Generalized linear model with explicit conditioning

Fix $n\ge 1$, $p\ge 1$, a finite design matrix $X\in {\mathbb{R}}^{n\times p}$ with rows $x_i^{\mathsf{T}}$, a dispersion value $\varphi$, and an open parameter domain $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$. A generalized linear model (GLM) on this design consists of the response family in Explicit exponential-family response contracts, a differentiable one-to-one link $g:\mathcal{M}\to g(\mathcal{M})$ on a specified mean domain $\mathcal{M}$, and a specified map $\tau :\mathcal{M}\to J$ satisfying $b^{\mathrm{\prime }}(\tau (\mu ))=\mu$. For $\beta \in \mathrm{\Theta }$, put

${\eta }_i=x_i^{\mathsf{T}}\beta ,{\mu }_i=g^{-1}({\eta }_i),{\theta }_i=\tau ({\mu }_i),$

and require ${\eta }_i\in g(\mathcal{M})$ and ${\theta }_i\in J$ for every $i$. The canonical link is the choice $g=\tau$, equivalently ${\theta }_i={\eta }_i$. When ${\theta }_i={\eta }_i$ and the component scales are common, write $a_i(\varphi )=a(\varphi )>0$; this common-scale restriction is part of every formula that uses $a(\varphi )$.

The joint sampling density is the product density from The product response law has the product density. For an observed $y=(y_1,\dots ,y_n)$ with $f_i(y_i;{\theta }_i,\varphi )>0$ for every $i$, define its log likelihood by

$\mathrm{\ell }(\beta ;y,\varphi )={\sum }_{i=1}^n\{\frac{y_i{\theta }_i-b({\theta }_i)}{a_i(\varphi )}+c_i(y_i,\varphi )\}\mathrm{.}$

For a differentiable log likelihood, the score is $U(\beta ;y,\varphi ):={\mathrm{\nabla }}_{\beta }\mathrm{\ell }(\beta ;y,\varphi )$. When its entries have finite second moments, the expected score outer-product information is $I(\beta ,\varphi ):={\mathbb{E}}_{\beta ,\varphi }[U{U}^{\mathsf{T}}]$. This page treats $X$ as fixed or conditions on its realized value; it supplies no random-design asymptotic conclusion.