Comparing scale parameters in several gamma distributions with known shapes
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Comparing scale parameters in several gamma distributions with known shapes Ali Akbar Jafari1
· Javad Shaabani1
Received: 20 July 2019 / Accepted: 25 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we present eleven approaches for testing the equality of scale parameters in gamma distributions when shape parameters are known. These approaches are applicable to other problems such as testing homogeneity of variances in normal distributions, verifying equality of scale-like parameters in inverse Gaussian distributions, and comparing scale parameters in two-parameter exponential distributions with censored data or K-record values. The performance of the proposed tests is compared in terms of empirical size and power using Monte Carlo simulation. Finally, the proposed methods are illustrated using two real data examples. Keywords Cornish–Fisher expansion · Generalized p-value · Jacobi expansion · Mellin transform · Saddlepoint · Satterthwaite approximation
1 Introduction The gamma distribution, as a generalization of exponential distribution, has been applied in various scientific studies such as climatology (Crow 1977) and hydrology (Gastwirth and Mahmoud 1986). The gamma distribution with shape parameter α and scale parameter λ has the probability density function (pdf) f X (x) =
x α−1 λα e−λx , x > 0, α > 0, λ > 0, Γ (α)
where Γ (α) is the gamma function. This distribution is denoted by Ga (α, λ). When comparing several gamma distributions, testing the equality of scale parameters is an important problem because it contains several well-known testing problems as special cases (see Sect. 3). Assuming that shape parameters are known, such special
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Ali Akbar Jafari [email protected] Department of Statistics, Yazd University, Yazd, Iran
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A. A. Jafari, J. Shaabani
cases are tests of equality of scale parameters or homogeneity of variances in several distributions. One example is testing the equality of scale parameters in one-parameter and two-parameter exponential distributions (see Nagarsenker and Nagarsenker 1986; Thiagarajh and Paul 1990; Nagarsenker and Nagarsenker 1991; Thiagarajah 1995; Kharrati-Kopaei and Malekzadeh 2019). Another example is testing the homogeneity of variances in the analysis of variance where it is assumed that the variances within populations are equal. When this assumption is violated, the F-test becomes overly conservative or liberal and no longer is an optimal test. Therefore, many tests for verifying the homogeneity of variances have been proposed in the literature, including the Bartlett’s test (Bartlett 1937), a generalized p-value test (Liu and Xu 2010), a bootstrap test (Cahoy 2010), two Wald tests (Allingham and Rayner 2012), an adjusted Bartlett’s test (Ma et al. 2015), a computational approach test (CAT) (Gökpınar and Gökpınar 2017) and a standardized likelihood ratio test (SLRT) (Gökpınar 2017). Testing the homogeneity of scale-like parameters between the inverse Gaussian distributions can also be considered as a s
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