Computing minimal signature of coherent systems through matrix-geometric distributions

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

SILR: A New Exact Test for Demonstrating That an Effect Exists in Binary Trials

Suppose N experimental participants each undergo t binary trials, and you want to show that at least some of the participants have higher success rates than would be expected if the true success rate for all participants on all trials were the same known value pnull. Under broad circumstances, a new test called SILR has substantially higher power than six other tests that might be used for this purpose. In one example, SILR yielded a significance level of 0.0032 while the other six tests all yielded values above 0.05. In one power analysis, SILR’s power exceeded that of the standard binomial test, applied to the results of all trials for all participants, even when the sample size for SILR was less than 30% that for the binomial.We provide a free R program for SILR. It can also find confidence limits on the number of experimental participants whose true hit rate exceeds the null value, and on the highest and lowest true hit rates of any participants. SILR does not show conclusively that any particular participant’s true hit rate differs significantly from the null rate. If such tests are desired, that can be done with Bonferroni-corrected binomial tests on individual participants. But SILR often has much higher power than these tests.

New maximal inequalities for N-demimartingales with scan statistic applications

In the present work, some new maximal inequalities for nonnegative N-demi(super)martingales are first developed. As an application, new bounds for the cumulative distribution function of the waiting time for the first occurrence of a scan statistic in a sequence of independent and identically distributed (i.i.d.) binary trials are obtained. A numerical study is also carried out for investigating the behavior of the new bounds.