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I would like to now what is the relation between the Hessian Matrix calculated at the maximum point of the log(likelihood) function, $H_{f}$ and the Hessian Matrix calculated at the minimum point of the -log(likelihood) function, $H_{-f}$.

The first should be definite negative and the second should be definite positive. But, is it possible to calculate the first hessian matrix from the second one?

It should be $H_{f}=-H_{-f}$, but I am not sure. Is there any proof of it? Thank you

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Let $\ell(\theta)$, for $\theta\in\Theta$, be the log-likelihood function. Then the maximum likelihood estimate is defined as

$$\hat\theta = \text{arg}\,\underset{\theta\in\Theta}{\min}-\ell(\theta)\\ \,\, = \text{arg}\, \underset{\theta\in\Theta}{\max}\ell(\theta)$$

Now, by the properties of the differentiation operator

$$\frac{d^2-\ell(\theta)}{d\theta d\theta^T} = -\frac{d^2\ell(\theta)}{d\theta d\theta^T}, $$

which holds for any $\theta$. In particular, for $\hat\theta$ we have that

$$ \frac{d^2-\ell(\theta)}{d\theta d\theta^T}|_{\theta=\hat\theta} = -\frac{d^2\ell(\theta)}{d\theta d\theta^T}\vert_{\theta=\hat\theta}. $$

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