A bootstrap method to test for the existence of finite moments

Abstract

This paper presents a simple bootstrap test to verify the existence of finite moments. The efficacy of the test relies on the fact that in the absence of a first moment and under certain general conditions, the arithmetic average of a sample grows at a rate greater than the growth rates of the arithmetic averages of the sub-samples. Firstly, we show test consistency analytically. Then, Monte-Carlo simulations are performed to compare our test with the Hill estimator.

Keywords:

• moment existence
• bootstrap test
• sub-sampling
• consistency
• Monte-Carlo simulations

AMS Subject Classification :

• 62F03
• 62F05
• 62F40
• 62G32

Acknowledgements

I would like to thank the participants at the 23rd Nordic Conference on Mathematical Statistics and the 10th International Vilnius Conference on Probability Theory and Mathematical Statistics (2010), especially Alfredas Ra\v ckauskas, Remigijus Leipus, and Vygantas Paulauskas. Furthermore, I wish to thank Marius Radavi\v cius and Irena Mikolajun for their help and consulting. Finally, I would like to express my appreciation to the anonymous referee for his/her useful comments.

Notes

Alternative estimators are discussed in detail by Haan and Peng (1998).

For simplicity, m∼log(n) will be assumed further in the analysis. A more general outcome is shown in Appendix 1.