On the robustness of the standard estimates of the exponential mean to contamination

TLDR: In this paper, the performance of the standard estimators of the mean of an exponential distribution is evaluated when this basic distribution is contaminated by another of the same type, and it is found that the sample mean is asymptotically fully efficient for the average mean of a mixture.

Abstract: SUMMARY The performances of the standard estimators of the mean of an exponential distribution are evaluated when this basic distribution is contaminated by another of the same type. Surprisingly, perhaps, it is found that the sample mean is asymptotically fully efficient for the mean of a mixture. For small samples, on the other hand, a comparison of the two standard statistics with the appropriate optimal linear statistics in terms of the mean squared error and bias shows that for relatively coarse mixtures there is a lot of room for improvement. With this in mind, other usable linear estimators are proposed and shown to be almost always preferable to the standard exponential statistics for the population mean. The emphasis is mainly on long-tailed contamination, but the behaviour of the two standard statistics and the new statistics in short-tailed mixtures is also discussed.