Title: Extreme Value Analysis Without the Largest Values: What Can Be Done?,
Abstract: In this paper we are concerned with the analysis of heavy-tailed data when a
portion of the extreme values is unavailable. This research was motivated by an
analysis of the degree distributions in a large social network. The degree
distributions of such networks tend to have power law behavior in the tails. We
focus on the Hill estimator, which plays a starring role in heavy-tailed
modeling. The Hill estimator for this data exhibited a smooth and increasing
"sample path" as a function of the number of upper order statistics used in
constructing the estimator. This behavior became more apparent as we
artificially removed more of the upper order statistics. Building on this
observation we introduce a new version of the Hill estimator. It is a function
of the number of the upper order statistics used in the estimation, but also
depends on the number of unavailable extreme values. We establish functional
convergence of the normalized Hill estimator to a Gaussian process. An
estimation procedure is developed based on the limit theory to estimate the
number of missing extremes and extreme value parameters including the tail
index and the bias of Hill's estimator. We illustrate how this approach works
in both simulations and real data examples.