On the Information in Extreme Measurements for Parameter Estimation

TL;DR

This paper introduces a new method to analyze the information content of extreme measurements, like minima and maxima, for parameter estimation, demonstrating that maxima carry most information and enabling significant data compression.

Contribution

A novel analytical approach for the Cramer-Rao Lower Bound based on extreme values, with practical application to exponential distributions and insights into information efficiency.

Findings

Maxima contain most of the information about the parameter.

Minimum values add negligible information.

Using maxima alone can compress data by a factor of 15 while retaining about 50% of the information.

Abstract

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.