Associate of the Society of Actuaries (ASA/FSA) Exam SRM — Statistics for Risk Modeling Syllabus

Exam SRM — Statistics for Risk Modeling syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exam SRM — Statistics for Risk Modeling in Associate of the Society of Actuaries (ASA/FSA), not a summary of it.

1. 1

   Foundations of Statistical Learning

   3 topics

   • Modeling Frameworks
     • Supervised vs. unsupervised learning
     • Regression vs. classification problems
   • Model Accuracy and the Bias-Variance Tradeoff
     • Training vs. test error
     • Overfitting and flexibility
   • Resampling Methods
     • Validation set approach
     • k-fold and leave-one-out cross-validation
     • The bootstrap
2. 2

   Linear Models

   3 topics

   • Simple and Multiple Linear Regression
     • Least squares estimation
     • Hypothesis tests and confidence intervals
     • R-squared and residual analysis
   • Model Diagnostics and Assumptions
     • Multicollinearity
     • Heteroscedasticity and influential points
   • Variable Selection and Regularization
     • Subset selection
     • Ridge regression
     • Lasso
3. 3

   Generalized Linear Models

   3 topics

   • GLM Structure
     • Exponential family distributions
     • Link functions
   • Common GLMs
     • Logistic regression
     • Poisson regression for counts
   • GLM Inference and Evaluation
     • Deviance and likelihood
     • Confusion matrices and classification metrics
4. 4

   Tree-Based and Other Predictive Methods

   3 topics

   • Decision Trees
     • Regression and classification trees
     • Pruning and cost-complexity
   • Ensemble Methods
     • Bagging
     • Random forests
     • Boosting
   • K-Nearest Neighbors
     • Distance metrics and choice of k
5. 5

   Unsupervised Learning and Time Series

   3 topics

   • Principal Components Analysis
     • Dimension reduction and loadings
     • Proportion of variance explained
   • Clustering
     • K-means clustering
     • Hierarchical clustering
   • Time Series Foundations
     • Trend, seasonality, and stationarity
     • Autoregressive and moving-average models

Exam SRM — Statistics for Risk Modeling flashcards for Associate of the Society of Actuaries (ASA/FSA)

19 of 62 cards from the Exam SRM — Statistics for Risk Modeling deck — real questions with worked answers.

1. In statistical learning, what is the distinction between supervised and unsupervised learning?

   Supervised learning has a response (target) variable $Y$ that the model predicts from features $X$ (e.g., regression, classification). Unsupervised learning has no response variable; it seeks structure in $X$ alone (e.g., PCA, clustering).

2. What is the difference between the goals of prediction and inference in modeling?

   Prediction aims to produce accurate $\hat{Y}$ values and treats $\hat{f}$ as a black box. Inference aims to understand the relationship between $Y$ and each predictor (which are important, sign/magnitude of effects, form of $f$).

3. What is the difference between parametric and non-parametric modeling approaches?

   Parametric methods assume a functional form for $f$ (e.g., linear) and estimate a finite set of parameters, reducing the problem to estimating coefficients. Non-parametric methods make no explicit assumption about $f$'s form, are more flexible, but require more data and risk overfitting.

4. What is the general additive error model that underlies statistical learning?

   $$Y = f(X) + \varepsilon$$ where $f$ is the systematic (reducible) relationship and $\varepsilon$ is an irreducible random error with $E[\varepsilon]=0$, independent of $X$.

5. Distinguish reducible error from irreducible error.

   Reducible error comes from $\hat{f}$ not matching $f$ and can be lowered with better models/more data. Irreducible error comes from $\varepsilon$ (variability $Y$ has even given $X$) and sets a lower bound on prediction error that no model can beat.

6. How is the test mean squared error (MSE) for a regression model defined?

   $$\text{MSE}_{\text{test}} = E\big[(Y_0 - \hat{f}(X_0))^{2}\big]$$ the expected squared prediction error on new (unseen) observations $(X_0, Y_0)$.

7. State the bias-variance decomposition of the expected test MSE at a point $x_0$.

   $$E\big[(y_0-\hat{f}(x_0))^{2}\big] = \operatorname{Var}(\hat{f}(x_0)) + \big[\operatorname{Bias}(\hat{f}(x_0))\big]^{2} + \operatorname{Var}(\varepsilon)$$

8. In the bias-variance tradeoff, how do bias and variance change as model flexibility increases?

   As flexibility increases, bias decreases (model captures more structure) and variance increases (model is more sensitive to the training data). Test error follows a U-shape; the optimal model balances the two.

9. Define bias and variance of an estimator $\hat{f}$ in words.

   Bias is the error from approximating a complex true $f$ with a simpler model — how far the average prediction is from the truth. Variance is how much $\hat{f}$ would change if estimated on a different training set.

10. What is overfitting, and how is it identified using training versus test error?

    Overfitting occurs when a model fits noise in the training data, giving low training error but high test error. It is identified when test (validation) error rises while training error keeps falling as flexibility increases.

11. For classification, what is the test error rate, and what is the Bayes classifier?

    Test error rate is $E[I(y_0 \neq \hat{y}_0)]$, the fraction misclassified. The Bayes classifier assigns each $x_0$ to the most probable class, $\arg\max_j \Pr(Y=j \mid X=x_0)$, and achieves the lowest possible error (the Bayes error rate).

12. Describe the validation set approach and its main drawbacks.

    Randomly split data into a training set and a validation (hold-out) set; fit on training, estimate test error on validation. Drawbacks: the error estimate is highly variable depending on the split, and using fewer observations to train tends to overestimate test error.

13. Describe k-fold cross-validation and give the CV error estimate.

    Split data into $k$ roughly equal folds; for each fold $i$, train on the other $k-1$ folds and validate on fold $i$. The estimate is $$\text{CV}_{(k)} = \frac{1}{k}\sum_{i=1}^{k} \text{MSE}_i.$$ Common choices are $k=5$ or $k=10$.

14. What is leave-one-out cross-validation (LOOCV) and how does it compare to k-fold CV?

    LOOCV is k-fold CV with $k=n$: each observation is its own validation set. It has very low bias but high variance and high computational cost; k-fold ($k=5,10$) offers a better bias-variance and computational tradeoff.

15. What is the bootstrap, and how are bootstrap samples drawn?

    The bootstrap estimates the sampling variability of a statistic by repeatedly drawing samples of size $n$ with replacement from the observed data, recomputing the statistic each time, and using the spread of those estimates (e.g., for standard errors and confidence intervals).

16. In a bootstrap sample of size $n$, what fraction of original observations is expected to be included?

    About $1 - (1-1/n)^{n} \to 1 - e^{-1} \approx 0.632$, so roughly two-thirds of observations appear in each bootstrap sample; the remaining ~$36.8\%$ are the out-of-bag observations.

17. State the simple linear regression model and its least-squares coefficient estimates.

    $$Y = \beta_0 + \beta_1 X + \varepsilon$$ with $$\hat{\beta}_1 = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^{2}}, \qquad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}.$$

18. How is the residual sum of squares (RSS) defined, and what does least squares do with it?

    $$\text{RSS} = \sum_{i=1}^{n}(y_i - \hat{y}_i)^{2}$$ Least squares chooses the coefficients that minimize RSS.

19. How do you interpret a slope coefficient $\beta_j$ in multiple linear regression?

    $\beta_j$ is the expected change in $Y$ for a one-unit increase in $X_j$, holding all other predictors fixed.

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