  This paper deals with parameter estimation from extreme measurements. While
being a special case of parameter estimation from partial data, in scenarios
where only one sample from a given set of K measurements can be extracted,
choosing only the minimum or the maximum (i.e., extreme) value from that set is
of special interest because of the ultra-low energy, storage, and processing
power required to extract extreme values from a given data set. We present a
new methodology to analyze the performance of parameter estimation from extreme
measurements. In particular, we present a general close-form approximation for
the Cramer-Rao Lower Bound on the parameter estimation error, based on extreme
values. We demonstrate our methodology on the case where the original
measurements are exponential distributed, which is related to many practical
applications. The analysis shows that the maximum values carry most of the
information about the parameter of interest and that the additional information
in the minimum is negligible. Moreover, it shows that for small sets of iid
measurements (e.g. K=15) the use of the maximum can provide data compression
with factor 15 while keeping about 50% of the information stored in the
complete set.
