Maximum Likelihood Regularity Conditions

  1. The support region for a selected model does not depend on $\theta$

  2. The parameters are identifiable (i.e., for $\theta_1\ne\theta_2,\; f(t|\theta_1)\ne f(t|\theta_2), \;\forall t$)

  3. The value of $\hat{\theta}{{MLE}}$ is on the interior of the parameter space $\Theta$

  4. $f(t|\underline{\theta})$ has a $3^{rd}$ mixed partial derivative

  5. $E\left[\frac{\partial^{2}\log(f(t|\theta))}{\partial\theta(\partial\theta)^T}\right]=\frac{\partial^2 E\left[\log(f(t|\theta))\right]}{\partial\theta(\partial\theta)^T}$

  6. Elements of $\mathscr{I}{\theta}$ are finite and $\mathscr{I}{\theta}$ is a positive-definite matrix

Properties of the Likelihood Function $\mathscr{L}$

  1. $\mathscr{L}(\theta|\underline{x})\ge 0$

  2. $\mathscr{L}(\theta|\underline{x})$ is not a pdf i.e. $\int \mathscr{L}(\theta|\underline{x})\;d\theta \ne 1$

  3. Suggests (relatively) which values of $\theta$ are more likely to have generated the observed data $\underline{x}$ (assuming the chosen parametric model is correct)

  4. 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 $\hat{\theta}{{MLE}}$)

  5. We often try to find $\hat{\theta}{{MLE}}$ by maximizing the log-likelihood function $$\mathcal{L}(\underline{\theta}|\underline{x})=\log\left[\mathscr{L}(\underline{\theta}|\underline{x})\right]$$

The likelihood is equal to the joint probability of the data

$$\mathscr{L}(\underline{\theta}|\underline{x})=\sum_{i=1}^{n}\mathscr{L}{i}(\underline{\theta}|x_i)=f(\underline{x}|\underline{\theta})=\prod{i=1}^{n}f(x_{i}|\underline{\theta}),\;\;\text{if}\;x_{i}\; iid$$


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teachingApps documentation built on July 1, 2020, 5:58 p.m.