Invariant exponential models applied to reliability theory and survival analysis

For a two-parameter exponential model with increasing failure rate (IFR) or decreasing failure rate (DFR) distributions necessary and sufficient condi tions of the existence of a solution of the likelihood equations are given. Also, all of the scale-invariant two-parameter statistical models closed b y raising to a power and by exponential tilting are introduced. The conditi ons of existence of a solution of the likelihood equations are studied for these invariant models, and the models are applied to obtain some uniformly most powerful unbiased tests of exponentially against alternatives in thes e models.