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.
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References
Arcones M. A. (1998) L p -estimators as estimates of a parameter of location for a sharp-pointed symmetric density. Scandinavian Journal of Statistics 25: 693–715
Arcones M. A. (1999) Rates of convergence of L p -estimators for a density with an infinite cusp. Journal of Statistical Planning and Inference 81: 33–50
Beirlant J., Vynckier P., Teugels J. L. (1996) Tail index estimation, Pareto quantile plots, and regression diagnostics. Journal of the American Statistical Association 91: 1659–1667
Caers J., van Dyck J. (1999) Nonparametric tail estimation using a double bootstrap method. Computational Statistics & Data Analysis 29: 191–211
Cheng S., Pan J. (1998) Asymptotic expansions of estimators for the tail index with applications. Scandinavian Journal of Statistics 25: 717–728
De Haan L., Peng L. (1998) Comparison of tail index estimators. Statistica Neerlandica 52: 60–70
De Haan L., Stadtmüller U. (1996) Generalized regular variation of second order. Journal of the Australian Mathematical Society Series A 61: 381–395
Geluk J. L., de Haan L. (1987) Regular variation, extensions and Tauberian theorems. CWI Tract (Vol. 40). Centre for Mathematics and Computer Science, Amsterdam
Goldie C.M., Smith R.L. (1987) Slow variation with remainder: theory and applications. Quarterly Journal of Mathematics 38: 45–71
Haeusler E., Teugels J.L. (1985) On asymptotic normality of Hill’s estimator for the exponent of regular variation. The Annals of Statistics 13: 743–756
Hall P. (1982) On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society Series B 44: 37–42
Hall P. (1990) Using the bootstrap to estimate mean squared error and select smoothing parameters in nonparametric problems. Journal of Multivariate Analysis 32: 177–203
Hall P., Welsh A.H. (1985) Adaptive estimates of parameters of regular variation. The Annals of Statistics 13: 331–341
Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3: 1163–1174
Lai P.Y., Lee S.M.S. (2005) An overview of asymptotic properties of L p regression under general classes of error distributions. Journal of the American Statistical Association 100: 446–458
Lai P.Y., Lee S.M.S. (2008) Ratewise efficient estimation of regression coefficients based on L p procedures. Statistica Sinica 18: 1619–1640
Politis D.N., Romano J.P., Wolf M. (1999) Subsampling. Springer, New York
Resnick S., Stărică C. (1997) Smoothing the Hill estimator. Advances in Applied Probability 29: 271–293
Resnick S., Stărică C. (1997) Asymptotic behaviour of Hill’s estimator for autoregressive data. Communications in Statistics: Stochastic Models 13: 703–721
Rogers A.J. (2001) Least absolute deviations regression under nonstandard conditions. Econometric Theory 17: 820–852
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Lai, P.Y., Lee, S.M.S. Estimation of central shapes of error distributions in linear regression problems. Ann Inst Stat Math 65, 105–124 (2013). https://doi.org/10.1007/s10463-012-0360-2
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DOI: https://doi.org/10.1007/s10463-012-0360-2