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

Suppose that a random variable has the probability density functionpv,σ(x) = υ/σΓ(1/υ)exp [-(x/σ)υ],0 ≤ x ≤ σ and ν may not be known. In order to estimate the probabilityPₑ(K)that the random variable exceeds a high thresholdK, an extrapolation can be made from counting estimatesP̂ₑ(x₁),P̂ₑ(x₂), ... ,P̂ₑ(xₘ), of the probabilities of exceedingmlower thresholds. Using the observation that a double logarithmic function ofPₑ(x), is approximately linear in log(x)for a useful range of the exponent, an estimate of In [-InPₑf(K)] 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.