The modified likelihood ratio test, which is based on penalized likelihood function, is usually used for testing homogeneity of the mixture models. The efficiency of this test is seriously affected by the shape of penalty function that is used in penalized likelihood function. The selection of penalty function is usually based on avoiding of complexity and increasing tractability, hence the results may be far from optimality. In this paper, we consider a more general form of penalty function that depends on a shape parameter. Then this shape parameter and the parameters of mixture models are estimated by using Bayesian paradigm. It is shown that the proposed Bayesian approach is more efficient in comparison to modified likelihood test. The proposed Bayesian approach is clearly more efficient, specially in nonidentifiability situation, where frequentist approaches are almost failed.

We propose an integrated approach for decision making about repair and maintenance of deteriorating systems whose failures are detected only by inspections. Inspections at periodic times reveal the true state of the system's components and preventive and corrective maintenance actions are carried out in response to the observed system state. Assuming a threshold-type policy, the paper aims at minimizing the long-run average maintenance cost per unit time by determining appropriate inspection intervals and a maintenance threshold. Using the renewal reward theorem, the expected cost per cycle and expected cycle length emerge as solutions of equations, and a recursive scheme is devised to solve them. We demonstrate the procedure and its outperformance over specific cases when the components' lifetime conforms to a Weibull distribution. Further, a sensitivity analysis is performed to determine the impact of the model's parameters. Attention has turned to perfect repair and inspection, but the structure allows different scenarios to be explored.