# Peak location of the maximum likelihood estimator's sampling distribution

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 $$t=\hat{\theta}_\mathrm{MLE}$$?

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 $$n-1$$.