On the uniform consistency of the Hill estimator.

We start by considering a kernel estimator g_{n,h}(t) for the regression function m_g(t):=E[g(Y)|X=t], where t is fixed and g:R->R is a measurable function with finite second moment. If h=h_n is a deterministic sequence such that h_n->0 and nh_n^d/log log n->\infty, it is well-known that g_{n,h_n}(t) estimates consistently m_g(t)f_X(t), where f_X is the density function of X. As an extension, we present a result in which additional assumptions are imposed to make g_{n,h}(t) a strongly consistent estimator,
uniformly for a certain range of bandwidths h. As an application, we consider random variables Y_1,...,Y_n for which the common
distribution function F has regularly varying upper tails of exponent -1/tauconsistency of this estimator, uniformly for a certain range of bandwidths tending to zero at particular rates

Publication year:2008

Keywords:Hill estimator, extreme values, nonparametric regression, kernel estimators, consistency, empirical processes, uniform in bandwidth