Detecting Outliers in Gamma Distribution

Comments on “Detecting Outliers in Gamma Distribution” by M. Jabbari Nooghabi et al. (2010)

“…This note shows that the results presented by Jabbari Nooghabi et al (2010) do not hold in all expected cases. With this, the technique proposed by Kumar and Lalhita (2012) for detecting upper outliers in Gamma samples is also not valid.…”

“…Jabbari Nooghabi et al (2010) also used the random variables Y j to find the pdf of the test statistic they proposed under the alternative (Theorem 3.1) and null (Corollary 3.1) hypotheses. Kumar and Lalhita (2012), followed the very same reasoning and methodology used in Theorem 3.1 of Jabbari Nooghabi et al (2010) to derive the pdf of Z k under the null hypothesis.…”

“…Data: Read m, R, and the pseudorandom number generator seed. Initialize Z of length R; Initialize r = 1; for 1 ≤ r ≤ R do Obtain X = (X 1 , X 2 ) from the Γ(m, 1) distribution; Sort X and obtain X (1) ≤ X (2) ; Compute Y 2 = X (2) − X (1) ; Store Z(r) = Y 2 ; Update r = r + 1; Analyze Z; Figure 1 presents the results obtained with this simulation with R = 10 4 : histograms of Y 2 and the densities proposed by Jabbari Nooghabi et al (2010) and Kumar and Lalhita (2012) (dashed lines), and the one we obtained and presented in Eq. (5) (solid lines).…”

“…(5) (solid lines). Figure 1: The pdf of Y 2 assumed by Jabbari Nooghabi et al (2010) and by Kumar and Lalhita (2012) in dashed lines, and in solid lines the pdf given in Eq. (5) Both densities coincide in the case m = 1, i.e., when X 1 , X 2 follow unitary mean Exponential distributions; cf.…”

A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested

“…For other distributions such as the Gamma distribution, procedures for detecting outliers were proposed [8], revised [9], and unfortunately proved to be inefficient [10].…”