Multiple Upper Outlier Detection Procedure in Generalized Exponential Sample

Abstract

Hawkins [6] defined an outlier as an observation that is significantly different from the remaining observations in a dataset so as to arouse suspicion that it was generated by different mechanism. Barnett and Lewis [2] defined an outlier as an observation that deviates significantly in the sample in which it occurs. Spatial outliers are different from outliers and many authors like Singh and Lalitha [9]. Outlier detection procedures for two parameter gamma distribution have been discussed by many authors. But one major disadvantage of the gamma distribution is that the distribution (or survival) function cannot be expressed in a closed form if the shape parameter is not an integer. Since it is in terms of an incomplete gamma function, one needs to obtain the distribution/survival function or the failure rate by numerical integration. This is a limitation in the usage of gamma distribution. It is observed that the generalized exponential distribution can be used as an alternative to the gamma distribution in many situations. Different properties like monotonicity of the hazard functions and tail behaviours of the gamma distribution and that of the generalized exponential distribution are quite similar in nature. But the latter one has a nice compact distribution (or survival) function. It is observed that for a given gamma distribution there exists a generalized exponential distribution so that the two distribution functions are almost identical. Since the gamma distribution function does not have a compact form, efficiently generating gamma random numbers is known to be problematic. It was observed that for all practical purposes it is possible to generate approximate gamma random numbers using generalized exponential distribution and the random samples thus obtained cannot be differentiated using any statistical tests. Many authors proposed a location and scale invariant test based on the test statistic Z_k for testing the upper outliers in two-parameter exponential sample. Kumar et. al. [7] and Singh and Lalitha [10] have proposed test statistics for testing multiple upper outlier detection in gamma sample. Various test statistics have been proposed to detect outliers in an exponential sample. Likes [8] also proposed a new test statistics to detect outlier in the exponential case. In this paper, the test statistic proposed by Likes has been used to detect outliers in a generalized exponential sample and the critical value of the test statistics has been obtained. A simulation study is carried out to compare the theoretical developments.