\[f(y_i,\theta_i,\phi)=\exp{[\phi\{y_i\theta_i-b(\theta_i)\}+c(y_i,\phi)]}\] As distribuições abaixo estão na forma da família exponencial.
\[f(y_i,\mu_i,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp \left[{-\frac{1}{2\sigma^2}(y_i-\mu_i)^2}\right ]\]
\[=\exp \left[{-\frac{1}{2\sigma^2}(y_i^2-2\mu_iy_i-\mu_i^2)-\frac{1}{2}\log(2\pi\sigma^2)}\right ]\]
\[ =\exp\left [ \frac{1}{\sigma^2}\left ( y_i\mu_i-\frac{\mu_i^2}{2} \right)- \frac{1}{2}\left(\frac{y_i^2}{\sigma^2}+\log(2\pi\sigma^2) \right ) \right ] \]
Deste modo, \(\phi=\frac{1}{\sigma^2}\), \(b'(\mu_i)=\frac{\mu_i^2}{2}\), \(c(y_i,\phi)=-1/2(\frac{y_i^2}{\sigma^2}+\log(2\pi\sigma^2)\), \(V(\mu_i)=\frac{\partial\mu_i}{\partial\theta_i}=\frac{\partial\mu_i}{\partial\mu_i}=1\), \(Var(y_i)=\phi^{-1}V(\mu_i)=\sigma^2 1=\sigma^2\).
\[f(y_i,\mu)=\frac{e^{-\mu}\mu^y}{y!}\]
\[f(y_i,\mu,\phi)=\exp\left [ -\mu+y\log(\mu)-\log(y!) \right ]\]
Assim, \(\theta=\log(\mu) \rightarrow e^{\theta}=e^{\log(\mu)} \rightarrow e^{\theta}=\mu\), \(\phi=1\), \(b(\theta)=\mu=e^{\theta}\), \(c(y_i,\phi)=-\log(y!)\), \(V(\mu)=\frac{\partial\mu}{\partial\theta}=e^{\theta}\), \(Var(Y_i)=\phi^{-1}V(\mu)=1*e^{\theta}=\mu\).
\[f(y,\mu,\phi)=\frac{1}{\Gamma(\phi)}\left ( \frac{\phi y}{\mu} \right )^{\phi}\exp\left \{ \frac{\phi y}{\mu}\right \}\frac{1}{y}\] Colocando na forma da família exponencial temos que:
\[f(y,\mu,\phi)=\exp\left \{ -\log\Gamma(\phi)+\phi\log(\phi y)-\phi\log(\mu)-\frac{\phi y}{\mu}-\log(y)\right \}\] \[=\exp\left \{ -\frac{\phi y}{\mu}+\phi\log(\phi y) -\log\Gamma(\phi)-\phi\log(\mu)-\log(y)\right \}\]
\[=\exp\left \{ \phi\left (-\frac{y}{\mu}-\log(\mu) \right )-\log\Gamma(\phi)+\phi\log(\phi y) -\log(y)\right \} \] Logo, \(\phi=\phi\),\(b(\theta)=\log(\mu) \rightarrow \theta=-\frac{1}{\mu}\rightarrow -\theta=\mu^{-1}\), \(\theta=-\frac{1}{\mu}\), \(c(y,\phi)=\log\Gamma(\phi)+\phi\log(\phi y)-\log(y)\), \(V(\mu)=\frac{\partial\mu}{\partial\theta}=\frac{1}{\theta^2}=\frac{1}{(-\frac{1}{\mu})^2}=\mu^2\), \(Var(y)=\phi^{-1}V(\mu)=\frac{\mu^2}{\phi}\).
Observação: \(\phi=\frac{\mu^2}{Var(y)}=\left ( \frac{\mu}{DP(y)} \right )^2=\left ( \frac{1}{CV}\right )^2\).
Casos particulares - \(\phi=1 \rightarrow\) exponencial; - \(\phi=\frac{k}{2}\) e \(\mu=k \rightarrow \chi^2_{(k)}\) - \(\phi= \infty \rightarrow\) normal; \(\phi\) grande \(\rightarrow Y\sim N(\mu,\frac{\mu^2}{\phi})\)
\[P[Y^{*}=y^{*}]=P[ny^{*}=Y^{*}]=\binom{n}{ny^{*}}\mu^{ny^{*}}(1-\mu)^{n-ny^{*}}=\binom{n}{ny^{*}}\left ( \frac{\mu}{1-\mu} \right )^{ny^{*}}(1-\mu)^{n}\]
Colocando na notação da família exponencial tem-se que
\[=\exp\left \{ n\left [ y^{*}\log\left ( \frac{\mu}{1-\mu} \right )+\log(1-\mu) \right]+\log\binom{n}{ny^{*}} \right \}\]
Assim,\(\phi=n\), \(\theta=\log\left ( \frac{\mu}{1-\mu} \right )\rightarrow e^{\theta}=\frac{\mu}{1-\mu} \rightarrow e^{\theta}(1-\mu)=\mu \rightarrow e^{\theta}-\mu e^{\theta}=\mu\). \(e^{\theta}=\mu+\mu e^{\theta} \rightarrow e^{\theta}=\mu(1+e^{\theta})\rightarrow \mu=\frac{e^{\theta}}{1+e^{\theta}}\), \(b(\theta)=-\log(1+\mu)=-\log\left (1-\frac{e^{\theta}}{1+e^{\theta}}\right )=-\log\left ( \frac{1}{1+e^{\theta}}\right )=\log(1+e^{\theta})\), \(c(y,\phi)=\log\binom{n}{ny^{*}}=\log\binom{\phi}{\phi y^{*}}\).
Função de variância
\[V(\mu)=\frac{\partial\mu}{\partial\theta}=\frac{e^{\theta}(1-e^{\theta})-e^{\theta}e^{\theta}}{(1+e^{\theta})^2}=\frac{e^{\theta}+e^{2\theta}-e^{2\theta}}{(1+e^{\theta})^2}=\frac{e^{\theta}}{(1+e^{\theta})^2}=\frac{e^{\theta}}{(1+e^{\theta})}\frac{1}{(1+e^{\theta})}=\mu(1-\mu)\\ Var(y^{*})=\phi^{-1}V(\mu)=\frac{1}{n}\mu(1-\mu)\]
\[f(y,\mu,\phi)=\frac{\phi^{\frac{1}{2}}}{\sqrt{2\pi y^3}}\exp\left \{ -\frac{\phi}{2\mu^2 y}(y-\mu)^2 \right \}\]
\[\exp\left \{-\frac{1}{2}\log(\phi)-\frac{1}{2}\log(2\pi y^3) -\frac{\phi}{2\mu^2 y}(y^2-2\mu y+\mu^2) \right \}\]
\[\exp\left \{-\frac{1}{2}\log(\phi)-\frac{1}{2}\log(2\pi y^3)-\frac{\phi y}{2\mu^2}+\frac{\phi}{\mu}-\frac{\phi}{2y}\right \}\]
\[\exp\left \{\phi\left (-\frac{y}{2\mu^2}+\frac{1}{\mu}\right )-\frac{\phi}{2y}+\frac{1}{2}\log(\phi)-\frac{1}{2}\log(2\pi y^3)\right \} \]
De modo que, \(\theta=-\frac{1}{2\mu^2}\rightarrow 2\mu^2=-\frac{1}{\theta}\rightarrow \mu^2=-\frac{1}{2\theta}\rightarrow \mu=(-2\theta)^{1/2}\), \(b(\theta)=\frac{1}{\mu}=-\frac{1}{(-2\theta)^{-1/2}}=-(-2\theta)^{-1/2}\), \(c(y,\phi)=-\frac{1}{2}\left [ \frac{\phi}{2y}-\log(\phi)+\log(2\pi y^3) \right]\).
Função Variância
\[V(\mu)=\frac{\partial\mu}{\partial \theta}=-\frac{1}{2}(-2\theta)^{-\frac{3}{2}}=\left [ -2\left ( -\frac{1}{2\mu^2} \right ) \right ]^{-\frac{3}{2}}=(\mu^{-2})^{-\frac{3}{2}}=\mu^{3}\]
\[f(T,\alpha,\beta)=\frac{1}{\sqrt{2\pi}}\exp\left \{ -\frac{1}{2\alpha^2}\left ( \frac{T}{\beta}+ \frac{\beta}{T}-2 \right ) \right \}\frac{T^{-\frac{3}{2}}(T+\beta)}{2\alpha\sqrt{\beta}}\]
Colocando na forma da família exponencial temos
\[\exp\left \{ -\frac{1}{2\alpha^2}\left ( \frac{T}{\beta}+ \frac{\beta}{T}-2 \right )+\log\left ( \frac{1}{\sqrt{2\pi}}\frac{T^{-\frac{3}{2}}(T+\beta)}{2\alpha\sqrt{\beta}} \right ) \right \}\]
\[\exp\left \{ -\frac{1}{2\alpha^2}\left ( \frac{T}{\beta}+ \frac{\beta}{T}-2 \right )+\log\left ( \frac{1}{\sqrt{2\pi}}\right )+\log \left ( \frac{T^{-\frac{3}{2}}(T+\beta)}{2\alpha\sqrt{\beta}} \right ) \right \}\]
\[\exp\left \{ -\frac{1}{2\alpha^2}\left ( \frac{T}{\beta}+ \frac{\beta}{T}-2 \right )-\log( \sqrt{2\pi})+\log ( {T^{-\frac{3}{2}}(T+\beta))}{-\log(2\alpha\sqrt{\beta})} \right \}\]
\[\exp\left \{ -\frac{1}{2\alpha^2}\left ( \frac{T}{\beta}+ \frac{\beta}{T}-2 \right )-\log( \sqrt{2\pi})+{-\frac{3}{2}}\log ( {T)+\log(T+\beta))}{-\log(2\alpha)+\log(\sqrt{\beta})} \right \}\]
\[\exp\left \{-\frac{T}{2\alpha^2\beta}-\frac{\beta}{2\alpha^2T}+ \frac{\alpha^2-\log(2\alpha)}{\alpha^2} -\frac{3}{2}\log(T)+\log(T+\beta)-\log(\sqrt{\beta})-\log(\sqrt{2\pi}) \right \}\]
\[\ldots\]