  Often, it is required to estimate the probability that a quantity such as
toxicity level, plutonium, temperature, rainfall, damage, wind speed, wave
size, earthquake magnitude, risk, etc., exceeds an unsafe high threshold. The
probability in question is then very small. To estimate such a probability,
information is needed about large values of the quantity of interest. However,
in many cases, the data only contain values below or even far below the
designated threshold, let alone exceedingly large values. It is shown that by
repeated fusion of the data with externally generated random data, more
information about small tail probabilities is obtained with the aid of certain
new statistical functions. This provides relatively short, yet reliable
interval estimates based on moderately large samples. A comparison of the
approach with a method from extreme values theory (Peaks over Threshold, or
POT), using both artificial and real data, points to the merit of repeated out
of sample fusion.
