FLAT LIKELIHOODS: THREE-PARAMETER WEIBULL MODEL CASE

Criticisms of maximum likelihood estimation frequently occur when likelihood function shape becomes flat. Although some research have been done regarding the possible causes of a flat likelihood, more work is needed to expand our knowledge on this subject. In this paper we analyze the origin of Weibull flat likelihoods. In particular, we study the severity of the likelihood flatness by examining the limit behaviour of the relative profile likelihood for the three-parameter Weibull threshold parameter, when this parameter goes to infinity. In the cases discussed here, flat likelihoods are not only related to sample size but also to an embedded model problem. Due to the widespread use of the likelihood function in inferential statistical methods, it is important not only to identify factors that can cause flat likelihoods, but also to study the severity of this flattening, in order to develop or apply ad hoc statistical and computational methods for making inferences.