Tests of independence in a bivariate exponential distribution

The problem of testing independence in a two component series system is considered. The joint distribution of component lifetimes is modeled by the Pickands bivariate exponential distribution, which includes the widely used Marshall and Olkin's distribution and the Gumbel's type II distribution. The case of identical components is first addressed. Uniformly most powerful unbiased test (UMPU) and likelihood ratio test are obtained. It is shown that inspite of a nuisance parameter, the UMPU test is unconditional and this test turns out to be the same as the likelihood ratio test. The case of nonidentical components is also addressed and both UMPU and likelihood ratio tests are obtained. A UMPU test is obtained to test the identical nature of the components and extensions to the type II censoring scheme and multiple component systems are also discussed. Some modifications to account for the difference in parameters under test and use conditions are also discussed.