Maximum Likelihood Estimation

$f(\mathbb{D} | \theta)$

$\hat{\theta}\text{ML} = \text{argmax}{\theta} g(x|\theta) = \text{argmax}_{\theta} \ln g(x | \theta)$

$\hat{\theta}\text{ML} = \text{argmax}{\theta} g(x_1,x_2,\cdots,~x_n | \theta)$ $= \text{argmax}{\theta} \prod\limits{k=1}^{n} g(x_k|\theta)$ -- i.i.d. / r.s. $= \text{argmax}{\theta} \sum\limits{k=1}^{n} \ln g(x_k|\theta)$

$\partial L \over \partial \theta} |{\theta = \hat\theta\text{ML}} = 0$

$\therefore$ Check ${\partial L \over \partial \theta} |{\theta = \hat\theta\text{ML}} < 0$

$\hat {h(\theta)}^\text{ML} = h(\hat\theta^\text{ML})$

$x_1, \cdots, x_n \sim \text{Geometric} (P)$ $\hat{\sigma^2}^\text{ML$

$\hat{1 \over p} = \overline{X_n} \Rightarrow \hat{p} = {1 \over \overline{x}}$

$\hat{\sigma^2}}^\text{ML} = {q \over p^2} = {\hat{{1-p} \over p^2}}^\text{ML} = 1 - {{1 \over \overline{x_n}} \over {{1 \over \overline{x_n}}^2}$

Max-likelihood: Tries to give the best PDF.

Max-likelihood parameter as $\hat \theta$

$$\hat \theta ^\text{ML} = \text{argmax}_{\theta} f(x_1, x_2, \cdots x^n | \theta) = \text{argmax}_{\theta} \ln f(x_1, x_2, \cdots x^n | \theta) = \text{argmax}_{\theta} L$$

Assuming IID

$$= \ln \prod_{k=1}^{n} f(x_k | \theta) = \text{argmax}_{\theta} \sum_{k=1}^{n} \ln f(x_k | \theta)$$

Maximum Likelihood Estimation

1. consistent (convergent in probability)
2. Asymptotically Normal
3. Invariance Principle $\hat{g(\theta){ML}} = g(\hat\theta{ML})$