Statistical inference for the lifetime performance index based on generalised order statistics from exponential distribution

In manufacturing industries, the lifetime of an item is usually characterised by a random variable X and considered to be satisfactory if X exceeds a given lower lifetime limit L. The probability of a satisfactory item is then ηL := P(X ≥ L), called conforming rate. In industrial companies, however, the lifetime performance index, proposed by Montgomery and denoted by CL, is widely used as a process capability index instead of the conforming rate. Assuming a parametric model for the random variable X, we show that there is a connection between the conforming rate and the lifetime performance index. Consequently, the statistical inferences about ηL and CL are equivalent. Hence, we restrict ourselves to statistical inference for CL based on generalised order statistics, which contains several ordered data models such as usual order statistics, progressively Type-II censored data and records. Various point and interval estimators for the parameter CL are obtained and optimal critical regions for the hypothesis testing problems concerning CL are proposed. Finally, two real data-sets on the lifetimes of insulating fluid and ball bearings, due to Nelson (1982) and Caroni (2002), respectively, and a simulated sample are analysed.