Comparative simulation study of likelihood ratio tests for homogeneity of the exponential distribution

The aim of this paper is to present and discuss the power of the exact likelihood ratio homogeneity testing procedure of the number of components k in the exponential mixture. First we present the likelihood ratio test for homogeneity (ELR), the likelihood ratio test for homogeneity against two-component exponential mixture (ELR2), and finally the likelihood ratio test for homogeneity against three-component exponential mixture (ELR3). Comparative power study of mentioned homogeneity tests against three-component subpopulation alternative is provided. Therein we concentrate on various setups of the scales and weights, which allow us to make conclusions for generic settings. The natural property is observed, namely increase of the power of exact likelihood ratio ELR, ELR2 and ELR3 tests with scale parameters considered in the alternative. We can state that the differences in power of ELR, ELR2 and ELR3 tests are small - therefore using of the computationally simpler ELR2 test is recommended for broad usage rather than computationally more expensive ELR3 test in the cases when unobserved heterogeneity is modelled. Anyhow caution should be taken before automatic usage of ELR3 in more informative settings, since the application of automatic methods hoping that the data will enforce its true structure is deceptive. Application of obtained results in reliability, finance or social sciences is straightforward.