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Consider translated Weibull distribution with probability density function:

$$ f(x ; k, \lambda, \theta) = \frac{k}{\lambda} \left( \frac{x-\theta}{\lambda} \right)^{k-1} \exp\left( - \left(\frac{x-\theta}{\lambda} \right)^k \right) \chi_{x \ge \theta}(x) $$

Consider $ \mathbb{E}\left( -\partial_\theta^2 \log f(x; k, \lambda, \theta) \right) = \mathbb{E}\left( \frac{k-1}{(x-\theta)^2 } \left(1 + \left( \frac{x-\theta}{\lambda} \right)^k \right) \right)$. Notice that the expectation is only convergent for $k > 2$.

Question: What is the significance, or interpretation, of infinite matrix element of the Fisher information matrix ? Does it mean that the maximum likelihood estimator for $\theta$ parameter is not asymptotically normal ?

Thank you for any light shed on the subject.

P.S. This is a cross post, and the same question has been asked on math.SE.

Consider translated Weibull distribution with probability density function:

$$ f(x ; k, \lambda, \theta) = \frac{k}{\lambda} \left( \frac{x-\theta}{\lambda} \right)^{k-1} \exp\left( - \left(\frac{x-\theta}{\lambda} \right)^k \right) \chi_{x \ge \theta}(x) $$

Consider $ \mathbb{E}\left( -\partial_\theta^2 \log f(x; k, \lambda, \theta) \right) = \mathbb{E}\left( \frac{k-1}{(x-\theta)^2 } \left(1 + \left( \frac{x-\theta}{\lambda} \right)^k \right) \right)$. Notice that the expectation is only convergent for $k > 2$.

Question: What is the significance, or interpretation, of infinite matrix element of the Fisher information matrix ? Does it mean that the maximum likelihood estimator for $\theta$ parameter is not asymptotically normal ?

Thank you for any light shed on the subject.