Testing for Upper Outliers in Gamma Sample

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

“…For instance, in the case here considered, H 0 states that data are generated as independent observations from a common distribution F . If F is a Gamma distribution, as in Jabbari Nooghabi et al (2010) and Kumar and Lalhita (2012), then H 0 : X 1 , X 2 , . .…”

“…The alternative hypothesis used in Jabbari Nooghabi et al (2010) and Kumar and Lalhita (2012) is the slippage alternative.…”

“…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.…”