Casualty Actuarial Society Credentials (ACAS/FCAS) MAS-II — Modern Actuarial Statistics II Syllabus

MAS-II — Modern Actuarial Statistics II syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for MAS-II — Modern Actuarial Statistics II in Casualty Actuarial Society Credentials (ACAS/FCAS), not a summary of it.

1. 1

   Bayesian Analysis and Markov Chain Monte Carlo

   3 topics

   • Bayesian Inference Fundamentals
     • Prior, likelihood, and posterior distributions
     • Conjugate priors and posterior summaries
     • Credible intervals and Bayesian point estimation
   • Computational Bayesian Methods
     • Gibbs sampling and Metropolis-Hastings algorithms
     • Convergence diagnostics and effective sample size
   • Bayesian Credibility Connections
     • Hierarchical models for credibility
     • Empirical vs. fully Bayesian approaches
2. 2

   Linear Models and Regression

   3 topics

   • Simple and Multiple Linear Regression
     • Ordinary least squares estimation and assumptions
     • Inference, confidence, and prediction intervals
     • Multicollinearity and diagnostics
   • Model Selection and Validation
     • Subset selection, stepwise methods, and shrinkage (ridge/lasso)
     • Cross-validation and the bias-variance tradeoff
   • Residual Analysis and Transformations
     • Diagnosing nonlinearity, heteroscedasticity, and outliers
     • Variable transformations and interaction terms
3. 3

   Generalized Linear Models

   3 topics

   • GLM Structure and Theory
     • Exponential family distributions
     • Link functions and the linear predictor
     • Maximum likelihood and iteratively reweighted least squares
   • GLMs for Insurance Applications
     • Poisson and negative binomial models for frequency
     • Gamma and Tweedie models for severity and pure premium
     • Offsets, weights, and exposure handling
   • GLM Diagnostics and Interpretation
     • Deviance, residuals, and dispersion
     • Interpreting coefficients and relativities
4. 4

   Time Series Analysis

   3 topics

   • Stationary Time Series
     • Autocorrelation and partial autocorrelation functions
     • AR, MA, and ARMA models
   • Nonstationary and Seasonal Models
     • Differencing and ARIMA models
     • Seasonal decomposition and SARIMA
   • Forecasting
     • Point and interval forecasts
     • Forecast accuracy evaluation
5. 5

   Statistical Learning and Predictive Modeling

   3 topics

   • Tree-Based Methods
     • Decision trees, bagging, and random forests
     • Boosting and gradient boosting machines
   • Unsupervised Learning
     • Principal component analysis
     • Clustering methods (k-means, hierarchical)
   • Model Evaluation in Practice
     • Classification metrics (ROC, AUC, lift)
     • Overfitting, regularization, and generalization

MAS-II — Modern Actuarial Statistics II flashcards for Casualty Actuarial Society Credentials (ACAS/FCAS)

21 of 72 cards from the MAS-II — Modern Actuarial Statistics II deck — real questions with worked answers.

1. State Bayes' Theorem for a parameter $\theta$ given data $x$, and name each component.

   $$\pi(\theta \mid x) = \frac{f(x \mid \theta)\,\pi(\theta)}{\int f(x \mid \theta)\,\pi(\theta)\,d\theta}$$ where $\pi(\theta)$ is the prior, $f(x\mid\theta)$ is the likelihood, $\pi(\theta\mid x)$ is the posterior, and the denominator is the marginal (normalizing constant). In words: posterior $\propto$ likelihood $\times$ prior.

2. What is a conjugate prior, and why is it useful?

   A prior is conjugate to a likelihood if the resulting posterior belongs to the same distributional family as the prior. It is useful because the posterior has a closed form, avoiding numerical integration. Example: a Beta prior with a Binomial likelihood yields a Beta posterior.

3. For a Binomial likelihood with $k$ successes in $n$ trials and a $\text{Beta}(\alpha,\beta)$ prior, what is the posterior distribution of $p$?

   $$p \mid k \sim \text{Beta}(\alpha + k,\ \beta + n - k)$$ The posterior mean is $\frac{\alpha+k}{\alpha+\beta+n}$.

4. For a Poisson likelihood with a $\text{Gamma}(\alpha,\beta)$ prior (rate parameterization), what is the posterior for $\lambda$ given observations summing to $\sum x_i$ over $n$ exposures?

   $$\lambda \mid x \sim \text{Gamma}\left(\alpha + \sum x_i,\ \beta + n\right)$$ Gamma is conjugate to the Poisson.

5. Define the loss function, posterior expected loss, and the Bayes estimator under squared-error loss.

   Loss $L(\theta,\hat\theta)$ measures the penalty for estimating $\theta$ by $\hat\theta$. Posterior expected loss is $E_{\theta\mid x}[L(\theta,\hat\theta)]$. Under squared-error loss $L=(\theta-\hat\theta)^2$, the Bayes estimator is the posterior mean $E[\theta\mid x]$.

6. Match each loss function to its corresponding Bayes estimator: squared-error, absolute-error, and zero-one loss.

   Squared-error loss $\to$ posterior mean; absolute-error loss $\to$ posterior median; zero-one loss $\to$ posterior mode (MAP estimate).

7. What is the difference between an informative (subjective) prior and a non-informative (vague) prior? Give an example of the latter.

   An informative prior encodes specific external knowledge with low variance; a non-informative prior is intended to let the data dominate, contributing little information. Examples of non-informative priors: a flat/uniform prior, or Jeffreys' prior $\pi(\theta)\propto\sqrt{I(\theta)}$ where $I(\theta)$ is Fisher information.

8. Define a credible interval and contrast it with a frequentist confidence interval.

   A $100(1-\alpha)\%$ credible interval is an interval containing $\theta$ with posterior probability $1-\alpha$, i.e. $P(\theta\in[a,b]\mid x)=1-\alpha$ — a direct probability statement about $\theta$. A confidence interval makes a statement about the long-run coverage of the procedure, not the probability that a fixed $\theta$ lies in a given interval.

9. What is the highest posterior density (HPD) credible interval and what property does it have?

   The HPD interval is the credible region such that every point inside has higher posterior density than any point outside. For a given credibility level it is the shortest such interval; for a unimodal symmetric posterior it coincides with the equal-tailed interval.

10. Explain the idea behind Markov Chain Monte Carlo (MCMC) and why it is used in Bayesian inference.

    MCMC constructs a Markov chain whose stationary distribution is the target posterior. By simulating the chain long enough, the draws approximate samples from the posterior, allowing estimation of posterior means, quantiles, and intervals when the posterior has no closed form (intractable normalizing constant).

11. Describe the steps of the Metropolis-Hastings algorithm and its acceptance probability.

    From current state $\theta$, propose $\theta^*$ from proposal $q(\theta^*\mid\theta)$. Accept $\theta^*$ with probability $$\alpha = \min\left(1,\ \frac{\pi(\theta^*)q(\theta\mid\theta^*)}{\pi(\theta)q(\theta^*\mid\theta)}\right)$$ Otherwise stay at $\theta$. For a symmetric proposal the ratio reduces to $\pi(\theta^*)/\pi(\theta)$ (Metropolis).

12. How does the Gibbs sampler work, and when is it applicable?

    Gibbs sampling updates each parameter (or block) in turn by drawing from its full conditional distribution given the current values of all others: $\theta_j^{(t+1)}\sim p(\theta_j\mid \theta_{-j}, x)$. It is applicable when the full conditionals are available in closed form and easy to sample, and is a special case of Metropolis-Hastings with acceptance probability 1.

13. What is the 'burn-in' period in MCMC, and why is it discarded?

    Burn-in is the set of initial iterations before the chain has converged to its stationary distribution. These early draws depend on starting values and are discarded so that only samples representative of the posterior are used for inference.

14. Name three diagnostics used to assess MCMC convergence and mixing.

    (1) Trace plots — visual inspection for stationarity and good mixing; (2) Autocorrelation plots / effective sample size — high autocorrelation means slow mixing; (3) The Gelman-Rubin statistic $\hat{R}$ from multiple chains, where $\hat{R}\approx 1$ indicates convergence.

15. State the relationship between Bayesian analysis and Bühlmann credibility in terms of the credibility estimate form.

    The Bühlmann credibility premium is the best linear approximation to the Bayesian posterior mean. Both take the form $Z\bar{X} + (1-Z)\mu$, a weighted average of the observed mean and the prior (collective) mean; Bühlmann is exact (equals the Bayes estimate) for linear-exponential conjugate families.

16. Give the Bühlmann credibility factor $Z$ and define its components.

    $$Z = \frac{n}{n + k}, \qquad k = \frac{v}{a} = \frac{\text{EPV}}{\text{VHM}}$$ where $n$ is the number of exposures, $v = E[\text{Var}(X\mid\Theta)]$ is the expected process variance (EPV), and $a = \text{Var}(E[X\mid\Theta])$ is the variance of the hypothetical means (VHM).

17. In Bühlmann-Straub credibility, how does the credibility factor change to handle varying exposures $m_i$?

    $$Z = \frac{m}{m + k}, \qquad m = \sum_i m_i, \qquad k = \frac{v}{a}$$ Exposures are summed rather than counted, and the credibility-weighted mean uses exposure-weighted averages of the observations.

18. How does the credibility factor $Z$ behave as $n\to\infty$ and as the VHM $a\to 0$? Interpret each.

    As $n\to\infty$, $Z\to 1$ (full credibility — data dominates). As $a\to 0$ (no heterogeneity between risks), $k\to\infty$ so $Z\to 0$, meaning the collective mean is used. Larger heterogeneity ($a$ up) or smaller process variance ($v$ down) raises $Z$.

19. Write the simple linear regression model and state the least-squares estimators for slope and intercept.

    Model: $y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$. Estimators: $$\hat\beta_1 = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2} = \frac{S_{xy}}{S_{xx}}, \qquad \hat\beta_0 = \bar{y} - \hat\beta_1 \bar{x}$$

20. List the standard assumptions of the ordinary least squares (OLS) linear regression model.

    (1) Linearity: $E[y\mid x]=X\beta$; (2) Independence of errors; (3) Homoscedasticity: $\text{Var}(\varepsilon_i)=\sigma^2$ constant; (4) Errors uncorrelated with predictors; (5) For inference, normality of errors $\varepsilon_i\sim N(0,\sigma^2)$; (6) No perfect multicollinearity among predictors.

21. Define $R^2$ and adjusted $R^2$, and explain why adjusted $R^2$ is preferred for model comparison.

    $$R^2 = 1 - \frac{SSE}{SST}, \qquad R^2_{adj} = 1 - \frac{SSE/(n-p-1)}{SST/(n-1)}$$ $R^2$ never decreases when predictors are added, so it cannot penalize complexity; adjusted $R^2$ penalizes for the number of parameters $p$ and can decrease when an unhelpful predictor is added.

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