Estimation of Small Probabilities by Linearization of the Tail of a Probability Distribution Function
Stephen B. Weinstein · IEEE Transactions on Communication Technology · 1971
Suppose that a random variable has the probability density functionp_{v,\sigma}(x) = \frac{\upsilon}{\sigma\Gamma(1/\upsilon)}exp [-(x/\sigma)^{\upsilon}],0 \leq x \leq \inftywhere σ and ν may not be known. In order to estimate the probabilityP_{e}(K)that the random variable exceeds a high thresholdK, 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 exceedingmlower thresholds. Using the observation that a double logarithmic function ofP_{e}(x), is approximately linear in log(x)for a useful range of the exponent, an estimate of In [-InP_{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.