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The existence of maximum likelihood estimates for a class of heteroscedastic regression models is considered. For a given dispersion function we show that, under a weak condition, the likelihood is singular at points corresponding to nonreplicated observations, causing unrestricted maximum likelihood estimation to break down, whilst for an alternative class of dispersion functions we obtain a much stronger linear independence condition for the likelihood to be unbounded.
Crisp et al. (Sat,) studied this question.