  In this paper, a novel approach to the problem of estimating the heavy-tail
exponent alpha>0 of a distribution is proposed. It is based on the fact that
block-maxima of size m of the independent and identically distributed data
scale at a rate of m^{1/alpha}. This scaling rate can be captured well by the
max-spectrum plot of the data that leads to regression based estimators.
Consistency and asymptotic normality of these estimators is established under
mild conditions on the behavior of the tail of the distribution. The results
are obtained by establishing bounds on the rate of convergence of moment-type
functionals of heavy-tailed maxima. Such bounds often yield exact rates of
convergence and are of independent interest. 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 proves competitive for both small and large sample sizes and
for a large range of tail exponents. The method is shown to be more robust than
the classical Hill plot and is illustrated on two data sets of insurance claims
and natural gas field sizes.
