Which distribution and link function is appropriate for studying the impact of miles driven on auto insurance claims?

The Poisson distribution with a log link function is particularly suitable for modeling count data, such as the number of insurance claims, which is inherently non-negative and often takes the form of counts (e.g., zero, one, two claims, etc.). In this context, the count of auto insurance claims can be influenced by continuous predictors, in this case, miles driven.

The choice of a log link function is beneficial because it helps to model the relationship between the predictor (miles driven) and the expected count of insurance claims in a way that maintains the non-negativity of the predicted counts. This linkage transforms the linear predictors into a rate scale, making it particularly effective when analyzing event counts over varying exposure levels like miles driven.

In contrast, distributions like the normal distribution might fall short because insurance claims can be skewed and not meet the assumptions of normality, especially if many observations are zero. Other distributions such as the binomial and exponential are also not as appropriate since they cater to different kinds of data. The binomial distribution focuses on proportions of successes in a fixed number of trials rather than direct counts, while the exponential distribution primarily models time until an event occurs.