# How to derive canonical link function of exponential distribution in GLM

I want to know the derivation, how to basically calculate it

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Commented Aug 1 at 18:27
• Welcome to Cross Validated! Do you mean something like, given the parametric family of the conditional distribution (Poisson, gamma, binomial, Gaussian, etc), what is the canonical link function, almost like a function mapping the likelihood to the canonical link?
– Dave
Commented Aug 1 at 19:12

GLM is essentially a conditional dispersed version of the exponential family distribution which can be written tersely as $$f(y;θ)=\exp(\frac{θy−b(θ)}{ϕ}+c(y,ϕ))$$, where $$ϕ$$ is the dispersion parameter, $$θ$$ is the usual canonical parameter, $$b(θ)$$ is the log partition function for normalization and $$c(y,ϕ)$$ is the base measure. On the other hand, for an exponential distribution as a special Gamma distribution with shape $$1$$ and rate $$𝜆$$ can be expressed as $$f(y;λ)=λ\exp(−λy)=\exp(\log(λ)−λy),y≥0$$ By matching the two terms it's apparent we have $$θ=−λ, b(θ)=−\log(−θ), ϕ=1, c(y,ϕ)=0$$ Furthermore it's well known the relation between the mean $$\mu$$ of the exponential distribution and its rate $$λ$$ is $$\mu=\frac{1}{λ}$$, and the link function $$g(μ)$$ links the mean $$μ$$ to the linear predictor $$η$$ which is nothing but the canonical parameter $$θ$$ when $$g$$ is a canonical link function. Therefore we finally arrive at $$g(μ)=η=θ=−λ=-\frac{1}{μ}$$