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Let's say we obtained a point maximum likelihood estimation $\hat{\theta}_\mathrm{MLE}\left(\mathbf{x}\right)$ from a set of measurements $\mathbf{x} = \left[x_1, x_2, \cdots, x_n \right]$ that follows a probability distribution $f_\mathbf{X}\left(\mathbf{x};\theta\right)$, where $\theta$ is the unknown population (true) parameter. The maximum likelihood estimator $T_\mathrm{MLE}\left(\mathbf{x}\right)$ is a derived random variable from $\mathbf{X}$ that follows a sampling distribution $f_{T_\mathrm{MLE}}\left(t;\theta\right)$.

In general, does the sampling distribution $f_{T_\mathrm{MLE}}\left(t;\hat{\theta}_\mathrm{MLE}\right)$ peak at $t=\hat{\theta}_\mathrm{MLE}$? In other words, is $\partial/\partial t f_{T_\mathrm{MLE}}\left(t;\hat{\theta}_\mathrm{MLE}\right) = 0$ at $t=\hat{\theta}_\mathrm{MLE}$?

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No, even if you assume the probability densities has every nice property you want (real-analytic, unimodal, all moments exists, ...).

For example, recall if iid $X_i\sim N(\mu,\sigma^2)$ (say unknown $\mu,\sigma^2$, same thing with known $\mu$), then the MLE is $\widehat{\sigma^2}_{MLE}=\frac1n\sum (X_i-\bar{X})^2$ (and $\hat{\mu}_{MLE}=\bar{X}$ which isn't what is interesting for this example). So in particular $n\widehat{\sigma^2}_{MLE}/\sigma^2\sim \chi^2_{n-1}$ which has mode at $n-3$ (assuming $n>3$), not $n$ or $n-1$.

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  • $\begingroup$ Great! Thank you. $\endgroup$
    – acoustica
    Oct 18, 2023 at 0:20

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