Estimation of Small Probabilities by Linearization of the Tail of a Probability Distribution Function

Suppose that a random variable has the probability density function , where σ and ν may not be known. In order to estimate the probability that the random variable exceeds a high threshold , an extrapolation can be made from counting estimates , , ... , , of the probabilities of exceeding lower thresholds. Using the observation that a double logarithmic function of , is approximately linear in log for a useful range of the exponent, an estimate of In [-In ] can be made by straightline extrapolation. In application to estimation of error rate in a digital communication system operating over an analog channel, only weak a-priori assumptions about the noise need be made, substantially fewer samples are required than for the usual counting estimate, and knowledge of the transmitted data sequence is unnecessary. A physical implementation of this technique in an error meter is described.