Modèle Binômial

$\bullet\hspace{2mm} Y_i \sim B(n_i,p_i) \Rightarrow f_i(y,\beta)=\frac{n_i!}{y!(n_i-y)!}p_i^y(1-p_i)^{n_i-y}$ avec $y\in\{0,...,n_i\}.$

$\bullet\hspace{2mm} \theta_i=\log\frac{p_i}{1-p_i}$, $b(\theta_i)=n_i\log(1+e^{\theta_i})$ et $\zeta=1.$

$\bullet\hspace{2mm} Var(Y_i)=n_ip_i(1-p_i) \hspace{5mm}\mbox{et}\hspace{5mm} \mu_i=n_ip_i.$

$\bullet\hspace{2mm}$Liens:
$\begin{array}{l} -\mbox{Mod\\lq ele Logit:} p_i=\frac{e^{<t_i,\beta>}}{1+e^{<t_i,... ...frac{\mu_i}{n_i}).\\ -\mbox{Mod\\lq ele Compl\'ementaire Log-Log:}\\ \end{array}$
$p_i=e^{-e^{<t_i,\beta>}}\hspace{3mm}\mbox{et}\hspace{3mm}l(\mu_i)=\log(-\log\frac{\mu_i}{n_i})$.

$\bullet\hspace{2mm}\hspace{2mm} \widehat{r_i}=\frac{y_i-n_i\widehat{p_i}}{\sqrt{n_i\widehat{p_i}(1-\widehat{p_i})}}$, $\widehat{y_i}=n_i\widehat{p_i}$ et $S.Dev=2\sum_i(y_i\log\frac{y_i}{\widehat{y_i}}+(n_i-y_i)\log\frac{n_i-y_i}{n_i-\widehat{y_i}}).$