pseudo-likelihood function

The present study proposes several methods for improving the parameter estimation of interval-censored exponential lifetime data. These methods can be regarded as the versions of substitution method which are based on the most probable time of occurring failures or the meantime of failures in each inspection interval. Based on simulation studies, the methods proposed in this present study can improve the existing estimators which are based on the midpoint of inspection intervals “or” the exponential probability plot.

Many distributions have been presented with bathtub-shaped failure rates for real-life data. A two-parameter distribution was defined by Chen (2000). This distribution can have a bathtub-shaped or increasing failure rate function. In this paper, we consider two bivariate models based on the proposed  distribution by Chen and use the proposed methods of Marshall and Olkin (1967) in the bivariate case and Marshall and Olkin (1997) in the univariate case. In the second case, their method is generalized to the bivariate case and a new bivariate distribution is introduced. These new bivariate distributions have natural interpretations, and they can be applied in fatal shock models or in competing risks models. We call these new distributions as the bivariate Chen (BCH) distribution and bivariate Chen-geometric (BCHG) distribution, respectively.  Moreover, the BCH can be obtained as a special case of the BCHG model. Then, the various properties of the new distributions are investigated.  The BCHG distribution has five parameters and the maximum likelihood estimators cannot be obtained in a closed form. We suggest using an EM algorithm that is very easy to implement. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze two real data sets for illustrative purposes.