The support region for a selected model does not depend on $\theta$
The parameters are identifiable (i.e., for $\theta_1\ne\theta_2,\; f(t|\theta_1)\ne f(t|\theta_2), \;\forall t$)
The value of $\hat{\theta}{{MLE}}$ is on the interior of the parameter space $\Theta$
$f(t|\underline{\theta})$ has a $3^{rd}$ mixed partial derivative
$E\left[\frac{\partial^{2}\log(f(t|\theta))}{\partial\theta(\partial\theta)^T}\right]=\frac{\partial^2 E\left[\log(f(t|\theta))\right]}{\partial\theta(\partial\theta)^T}$
Elements of $\mathscr{I}{\theta}$ are finite and $\mathscr{I}{\theta}$ is a positive-definite matrix
$\mathscr{L}(\theta|\underline{x})\ge 0$
$\mathscr{L}(\theta|\underline{x})$ is not a pdf i.e. $\int \mathscr{L}(\theta|\underline{x})\;d\theta \ne 1$
Suggests (relatively) which values of $\theta$ are more likely to have generated the observed data $\underline{x}$ (assuming the chosen parametric model is correct)
If it exists, we say that the value of $\underline{\theta}$ that maximizes $\mathscr{L}(\underline{\theta}|\underline{x})$ is the maximum likelihood estimator (denoted $\hat{\theta}{{MLE}}$)
We often try to find $\hat{\theta}{{MLE}}$ by maximizing the log-likelihood function $$\mathcal{L}(\underline{\theta}|\underline{x})=\log\left[\mathscr{L}(\underline{\theta}|\underline{x})\right]$$
$$\mathscr{L}(\underline{\theta}|\underline{x})=\sum_{i=1}^{n}\mathscr{L}{i}(\underline{\theta}|x_i)=f(\underline{x}|\underline{\theta})=\prod{i=1}^{n}f(x_{i}|\underline{\theta}),\;\;\text{if}\;x_{i}\; iid$$
Both $f(t|\underline{\theta})$ and $\mathscr(\underline{\theta})$ start with a distributional assumption
$f(\underline{x}|\underline{\theta})$ relates
Is a function of $\underline{x}$ assuming $\underline{\theta}=\theta_{1},\theta_{2},...$ are
What data $\underline{x}$ are most likely to be produced by a distribution with parameters $\underline{\theta}$?
$f(x_i|\underline{\theta})$ is the probability density associated with observation $x_i$
But, this statement comes with the following assumptions
1) We know (or at least have specified) a functional form values for $\theta$
2) We know (or at least have specified) values for $\theta$
$\mathscr{L}(\underline{\theta}|\underline{x})$
Is a function of $\underline{\theta}$ assuming $\underline{x}=x_{1},...,x_{n}$ has already been observed
What values of $\underline{\theta}$ are most likely to have produced $\underline{x}$?
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.