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Estimation of central shapes of error distributions in linear regression problems

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  • P. Lai
  • Stephen Lee

Abstract

Consider a linear regression model subject to an error distribution which is symmetric about 0 and varies regularly at 0 with exponent ζ. We propose two estimators of ζ, which characterizes the central shape of the error distribution. Both methods are motivated by the well-known Hill estimator, which has been extensively studied in the related problem of estimating tail indices, but substitute reciprocals of small L p residuals for the extreme order statistics in its original definition. The first method requires careful choices of p and the number k of smallest residuals employed for calculating the estimator. The second method is based on subsampling and works under less restrictive conditions on p and k. Both estimators are shown to be consistent for ζ and asymptotically normal. A simulation study is conducted to compare our proposed procedures with alternative estimates of ζ constructed using resampling methods designed for convergence rate estimation. Copyright The Institute of Statistical Mathematics, Tokyo 2013

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  • P. Lai & Stephen Lee, 2013. "Estimation of central shapes of error distributions in linear regression problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 105-124, February.
    • Handle: RePEc:spr:aistmt:v:65:y:2013:i:1:p:105-124
    DOI: 10.1007/s10463-012-0360-2
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    References listed on IDEAS

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    1. Rogers, Alan J., 2001. "Least Absolute Deviations Regression Under Nonstandard Conditions," Econometric Theory, Cambridge University Press, vol. 17(4), pages 820-852, August.
    2. Lai, P.Y. & Lee, Stephen M.S., 2005. "An Overview of Asymptotic Properties of Lp Regression Under General Classes of Error Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 446-458, June.
    3. L. De Haan & L. Peng, 1998. "Comparison of tail index estimators," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 52(1), pages 60-70, March.
    4. Hall, Peter, 1990. "Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 177-203, February.
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