Adaptive Estimates of Parameters of Regular Variation

The problem of estimating shape and scale parameters for a distribution with regularly varying tails is related to that of nonparametrically estimating a density at a fixed point, in that optimal construction of the estimators depends substantially upon unknown features of the distribution. We show how to overcome this problem by using adaptive methods. Our main results hold very generally, for a large class of adaptive estimators. Later we consider specific versions of adaptive estimators, and describe their performance both in theory and by means of simulation studies. We also examine a technique proposed by Hill (1975) for solving similar problems.