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

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.