A new statistic for detecting outliers in Rayligh distribution
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ORIGINAL RESEARCH
A new statistic for detecting outliers in Rayligh distribution Einolah Deiri1 Received: 18 January 2019 / Accepted: 8 September 2020 © The Author(s) 2020
Abstract Zerbet and Nikulin (Commun Statist Theor Meth 32(3): 573–583, 2003) presented the new statistic Zk for detecting outliers in exponential distribution. They also compared this statistic with Dixon’s statistic Dk . Jabbari et al. (Commun Statist Theor Meth 39(4): 698–706, 2010) expend this statistic ( Zk ) for Gamma distribution. In this paper, we generalize statistics Zk–Zk∗ , for detecting outliers in Rayligh distribution and compare the results with the generalized Dixon’s statistic. Distribution of the test based on the statistic Zk∗ under slippage alternatives is obtained. The criterion value and power of the new test are also calculated and compared with the criterion value of the Dixon’s statistic. The results show that the test based on statistic Zk∗ is more powerful than the test based on the statistic Dk. Keywords Dixon’s statistic · Rayligh sample · Outliers · Power of the test · Slippage hypothesis · Test of chauvenet · Upper outlier · Z statistic
1 Introduction Bol’shev (1969) generalized the Chauvenet’s test for rejecting outlier observations (see Bol’shev 1969; Voinov and Nikulin 1993, 1996). This method is suitable for detecting k outliers for univariate data set. The Chauvenet’s test can be used for exponential case. Ibragimov and Khalna (1978) considered various modification of this test. Several authors considered the problem of testing one outlier in exponential distribution (Chikkagoudar and Kunchur 1983; Kabe 1970; Lewis and Fiellerm 1979; Likes 1966). Only two types of statistics for testing multiple outliers are exist. First is Dixon’s while the second is based on the ratio of the some of the observations suspected to be outliers to the sum of all observations of the sample. Most of these authors have considered a general case of gamma model and the results for exponential model are given as a special case. This approach is focused on alternative models, namely slippage alternatives in exponential samples (see Barnett and Lewis 1978). Zerbet and Nikulin (2003) proposed a statistic which is different to the well-known Dixon’s statistic Dk to detect multiple outliers. In this paper, we generalize the statistics Zk–Zk∗ * Einolah Deiri [email protected] 1
for detecting outliers in Rayligh distribution. Distribution of the test based on these statistics under slippage alternatives is obtained and the tables of critical values are given for various sample size n and number of outliers k. The power of these tests are also calculated and compared. The results show that the test based on statistic Zk∗ is more powerful than the test based on statistic Dk.
2 Statistical inference Let X1 , X2 , … , Xn are arbitrary independent random variables. In this paper, we want to test the hypothesis: H0 ∶ X 1 , X2 , … , Xn derive from a Rayligh distribution as. ( 2) x , 𝜃 > 0, 𝜃 is unknown Pr{X ≤ x|H0 } = F(x;0) = 1 − ex
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