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In Generalized Linear Models by Peter McCullagh, John Ashworth Nelder,

  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!

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For the title, yes, any likelihood function is in terms of parameters and input variables.

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 $1/\mu^2$.

In this way, parameters can be estimated via MLE with derivatives of likelihood function like score function, not as easy as your formula. I believe you can find the answer in the book.

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  • $\begingroup$ 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?) $\endgroup$ – Tim Aug 19 '13 at 0:47
  • $\begingroup$ As I said, "any likelihood function is in terms of parameters and input variables." You can find parameter estimation in the book, for example, Section 4.4 staring from page 114 for binary data. $\endgroup$ – Randel Aug 19 '13 at 1:13

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