What is the canonical link for the gamma/exponential distribution?

What is the canonical link for the gamma/exponential distribution?

• A. Identity
• B. Log
• C. Inverse
• D. Inverse squared

Answer: (C)

In GLMs, the canonical link is the one that makes the mean μ enter the linear predictor exactly as the natural parameter of the exponential-family form. For the gamma distribution (including the exponential as a special case), the natural parameter that pairs with the data is proportional to -1/μ. That means the link between μ and the linear predictor is g(μ) = 1/μ, i.e., the inverse link. This alignment with the natural parameter simplifies the model’s likelihood and estimation. Other links may be used, but they do not preserve the natural-parameter form.

In GLMs, the canonical link is the one that makes the mean μ enter the linear predictor exactly as the natural parameter of the exponential-family form. For the gamma distribution (including the exponential as a special case), the natural parameter that pairs with the data is proportional to -1/μ. That means the link between μ and the linear predictor is g(μ) = 1/μ, i.e., the inverse link. This alignment with the natural parameter simplifies the model’s likelihood and estimation. Other links may be used, but they do not preserve the natural-parameter form.