Distribution Families

Definition

A distribution family (density family) on $X$ is a family of distributions (densities) $\set{p^{(\theta )}}_{\theta \in \Theta }$ on $X$. We call the index set $\Theta $ (see Families) the parameters. Frequently $\Theta \subset \R ^p$ where $p \in \N $.

Similarly, a conditional distribution family (conditional density family) on $Z$ from $X$ is a family $\set{q^{(\theta )}}_{\theta \in \Theta }$ whose terms $q^{(\theta )}: Z \times X \to \R $ are such that $q^{(\theta )}(\cdot , \xi ): Z \to \R $ is a distribution (density) for every $\xi \in X$.

Examples

For example, let $\Theta = [0, 1]$ and consider the family of distributions $\set{p^{(\theta )}: \set{0, 1} \to [0,1]}_{\theta \in [0,1]}$ defined by, for each $\theta \in [0,1]$,

\[ p^{(\theta )}(1) = \theta \text{ and } p^{(\theta )}(0) = 1-\theta . \]

This family is called the Bernoulli family and $p^{(\theta )}$ is called a Bernoulli distribution with parameter $\theta $.

For a second example, let $\Theta = \R \times \R _{+}$ and consider the family of densities $\set{f^{(\theta )}: \R \to \R }_{\theta \in \Theta }$ defined by, for each $\theta = (\mu , \sigma ) \in \Theta ,$

\[ f^{(\theta )}(x) = (1/\sqrt{2\pi }\sigma )\exp((x - \mu )/\sigma ^2). \]

This family is called the normal family and $f^{(\theta )}$ with $\theta = (\mu , \sigma )$ is called a normal density with mean $\mu $ and variance $\sigma ^2$.