Two groups are assumed to follow Weibull distributions with different parameters, which are unknown. Maximum likelihood estimates of unknown parameters have been obtained.
We studied the problem, see the paper:
https://www.researchgate.net/publication/238001712_On_comparing_the_shape_parameters_of_two_Weibull_distributions?ev=srch_pub
Article On comparing the shape parameters of two Weibull distributions
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