Inverse, or negative binomial, sampling is often used when the observation of interest occurs extremely infrequently. As this is the case in bit error rate (BER) simulations, especially in high signal-to-noise ratio cases, negative binomial sampling can be advantageously employed in a computationally economic fashion to compare bit error rates between different systems. When the results of two negative binomial sampling tests are compared, point estimates and interval estimates quantify the performance relationship between the results of the tests. This paper derives a new, optimal, logarithmically symmetric confidence interval estimator for the ratio of BER estimates derived from two negative binomial tests. In addition, a three-sided hypothesis test with a single significance level is derived to quantify the confidence of the relationship between the two systems. Low-BER approximations for the confidence interval and decision thresholds are derived based on the F-distribution. The approximation is shown to work with BERs as high as 10-2. An example inspired by bit interleaved coded modulation shows how the technique can be used to reduce simulation time by an order of magnitude and facilitate straightforward interpretation and comparison between different systems. Negative binomial sampling is recommended for comparison experiments where BER is the key metric.

This paper examines the binomial and negative binomial tests for estimating the bit error rate using Monte Carlo simulations. Whereas the estimator error variances and simulation times do not recommend one test over the other, confidence interval analysis and test design criteria recommend the negative binomial test for estimating a small bit error rate based on a limited number of error events. The Clopper-Pearson interval length is much more consistent for the negative binomial test in the small bit error rate case. The logarithmically centered confidence interval is introduced and analyzed. For a given confidence interval length and confidence probability, the negative binomial test requires fewer error events than the binomial test. The asymptotic analyses of the Clopper-Pearson and logarithmically-centered confidence intervals yield useful design criteria for the negative binomial test in a way that is not possible for the binomial test.

Simulation, point estimation, and interval estimation of the bit error rate are rigorously examined. Exact confidence intervals based on the F-distribution approximation for the binomial and negative binomial distributions are derived. Simulations demonstrate 1) that the analysis gives accurate confidence intervals for bit error rate estimators and 2) that the performance of the binomial and negative binomial estimators give similar performance. The negative binomial estimator has the advantage that an a priori estimate or guess of the true bit error probability is not required.