Exponential Family Distributions for GLM Applications
Study the exponential family of distributions and its role in generalized linear models for Exam MAS-I.
What Is the Exponential Family
The exponential family of distributions is a broad class that includes most of the distributions used in actuarial GLM modeling. A distribution belongs to the exponential family if its probability density (or mass) function can be written in the form f(y; theta, phi) = a(y, phi) * exp((y * theta - b(theta)) / phi). Here theta is the natural (canonical) parameter, phi is the dispersion parameter, b(theta) is the cumulant function (also called the log-partition function), and a(y, phi) is a normalization factor.
This form may look abstract, but it unifies the theory of generalized linear models. Every distribution in the exponential family shares the same estimation framework, the same relationship between mean and variance, and the same deviance structure. Learning the exponential family once gives you a unified understanding of all GLM models.
Key Properties from the Cumulant Function
The cumulant function b(theta) determines the mean and variance of the distribution through its derivatives. The mean is E[Y] = b'(theta), the first derivative of b with respect to theta. The variance is Var(Y) = phi * b''(theta), where b''(theta) is the second derivative. Since b''(theta) is a function of theta, and theta is related to the mean mu through mu = b'(theta), the variance is a function of the mean: Var(Y) = phi * V(mu), where V(mu) = b''(theta) evaluated at the theta corresponding to mu.
The function V(mu) is called the variance function. It characterizes each exponential family member and determines how the variance changes with the mean. This is a powerful concept: knowing V(mu) tells you which distribution family you are working with, and it directly affects how observations are weighted in the GLM estimation process.
Common Exponential Family Members
The Normal distribution has theta = mu, b(theta) = theta^2/2, V(mu) = 1, and phi = sigma^2. The constant variance function V(mu) = 1 means the variance does not depend on the mean. The canonical link is the identity: g(mu) = mu.
The Poisson distribution has theta = ln(mu), b(theta) = exp(theta) = mu, V(mu) = mu, and phi = 1. The variance equals the mean, and the canonical link is the log: g(mu) = ln(mu). This is the most common model for claim counts.
The Gamma distribution has theta = -1/mu, b(theta) = -ln(-theta) = ln(mu), V(mu) = mu^2, and phi = 1/alpha (where alpha is the shape parameter). The variance is proportional to the square of the mean, and the canonical link is the reciprocal: g(mu) = 1/mu. The Gamma is commonly used for modeling claim severities because V(mu) = mu^2 implies that larger claims have more variability, which matches empirical patterns.
Additional Family Members
The Binomial distribution (for a single trial, i.e., Bernoulli) has theta = ln(mu/(1-mu)) = logit(mu), b(theta) = ln(1 + exp(theta)), V(mu) = mu*(1-mu), and phi = 1. The canonical link is the logit: g(mu) = ln(mu/(1-mu)). This is the standard model for binary outcomes such as claim occurrence indicators.
The Inverse Gaussian distribution has V(mu) = mu^3 and canonical link g(mu) = 1/mu^2. It has an even heavier variance-mean relationship than the Gamma, making it suitable for very heavy-tailed severity data. The Tweedie family generalizes these relationships with V(mu) = mu^p, where p determines the specific family member: p = 0 (Normal), p = 1 (Poisson), p = 2 (Gamma), p = 3 (Inverse Gaussian). For 1 < p < 2, the Tweedie distribution is a compound Poisson-Gamma, useful for modeling aggregate claims that include zeros.
Canonical Link Functions
Each exponential family member has a canonical link function that connects the mean mu to the linear predictor eta = x'*beta. The canonical link is the function g such that g(mu) = theta, the natural parameter. Using the canonical link simplifies the GLM estimation because the sufficient statistic for beta becomes X'*y (the matrix of covariates transposed times the response vector), and the Fisher information matrix takes a particularly simple form.
The canonical links are: identity for Normal, log for Poisson, reciprocal for Gamma, logit for Binomial, and 1/mu^2 for Inverse Gaussian. While canonical links have theoretical advantages, non-canonical links are sometimes preferred in practice. For example, the log link is commonly used with the Gamma distribution for severity modeling because it ensures positive fitted values and provides multiplicative rating factors, even though the canonical Gamma link is the reciprocal.
Sufficient Statistics and Maximum Likelihood
Exponential family distributions have sufficient statistics that can be expressed in a fixed-dimensional form regardless of sample size. For a GLM with an exponential family response, the sufficient statistics for the regression coefficients beta are the p quantities X' * y (one for each covariate). This is a direct consequence of the exponential family form and explains why GLMs can be efficiently estimated even with very large data sets.
The score equation for maximum likelihood estimation is X' * W * (y - mu) = 0, where W is a weight matrix depending on the variance function and the link function. The iteratively reweighted least squares (IRLS) algorithm solves this equation by repeatedly updating the weights and fitting a weighted least squares problem. This algorithm is the standard method for fitting GLMs and converges quickly for well-specified models.
Common Pitfalls
A frequent mistake is confusing the natural parameter theta with the mean mu. For the Poisson distribution, theta = ln(mu), not mu itself. For the Gamma, theta = -1/mu. When verifying that a distribution belongs to the exponential family, students sometimes try to force a distribution into the canonical form incorrectly. Practice writing out each distribution's density and matching terms to the exponential family formula.
Another error is assuming that the dispersion parameter phi is always 1. For Poisson and Binomial distributions, phi = 1 (these are called "known dispersion" families). For Normal, Gamma, and Inverse Gaussian, phi is an additional parameter that must be estimated. The distinction matters for computing standard errors and confidence intervals.
Study Tips
Exam MAS-I tests identification of exponential family members, their canonical links, and variance functions. Memorize the table: Normal (V=1, identity link), Poisson (V=mu, log link), Gamma (V=mu^2, reciprocal link), Binomial (V=mu(1-mu), logit link), Inverse Gaussian (V=mu^3, 1/mu^2 link). Practice deriving E[Y] and Var(Y) from b(theta) for each family member. Know the relationship Var(Y) = phi * V(mu) and understand how V(mu) determines the weighting in GLM estimation. Be prepared to verify that a given distribution belongs to the exponential family by rewriting its density in canonical form.