Extrapolation of Threshold-Limited Null Measurement Frequencies

Abstract

The total measurable level of a pathogen is due to many sources, which produce a variety of pulses, overlapping in time, that rise suddenly and then decay. What is measured is the level of the total contribution of the sources at a given time. But since we are only capable of measuring the total level above some threshold $x_0$, we would like to predict the distribution below this level. Our principal model assumption is that of the asymptotic exponential decay of all pulses. We show that this implies a power law distribution for the frequencies of low amplitude observations. As a consequence, there is a simple extrapolation procedure for carrying the data to the region below $x_0$.