Formula sheets

Poisson process with rate $\lambda$: $N(t)\sim\text{Poisson}(\lambda t)$; inter-arrivals $\sim\text{Exp}(1/\lambda)$; thinning gives independent Poisson processes with rates $p_i\lambda$.

MLE asymptotics $\;\hat\theta\dot\sim N\!\left(\theta,\;I(\theta)^{-1}\right)$, $I(\theta)=-E\!\left[\partial^2\ell/\partial\theta^2\right]$.

Likelihood ratio test $\;2(\ell_1-\ell_0)\dot\sim\chi^2_{\Delta p}$.

GLM $\;g(\mu_i)=\mathbf{x}_i^\top\boldsymbol\beta$, deviance $D=2\phi(\ell_{sat}-\ell_{model})$; AIC $=2p-2\ell$, BIC $=p\ln n-2\ell$.

AR(1) $\;X_t=\phi X_{t-1}+\varepsilon_t$: $\operatorname{Var}(X)=\sigma^2_\varepsilon/(1-\phi^2)$, $\rho_k=\phi^{k}$.