Exact simultaneous location-scale tests for two shifted exponential samples

The failure time distribution for various items often follows a shifted (two-parameter) expo- nential model and not the traditional (one-parameter) exponential model. The shifted exponen- tial is very useful in practice, in particular in the engineering, biomedical sciences and industrial quality control when modeling time to event or survival data. The open problem of simultane- ous testing for differences in origin and scale parameters of two shifted exponential distributions is addressed. Two exact tests are proposed using maximum likelihood estimators. They are based on the combination of two statistics following a maximum-type and a distance-type ap- proach. The exact null distributions of the respective test statistics are derived analytically. Small sample type-one error rate and power of the tests are studied numerically. We showed that the test based on the maximum type combination (the Max test) should be preferred being generally more powerful than the test based on the distance type combination (the Distance test). An application to a biomedical experiment is discussed.