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

TLDR: In this article, the authors proposed to estimate the probability that a random variable exceeds a high threshold by counting estimates of the probabilities of exceeding m lower thresholds, where σ and ν may not be known.

Abstract: Suppose that a random variable has the probability density function p_{v,\sigma}(x) = \frac{\upsilon}{\sigma\Gamma(1/\upsilon)}exp [-(x/\sigma)^{\upsilon}] , 0 \leq x \leq \infty where σ and ν may not be known. In order to estimate the probability P_{e}(K) that the random variable exceeds a high threshold K , an extrapolation can be made from counting estimates \hat{P}_{e}(x_{1}) , \hat{P}_{e}(x_{2}) , ... , \hat{P}_{e}(x_{m}) , of the probabilities of exceeding m lower thresholds. Using the observation that a double logarithmic function of P_{e}(x) , is approximately linear in log (x) for a useful range of the exponent, an estimate of In [-In P_{e}f(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.