  A general rate estimation method is proposed that is based on studying the
in-sample evolution of appropriately chosen diverging/converging statistics.
The proposed rate estimators are based on simple least squares arguments, and
are shown to be accurate in a very general setting without requiring the choice
of a tuning parameter. The notion of scanning is introduced with the purpose of
extracting useful subsamples of the data series; the proposed rate estimation
method is applied to different scans, and the resulting estimators are then
combined to improve accuracy. Applications to heavy tail index estimation as
well as to the problem of estimating the long memory parameter are discussed; a
small simulation study complements our theoretical results.
