# Exact distributions of tests of outliers for exponential samples

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## Abstract

In this paper, we propose an algorithm to derive the exact distributions of discordancy tests for exponential samples under the slippage alternative providing that their survival functions involve the linear combinations of independent and identically distributed exponential random variables with arbitrary real coefficients. In addition, we define the various performance measures in terms of conditional probabilities that the observed value of the test statistic exceeds the critical value given that the contaminants have the specific position numbers in the ordered sample. These make possible to calculate various performance measures of discordancy tests for the exponential samples to any desired degree of accuracy. For the purpose of illustration, we derive the distributions of the maximum likelihood ratio tests for testing single and multiple outliers in the exponential samples and then we calculate their performance measures accurately to six decimal places. Moreover, the definitions of the performance criteria are not restricted to the discordancy tests for exponential samples only, they are also equally applicable to the discordancy tests for samples from other distributions.

## Keywords

Exponential distribution Slippage alternative Contaminants Performance measures Tests of discordancy Spurious and non-spurious power## Notes

### Acknowledgements

The author would like to thank two anonymous reviewers and the editor for their helpful and constructive comments that have improved the paper.

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