A note on the bootstrap method for testing the existence of finite moments
Keywords:Bootstrap, finite moments, heavy tails, tail index estimator, test
AbstractThis paper discusses a bootstrap-based test, which checks if finite moments exist, and indicates cases of possible misapplication. It notes, that a procedure for finding the smallest power to which observations need to be raised, such that the test rejects a hypothesis that the corresponding moment is finite, works poorly as an estimator of the tail index or moment estimator. This is the case especially for very low- and high-order moments. Several examples of correct usage of the test are also shown. The main result is derived analytically, and a Monte-Carlo experiment is presented.
P. DEHEUVELS, E. HAEUSLER, M. D. MASON (1988). Almost sure convergence of the Hill estimator. Mathematical Proceedings of the Cambridge Philosophical Society, 104, no. 2, pp. 371–381.
A. L. M. DEKKERS, J. H. J. EINMAHL, L. DE HAAN (1989). A moment estimator for the index of an extreme-value distribution. The Annals of Statistics, 17, no. 4, pp. 209–222.
C. DERMAN, H. ROBBINS (1955). The strong law of large numbers when the first moment does not exist. Proceedings of the National Academy of Sciences, 41, no. 8, pp. 586–587.
H. DREES, S. RESNICK, L. DE HAAN (2000). How to make a Hill plot. The Annals of Statistics, 28, no. 1, pp. 254–274.
I. Fedotenkov (2013). A bootstrap method to test for the existence of finite moments. Journal of Nonparametic Statistics, 25, no. 2, pp. 315–322.
B. HILL (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3, no. 5, pp. 1163–1174.
S. RESNICK, C. STĂRICĂ (1997). Smoothing the Hill estimator. Advances in Applied Probability, 29, no. 1, pp. 271–293.
S. RESNICK, C. STĂRICĂ (1998). Tail index estimation for dependent data. Annals of Applied Probability, 8, no. 4, pp. 1156–1183.