A new statistic for detecting outliers in Rayligh distribution

Zerbet and Nikulin (Commun Statist Theor Meth 32(3): 573–583, 2003) presented the new statistic Zk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z}_{k}$$\end{document} for detecting outliers in exponential distribution. They also compared this statistic with Dixon's statistic Dk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D}_{k}$$\end{document}. Jabbari et al. (Commun Statist Theor Meth 39(4): 698–706, 2010) expend this statistic (Zk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z}_{k}$$\end{document}) for Gamma distribution. In this paper, we generalize statistics Zk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z}_{k}$$\end{document}–Zk∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z}_{k}^{*}$$\end{document}, 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∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z}_{k}^{*}$$\end{document} 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∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z}_{k}^{*}$$\end{document} is more powerful than the test based on the statistic Dk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D}_{k}$$\end{document}.


Introduction
generalized the Chauvenet's test for rejecting outlier observations (see Bol'shev 1969;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 D k to detect multiple outliers. In this paper, we generalize the statistics Z k -Z * k 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 Z * k is more powerful than the test based on statistic D k .
where ≥ 1, is unknown and X (1) , X (2) , … , X (n) are the order statistics corresponding to the observa-tionsX 1 , X 2 , … , X n . This hypothesis could be considered as an important sub-hypothesis of the one saying that k of n observations are suspected to be outliers (for > 1 , these k observations are called upper outliers). The hypothesis H 0 corresponds to the = 1 To testH 0 , we propose the following statistics: For m = 1 , the above statistics ( Z * k ) proposed by Zerbet and Nikulin (2003).
Following the idea of the Chauvenet's test, we assume that the decision criterion is: the hypothesis H 0 is rejected when Z * k >z c with Z c = Z c ( ) being the critical value corresponding to the significance level .

The distribution of the statistics Z * k under alternatives
In this section, we find the distribution of the statistics Z * k , according to Zerbet and Nikulin (2003) method. Then the distributions of these statistics under the slippage alternative hypothesis H k are obtained by the following theorem.
Theorem 3.1. The distribution of the statistic Z * k , under H k is given by: where Proof. To proof this theorem, we must obtain the distribution of the statistic Z * k under the alternative hypothesis H k .
We first compute the corresponding alternative distribution of the statistic: we obviously obtain that: The characteristic function of (v, w) is Knowing that Y i , j = 1,2, … , n − k follows the Rayligh distribution of parameters 1 and (k + n − k − j + 1) −1 , and Y n−k+j , j = 1,2, … , k have the same distribution but with parameters 1 and (k − j + 1) −1 (see Chikkagoudar and Kunchur (1983)), then the characteristic function (V,W) is Therefore, the joint density function of (v, w) can be obtained as follows: To find the joint probability density function of variables (v, w) , we first calculate the following products: By replacement Eqs. 2-5 in Eq. (1), the joint pdf of variables (v, w) will be as follows: in the process to find the joint pdf of variables (V,W), we know that As a conclusion, the pdf of U k is then, Then the distribution function of Z * k can be found from (1) using the relation and the proof is complete.

Corollary:
Under H 0 the distribution of statistic Z * k is obtained from the Theorem 3.1 using = 1.
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