Introduction to Generalized Linear Models (GLMs)

Generalized Linear Models (GLMs) are a flexible generalization of ordinary least squares regression. They allow for response variables that have error distribution models other than a normal distribution and for the linear predictor to be related to the response variable via a link function.

Why GLMs?

Traditional linear regression assumes that the response variable is normally distributed and that the variance is constant. However, many real-world phenomena, especially in actuarial science, violate these assumptions. For instance, claim counts are often non-negative integers (Poisson or Negative Binomial), and binary outcomes (e.g., policy acceptance) follow a Bernoulli distribution. GLMs provide a unified framework to model such data.

Components of a GLM

A GLM consists of three key components:

Common Distributions and Link Functions

Distribution	Common Link Function	Canonical Link	Response Variable Type
Normal	Identity ( $g(\mu) = \mu$ )	Identity	Continuous
Binomial	Logit ( $g(\mu) = \log(\mu / (1-\mu))$ )	Logit	Proportions or Counts (out of n trials)
Poisson	Log ( $g(\mu) = \log(\mu)$ )	Log	Counts (non-negative integers)
Gamma	Inverse ( $g(\mu) = 1/\mu$ )	Inverse	Positive continuous (e.g., claim amounts)

The canonical link function is a specific link function that simplifies the mathematical properties of the GLM and is often the default choice. For example, the logit link is canonical for the Binomial distribution, and the log link is canonical for the Poisson distribution.

Model Fitting and Interpretation

GLMs are typically fitted using Maximum Likelihood Estimation (MLE). The process involves iteratively estimating the model parameters until convergence. Interpretation of coefficients depends on the link function. For a logit link, coefficients represent the change in the log-odds of the outcome for a one-unit change in the predictor. For a log link, coefficients represent the change in the log of the expected count/rate.

Remember: The interpretation of coefficients in GLMs is tied to the chosen link function. Always consider the link function when explaining the model's results.

GLMs in Actuarial Science

GLMs are fundamental in actuarial modeling for various applications, including:

Pricing Insurance Products: Modeling claim frequencies and severities.
Reserving: Estimating future claim payments.
Risk Management: Assessing and quantifying various risks.
Fraud Detection: Identifying unusual patterns in claims data.

What are the three essential components of a Generalized Linear Model?

The three components are the random component (distribution of the response), the systematic component (linear predictor), and the link function (connecting the mean to the linear predictor).

This diagram illustrates the flow of information within a Generalized Linear Model. The predictor variables ( $X_i$ ) are combined linearly to form the linear predictor ( $\eta$ ). This linear predictor is then transformed by the link function ( $g$ ) to model the expected value of the response variable ( $\mu$ ). Finally, the response variable ( $Y$ ) is assumed to follow a specific distribution from the exponential family, with its mean related to $\mu$ .

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Key Takeaways for Actuarial Exams

For actuarial exams, understanding the theoretical underpinnings of GLMs is crucial. Be prepared to:

Identify the appropriate distribution and link function for a given problem.
Interpret model coefficients in the context of the link function.
Understand the assumptions and limitations of GLMs.
Differentiate GLMs from standard linear regression.

Learning Resources

Generalized Linear Models - Society of Actuaries(documentation)

Official study notes from the Society of Actuaries that cover GLMs as part of the Exam P syllabus. This is a primary resource for exam preparation.

Introduction to Generalized Linear Models - StatQuest with Josh Starmer(video)

A highly visual and intuitive explanation of GLMs, breaking down the concepts of distributions, link functions, and the overall model structure.

Generalized Linear Models - Wikipedia(wikipedia)

A comprehensive overview of GLMs, including their history, mathematical formulation, and applications. Useful for a broad understanding.

An Introduction to Generalized Linear Models - UCLA Statistical Consulting(blog)

A clear and accessible explanation of GLMs, focusing on the intuition behind the components and how they differ from linear regression.

GLM Theory and Applications - R Documentation(documentation)

A detailed technical document on Generalized Linear Models in R, covering the underlying theory, implementation, and common usage patterns.

Actuarial Mathematics for Life Contingent Risks - Chapter 10: Generalized Linear Models(paper)

A sample chapter from a widely used actuarial textbook, providing a rigorous treatment of GLMs within the context of life contingencies.

Understanding Generalized Linear Models - Towards Data Science(blog)

An article that explains GLMs with practical examples and code snippets, making the concepts more tangible for learners.

Generalized Linear Models - Coursera (e.g., from a statistics course)(video)

A lecture from a university-level statistics course on Coursera, offering a structured approach to learning GLMs.

GLM: Poisson Regression - DataCamp(tutorial)

A tutorial focusing on Poisson regression, a common type of GLM, with practical implementation in R and explanations of its components.

The Theory of Generalized Linear Models - Cambridge University Press(paper)

While this links to a book, the abstract and table of contents often provide a good overview of the theoretical depth of GLMs, suitable for advanced study.

Introduction to GLMs