Exam MAS-II

The Modern Actuarial Statistics II exam is a 4-hour computer-based exam covering Credibility Theory, Linear Mixed Models, Statistical Learning, and Time Series as part of the ACAS credentialing pathway.

Prerequisite knowledge

• Calculus and Probability at a thorough level.
• Linear Algebra at the level assumed as a prerequisite for an undergraduate regression course.
• All concepts from MAS-I, including Generalized Linear Models and Statistical Learning fundamentals.

Learning Objectives

A. Introduction to Credibility (15–25%)

Candidates should understand and be able to calculate credibility weighted estimates.

1. Calculate classical (Limited Fluctuation Credibility), Bühlmann Credibility, Bühlmann-Straub Credibility, and Bayesian Credibility-weighted estimates for frequency, severity, and aggregate loss
2. Understand the framework used for the classical (limited fluctuation), Bühlmann, Bühlmann-Straub, and Bayesian credibility procedures
3. Determine the standard for full credibility and apply the square-root rule for partial credibility
4. Compute the Bühlmann structural parameters — the Expected Value of Process Variance (EPV) and the Variance of Hypothetical Means (VHM) — and the credibility constant k they define
5. Derive the posterior and predictive distributions under a conjugate prior, and recognize when the Bayesian and Bühlmann estimates coincide
6. Estimate the structural parameters from data using empirical Bayes (non-parametric and semi-parametric) methods
7. Select an appropriate complement of credibility

Credibility procedures

• Limited Fluctuation Credibility — Full Credibility Standard, Partial Credibility
• Bühlmann Credibility — Expected Value of Process Variance, Variance of Hypothetical Means
• Bühlmann-Straub Credibility — unequal exposures
• Bayesian Credibility — Conjugate Prior, Predictive Distribution
• Empirical Bayes Credibility — structural parameters estimated from experience

Readings: Tse

B. Linear Mixed Models (10–20%)

Candidates should understand the structure of Linear Mixed Models, including how to accommodate models with correlated observations or models where the variance is either not assumed to be constant or a function of the mean.

1. Understand the assumptions behind the Linear Mixed Model design, and distinguish fixed effects from random effects
2. Understand how to use a Hierarchical Model, including random intercept and random slope specifications and nested versus crossed grouping factors
3. Interpret the variance components and the implied covariance structure of the residuals, and compute the Intraclass Correlation
4. Fit a mixed model by Restricted Maximum Likelihood (REML) or maximum likelihood, and know which of the two a given comparison requires
5. Interpret output from a Linear Mixed Model and make appropriate choices when evaluating modeling options, using the Likelihood Ratio Test, AIC and BIC
6. Interpret linear mixed model diagnostics and summary statistics to evaluate the model structure and variable selection
7. Recognize the Best Linear Unbiased Predictor (BLUP) of a random effect as a credibility-weighted estimate

Model components

• Fixed Effects — the population-average part, X\beta
• Random Effects — group-level departures, Zb
• Variance Components · Covariance Structure · Intraclass Correlation
• Random Intercept and Slope · Hierarchical Model
• Restricted Maximum Likelihood · Likelihood Ratio Test · Best Linear Unbiased Predictor

Readings: West

C. Statistical Learning (40–50%)

Candidates should understand the mechanics of the algorithms identified in the tasks below and recognize their inherent strengths and weaknesses to be able to select the most appropriate procedure for the learning task at hand.

1. Distinguish supervised from unsupervised learning, and explain the bias-variance tradeoff behind test-error behaviour
2. Estimate test error by cross-validation and estimate sampling variability by the bootstrap
3. Compute K-Nearest Neighbors (KNN)
4. Prune Decision Trees by cost-complexity pruning
5. Calculate summary statistics for a set of decision trees (e.g., Gini Index, Entropy, Residual Sum of Squares)
6. Understand the assumptions underlying different Tree Ensemble methods — bagging, random forests and boosting — and the improvements they can make to decision trees
7. Read out-of-bag error and variable importance from a fitted ensemble
8. Compute elements of Principal Components Analysis (PCA) (e.g., loading vectors, variance explained)
9. Interpret Principal Components Analysis (PCA) software outputs, including the scree plot
10. Perform the computations behind Clustering procedures (e.g., K-Means Clustering, hierarchical clustering)
11. Interpret clustering procedure outputs, including the dendrogram and the choice of linkage
12. Interpret Neural Network results, including the role of the activation function, hidden layers, and backpropagation
13. Apply regularization to control model complexity in a linear or extended linear model
14. Calculate measures of model predictive accuracy (e.g., Lift, Gini Index, AUROC, the confusion matrix)
15. Compare models via predictive performance measures (e.g., double lift chart, quantile plot)

Learning framework

• Statistical Learning — the overall framework these methods sit in
• Supervised Learning · Unsupervised Learning
• Bias-Variance Tradeoff · Cross-Validation · Bootstrap · Regularization
• Test error measured by the Mean Square Error; bootstrap variability by the Standard Deviation of the resampled estimates

Supervised methods

• K-Nearest Neighbors
• Decision Tree — Tree Pruning, Gini Index, Entropy, Residual Sum of Squares
• Tree Ensemble — Bagging, Random Forest, Boosting, Out-of-Bag Error, Variable Importance
• Neural Network — Activation Function, Backpropagation

Unsupervised methods

• Principal Components Analysis — Loading Vector, Proportion of Variance Explained, Scree Plot
• Clustering — K-Means Clustering, Hierarchical Clustering, Dendrogram, Linkage

Model evaluation

• Confusion Matrix · AUROC · Gini Index
• Lift · Double Lift Chart · Quantile Plot

Readings: James et al. · GLM · Salis

D. Time Series with Constant Variance (15–25%)

Candidates should understand the basic applications of the Auto Regressive Integrated Moving Average (ARIMA) time series model.

1. Model relationships of current and past values of a statistic/metric using autoregressive and moving average terms
2. Test a series for stationarity and achieve it by differencing; recognize white noise and the random walk as limiting cases
3. Identify a model from the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF), and check residuals against the same plots
4. Understand the framework of ARIMA models (e.g., trends and seasonality)
5. Separate a series into trend, seasonal and irregular parts by decomposition, and smooth it with exponential smoothing
6. Calculate trends and seasonality using Time Series with regression (e.g., deterministic vs. stochastic trend)
7. Interpret time series output to make forecasts, with prediction intervals

Building blocks

• Stationarity · White Noise · Random Walk · Differencing
• Autocorrelation Function · Partial Autocorrelation Function
• Autoregressive Model · Moving Average Model · ARIMA

Trend and seasonality

• Deterministic and Stochastic Trend · Seasonality
• Time Series Decomposition · Exponential Smoothing · Time Series Forecast

Readings: Cowpertwait

Source Material

Source Material

• Introductory Time Series with R (Cowpertwait - 2009)
  • D1–D4 — Chapters 1–5 (excluding Sections 3.3 and 3.4), 6, 7 (Sections 7.1, 7.2 and 7.3)
• Generalized Linear Models for Insurance Rating (Goldburd et al. - 2020)
  • C10–C11 — CAS Monograph #5, 2nd edition, Chapter 7
• An Introduction to Statistical Learning (James et al. - 2021)
  • C1–C9 — 2nd edition, Chapters 1 (Background reading only), 2.2, 4.4.2 (Confusion Matrix Only), 8, 10, 12. Exam questions will not be sourced directly from Chapter 1.
• Nonlife Actuarial Models (Tse - 2009)
  • A1–A2 — Chapters 6.1–6.3, 7.1–7.4, 8.1–8.2, and 9.1–9.2
• Linear Mixed Models (West et al. - 2022)
  • B1–B4 — 3rd Edition, all chapters, excluding coding examples; Additional Notes on Shrinkage Means (http://www-personal.umich.edu/~bwest/shrinkage.doc)