Statistical Model

A statistical model is a collection of probability distributions that describes the behavior of a (set of) random variable(s) in a statistical experiment. The collection is denoted by

$$\mathcal{P}=\{P_{\theta }:\theta \in \Theta ,P_{\theta }\text{ is a probability distribution on }\mathcal{X}\},$$

where $\mathcal{X}$ is the sample space, and $\Theta$ is the parameter set.

Notions Around Statistical Model

• If $\Theta$ lies in a finite-dimensional space, then the model is parametric.
• If $\Theta$ is of infinite dimension, then the model is non-parametric.
• If the underlying probability of the experiment belongs to $\mathcal{P}$, then the model is well-specified.
• The parameter $\theta$ is called identifiable if $\Theta \ni \theta \mapsto P_{\theta }$ is injective, i.e., $\theta \ne {\theta }^{\prime }⟹P_{\theta }\ne P_{{\theta }^{\prime }}$