Exact distributions of tests of outliers for exponential samples

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. KeywordsExponential distribution • Slippage alternative • Contaminants • Performance measures • Tests of discordancy • Spurious and non-spurious power B Nirpeksh Kumar

Use of median‐based estimator to construct Phase II exponential chart

“…Clearly, the incorrect parameter estimation will affect the Phase II control chart's performance. However, to identify the spurious observations in exponential samples, several testing procedures have been proposed in literature 29-32 . The reader is referred to Bernet and Lewis 28 for a detailed discussion on different testing procedures to detect outliers.…”

“…However, to identify the spurious observations in exponential samples, several testing procedures have been proposed in literature. [29][30][31][32] The reader is referred to Bernet and Lewis 28 for a detailed discussion on different testing procedures to detect outliers. We should mention here that these tests have less power to discriminate between spurious and regular observations, especially, when the sample size is small and the spurious observation is not so far from the regular observations.…”

“…We should mention here that these tests have less power to discriminate between spurious and regular observations, especially, when the sample size is small and the spurious observation is not so far from the regular observations. 32,33 Please note that the MLE and MVUE are the functions of sample mean that is severely affected by the presence of spurious observations and, hence, the Phase II control limits using these estimators may lead to wrong conclusions. On contrary, the sample median is a robust estimator and much less affected by the extreme observations or outliers in the sample.…”