil–Sen; weighted balance exponential generalized beta prime; generalized beta prime; hyperbolic balance; least absolute deviation; least trimmed squares; Pareto distribution; right median; Theil–Sen; weighted balance
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MDPI and ACS Style

Balkema, G.; Embrechts, P. Linear Regression for Heavy Tails. Risks 2018, 6, 93. https://0-doi-org.brum.beds.ac.uk/10.3390/risks6030093

AMA Style

Balkema G, Embrechts P. Linear Regression for Heavy Tails. Risks. 2018; 6(3):93. https://0-doi-org.brum.beds.ac.uk/10.3390/risks6030093

Chicago/Turabian Style

Balkema, Guus, and Paul Embrechts. 2018. "Linear Regression for Heavy Tails" Risks 6, no. 3: 93. https://0-doi-org.brum.beds.ac.uk/10.3390/risks6030093

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