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The cumulant generating function of the statistic
W
defined in Chang et al. (Stat Probab Lett 82:1755–1760,
2012
) for testing homogeneity of the scale-like parameters of several inverse Gaussian distributions is derived in this paper. Then two saddlepoint-based methods are proposed to obtain the asymptotic distribution function of
W
. Simulation results illustrate that the proposed methods are very accurate and powerful even for very small sample sizes and large number of populations.