Estimation of parameters for the truncated exponential distribution

We consider the right truncated exponential distribution where the truncation point is unknown and show that the ML equation has a unique solution over an extended parameter space. In the case of the estimation of the truncation point T we show that the asymptotic distribution of the MLE is not centered at T. A modified MLE is introduced which outperforms all other considered estimators including the minimum variance unbiased estimator. Asymptotic as well as small sample properties of different estimators are investigated and compared. The truncated exponential distribution has an increasing failure rate, ideally suited for use as a survival distribution for biological and industrial data.