Assessing the lifetime performance of products is one of the most important topics in the man- ufacturing industries. In this paper, we assume that the lifetimes of products are independent and have a common two-parameter exponential distribution. The lifetime performance index -CL- pro- vides a means for evaluating the performance of a process under a known lower lifetime limit L. We consider a sample of generalized order statistics -GOS-, introduced by Kamps [1], which contains several models of ordered random variables, e.g. ordinary order statistics, progressively censored order statistics and record values. Then, we obtain the maximum likelihood estimator -MLE- and the uniformly minimum variance unbiased estimator -UMVUE- of CL on the basis of a GOS sample. These estimators are compared in terms of mean squared error -MSE- and Pitman measure of close- ness -PMC- criteria. The UMVUE of CL is utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, in order to illustrate the results, two real data sets due to Lawless [2] and Proschan [3], and a simulated sample are analyzed.

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