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What is the intuition behind infinite Cramer-Rao Lower Bound and its connection with the unbiasedness?

I understand, that if the likelihood $p(x|\theta)$ does not vary sufficiently around $\theta$, then, we get small Fisher Information and the bound on the variance of any unbiased estimator $\hat{\theta}$ grows. But why this grow is unbounded? It seems natural to expect that it should be somehow bounded by the size of the region with small likelihood variability.

In another words, why does CRLB rely on the local variability, but the bound on the estimator's variance could grow infinitely?

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  • $\begingroup$ Well, the Fisher information is defined via derivatives, so uses a local linear (well, quadratic since second derivatives) approx of the likelihood function. If you have an irregular likelihood function where this local approximation ideas works badly, then the CRLB could be bad ... Then you could turn to alternatives to the Fisher info/CRLB based on finite differences. $\endgroup$ – kjetil b halvorsen Jul 31 '18 at 11:00

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