The GLM consists of methods which model responses following a distribution from the exponential family. These distributions have a probability function which can be expressed as:
$$ f(x; \theta) = a(\theta)g(x)\exp\left[b(\theta)R(x)\right] $$
Therefore, the $t$-distribution is not a member of the exponential family.
My understanding is that the $t$ test assumes that responses are $t$-distributed, and therefore the $t$ test is not a GLM.
However, some sources (example) indicate that the $t$ test assumes that the responses are normally-distributed and thus belongs in the GLM.
- If responses are normally-distributed, wouldn't a $z$ test be more appropriate?
- Can the $t$ distribution be used as the probability distribution of the response $Y$ in a GLM?