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This is the empirical Fischer Information. Also consider the outer product with itself of the first derivative of the log likelihood with respect to the parameters. This will always be semi-negative definite.

As the two has same expected value, negative of empirical second derivative of the log likelihood with respect to the parameters should be semi-positive definite for $n$ large. But unlike the outer product with itself of the first derivative of the log likelihood with respect to the parameters, can it not be semi-positive definite for small $n$?

Thanks.

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  • $\begingroup$ It's hard to see why $n$ would play any role in the answer. This question is reminiscent of the one at stats.stackexchange.com/questions/7308, which perhaps might be relevant. $\endgroup$ – whuber Oct 29 '18 at 20:55

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