# Can the likelihood function in generalized linear model be written in terms of the model parameter and the input variable?

1. the exponential family distribution has the following form: $$f_Y(y; \theta, \phi) = \exp \{ (y\theta - b(\theta))/a(\phi) + c(y, \phi)\} (2.4)$$ Then the mean is $$EY = b'(\theta)$$

I was wondering if in general, is $b'$ assumed to be invertible or injective in generalized linear models, so that we can write $\theta = b'^{-1}(EY)$?

2. The mean is related to the linear predictor via $g(EY) = X \beta$. Is the link function $g$ assumed to be injective or invertible in generalized linear models, so that $EY= g^{-1}(X\beta)$?

The purpose of the two questions is to help me to understand how the parameter $\beta$ in generalized linear models is estimated via MLE. In particular, can the pdf of $Y$ be written as $f_Y(y; \beta, X, \phi)$, for example, by $\theta = b'^{-1}(g^{-1}(X\beta))$, so that we can apply MLE on $f_Y(y; \beta, X, \phi)$? Thanks!

The answers to your next two questions are generally "No". You can take a look at Table 2.1 on page 30 of the book McCullagh and Nelder (1989). There are two rows for $b(\theta)$ and link function of common distributions. For example, $b(\theta)=\theta^2/2$ for normal distribution and $b(\theta)=-(-2\theta)^{1/2}$ for inverse Gaussian. The canonical link function for inverse Gaussian is $1/\mu^2$.
• Thanks! Do you mean the likelihood function may not be able to be rewritten in terms of $\beta$ and $X$? How does MLE estimate $\beta$ then? (or if the book has the answer, where is it?) – Tim Aug 19 '13 at 0:47