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I first characterize these cases in models with densities linear in an infinite-dimensional parameter. I then introduce a novel \u201cgeneralized Fisher information.\u201d If positive, it implies (possibly irregular) identification when other conditions hold. If zero, it implies impossibility results on rates of estimation. Three examples illustrate the applicability of the general results. First, I consider the canonical example of average densities. Second, I show irregular identification of the median willingness to pay in contingent valuation studies. 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