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

Abstract

Suppose that a random variable has the probability density function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{v,\sigma}(x) = \frac{\upsilon}{\sigma\Gamma(1/\upsilon)}exp [-(x/\sigma)^{\upsilon}]</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 \leq x \leq \infty</tex> where σ and ν may not be known. In order to estimate the probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{e}(K)</tex> that the random variable exceeds a high threshold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> , an extrapolation can be made from counting estimates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{P}_{e}(x_{1})</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{P}_{e}(x_{2})</tex> , ... , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{P}_{e}(x_{m})</tex> , of the probabilities of exceeding <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> lower thresholds. Using the observation that a double logarithmic function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{e}(x)</tex> , is approximately linear in log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(x)</tex> for a useful range of the exponent, an estimate of In [-In <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{e}f(K)</tex> ] 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.