Society of Actuaries (SOA) PA Practice Exam

What is the canonical link function for a binomial distribution in GLMs?

1. Probit

2. Logit

3. Cauchit

4. Cloglog

The correct answer is: Logit

The canonical link function for a binomial distribution in Generalized Linear Models (GLMs) is the logit link function. This is because the logit link function is specifically designed to relate the linear predictor to the probability of success in a binomial model. When dealing with a binomial distribution, we are often interested in modeling the probability of success (e.g., the probability that an event occurs) as a function of explanatory variables. The logit function transforms the probability \( p \) into an odds ratio by taking the natural logarithm of the ratio of the probability of success to the probability of failure, given by \( \log\left(\frac{p}{1-p}\right) \). This transformation ensures that the predicted probabilities remain within the bounds of 0 and 1, which is essential for probabilities. The other link functions, such as probit, Cauchit, and cloglog, although they can also be used in binomial GLMs, are not considered the canonical link function. They serve different purposes or have different interpretations but do not adhere to the preferred logit link that typically provides better interpretability and convergence properties in many practical applications. Thus, the logit link stands out as the canonical choice in this