Estimating Heavy-Tail Exponents Through Max Self–Similarity

In this paper, a novel approach to the problem of estimating the heavy-tail exponent α >; 0 of a distribution is proposed. It is based on the fact that block-maxima of size m scale at a rate m^1/α for independent, as well as for a number of dependent data. This scaling rate can be captured well by the max-spectrum plot of the data that leads to regression based estimators for α. Consistency and asymptotic normality of these estimators is established for independent data under mild conditions on the behavior of the tail of the distribution. The proposed estimators have an important computational advantage over existing methods; namely, they can be calculated and updated sequentially in an on-line fashion without having to store the entire data set. Practical issues on the automatic selection of tuning parameters for the estimators and corresponding confidence intervals are also addressed. Extensive numerical simulations show that the proposed method is competitive for both small and large sample sizes, robust to contaminants and continues to work under the presence of substantial amount of dependence. The proposed estimators are used to illustrate the close connection between long-range dependence and heavy tails over an Internet traffic trace.