We are given
1) Y = $(Y_1,Y_2,...,Y_n)^T$ ~ Exponential
2) E[Y] = $\mu$ = X$\beta$, where X $\in R^{nxr}$ and $\beta \in R^r$
My question is can we apply the glm in this case? The case where the canonical link does not relate $\mu$ and $X\beta$. If the answer is Yes, then how? if no, then what is the suitable modelling technique instead?
In addition, if we are given Y = $(Y_1,Y_2,...,Y_n)^T$ ~ Poisson or Inverse Gaussian instead, how can we apply glm?
Since you have $\mu = X\beta$ this is the identity link.
To fit an exponential GLM, you fit a gamma GLM but instead of estimating the dispersion parameter $\phi$ from the data, you specify that it's 1.
This doesn't affect the parameter estimation, only the standard errors.
[Consequently in R you simply specify the dispersion parameter when calling summary (summary(myExpGLMfitobject,dispersion=1)) . See ?summary.glm.]
For the Poisson and Inverse Gaussian cases you simply specify the relevant family. [In R, see ?family.]
family contains an example for the binomial. Beware, however that as you move away from the canonical link (in some sense), the easier it becomes to encounter difficulties in optimizing the likelihood. (Good starting points will often help, but some of the potential problems are inherent).
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