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

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.

In multiple regression, give the matrix form of the OLS estimator and the fitted/hat relationship.

$$\hat\beta = (X^\top X)^{-1} X^\top y, \qquad \hat{y} = Hy, \quad H = X(X^\top X)^{-1}X^\top$$ $H$ is the hat (projection) matrix; its diagonal entries $h_{ii}$ are the leverages.

What is the F-test in regression used for, and what is its general form?

It tests whether a set of coefficients is jointly zero (overall significance or comparing nested models). $$F = \frac{(SSE_{reduced}-SSE_{full})/(df_R - df_F)}{SSE_{full}/df_F}$$ A large $F$ (small p-value) rejects the reduced model in favor of the full model.

Define multicollinearity and the Variance Inflation Factor (VIF). What VIF value is commonly a concern?

Multicollinearity is high correlation among predictors, inflating coefficient standard errors. $$VIF_j = \frac{1}{1 - R_j^2}$$ where $R_j^2$ is from regressing predictor $j$ on the others. $VIF_j > 10$ (sometimes $>5$) is a common rule-of-thumb threshold for concern.

Compare AIC and BIC for model selection, including their formulas and how they differ in penalizing complexity.

$$\text{AIC} = -2\ln L + 2p, \qquad \text{BIC} = -2\ln L + p\ln n$$ Both reward fit and penalize parameters $p$; lower is better. BIC's penalty $\ln n$ exceeds AIC's $2$ for $n>7$, so BIC favors simpler models and is consistent (selects the true model as $n\to\infty$), while AIC targets predictive accuracy and tends to select larger models.

Describe $k$-fold cross-validation and its purpose.

Split data into $k$ roughly equal folds. For each fold $i$, train on the other $k-1$ folds and validate on fold $i$; average the validation errors across folds. It estimates out-of-sample (test) prediction error, helping select models/tuning parameters while using data efficiently. $k=5$ or $k=10$ are common; $k=n$ is leave-one-out (LOOCV).

Explain the bias-variance tradeoff for expected test error.

Expected test MSE decomposes as $$E[(y-\hat{f}(x))^2] = \text{Bias}[\hat{f}(x)]^2 + \text{Var}[\hat{f}(x)] + \sigma^2$$ Increasing model flexibility lowers bias but raises variance; $\sigma^2$ is irreducible. The optimal model minimizes the sum, balancing underfitting (high bias) against overfitting (high variance).

Compare forward selection, backward elimination, and stepwise selection.

Forward selection starts with no predictors and adds the best one at each step. Backward elimination starts with all predictors and removes the least useful at each step. Stepwise combines both, adding and removing at each step. All are greedy and not guaranteed to find the globally best subset; backward requires $n>p$.

What are ridge and lasso regression, and how do their penalties differ in effect?

Both add a penalty to the least-squares objective. Ridge uses $\lambda\sum\beta_j^2$ ($L_2$), shrinking coefficients toward zero but keeping all. Lasso uses $\lambda\sum|\beta_j|$ ($L_1$), which can shrink coefficients exactly to zero, performing variable selection. Both reduce variance at the cost of some bias; $\lambda$ is tuned (e.g. by CV).

Define a residual and a standardized (studentized) residual in regression.

Residual: $e_i = y_i - \hat{y}_i$. Standardized/internally studentized residual: $$r_i = \frac{e_i}{\hat\sigma\sqrt{1-h_{ii}}}$$ where $h_{ii}$ is the leverage. It rescales residuals to roughly constant variance so unusually large values are comparable across observations.

What does a residuals-vs-fitted plot reveal, and what patterns indicate problems?

It checks linearity and homoscedasticity. A random horizontal band around zero is ideal. A curved pattern indicates nonlinearity (model misspecification); a funnel/fan shape indicates heteroscedasticity (non-constant variance).

Define leverage and Cook's distance, and state their roles.

Leverage $h_{ii}$ (diagonal of the hat matrix) measures how far observation $i$'s predictors are from the mean of predictors; high leverage points can strongly influence the fit. Cook's distance $$D_i = \frac{r_i^2}{p}\cdot\frac{h_{ii}}{1-h_{ii}}$$ measures the overall influence of observation $i$ on all fitted values; $D_i>1$ (or $>4/n$) flags influential points.

What is a Q-Q plot used for in residual analysis, and what does departure from the line indicate?

A normal Q-Q plot compares ordered residuals (sample quantiles) against theoretical normal quantiles to assess the normality assumption. Points on the 45-degree line indicate normality; S-shaped curves indicate heavy or light tails, and systematic curvature indicates skewness.

Describe the Box-Cox transformation and its purpose.

$$y^{(\lambda)} = \begin{cases} \frac{y^\lambda - 1}{\lambda}, & \lambda \neq 0 \\ \ln y, & \lambda = 0 \end{cases}$$ It is a family of power transformations of a positive response chosen (via maximum likelihood over $\lambda$) to stabilize variance and make residuals more nearly normal and linear.

State the three components that define a Generalized Linear Model (GLM).

(1) Random component: the response distribution from the exponential dispersion family (e.g. Normal, Poisson, Gamma, Binomial); (2) Systematic component: the linear predictor $\eta = X\beta$; (3) Link function $g(\cdot)$ connecting them via $g(\mu) = \eta$ where $\mu = E[y]$.

Write the general form of the exponential dispersion family density used in GLMs and identify the mean and variance.

$$f(y;\theta,\phi) = \exp\left\{\frac{y\theta - b(\theta)}{a(\phi)} + c(y,\phi)\right\}$$ The mean is $E[y]=b'(\theta)=\mu$ and the variance is $\text{Var}(y)=a(\phi)\,b''(\theta)=a(\phi)V(\mu)$, where $V(\mu)=b''(\theta)$ is the variance function and $\phi$ is the dispersion parameter.

What is the canonical link function, and give the canonical links for Normal, Poisson, and Binomial responses.

The canonical link is $g$ such that $g(\mu)=\theta$ (the natural parameter), giving desirable estimation properties. Normal: identity $g(\mu)=\mu$; Poisson: log $g(\mu)=\ln\mu$; Binomial: logit $g(\mu)=\ln\frac{\mu}{1-\mu}$.

Match the GLM response distribution to its variance function $V(\mu)$: Normal, Poisson, Gamma, Binomial.

Normal: $V(\mu)=1$; Poisson: $V(\mu)=\mu$; Gamma: $V(\mu)=\mu^2$; Binomial (proportion): $V(\mu)=\mu(1-\mu)$.

How are GLM coefficients estimated, and why isn't OLS used?

By maximum likelihood, solved numerically via iteratively reweighted least squares (IRLS) because the likelihood equations are generally nonlinear with no closed form. OLS assumes constant-variance normal errors, which does not hold for non-normal responses with mean-dependent variance.

In insurance pricing, why is a log link with a Poisson or Gamma GLM commonly used, and what multiplicative structure results?

The log link makes the model multiplicative: $\mu = e^{\beta_0}e^{\beta_1 x_1}\cdots$, so each rating factor acts as a multiplicative relativity on the base rate — matching how rating plans are structured. Poisson is natural for claim frequency (counts) and Gamma for claim severity (positive, right-skewed).

What is the Tweedie distribution, and why is it useful for modeling pure premium / loss cost directly?

The Tweedie is an exponential-family distribution with variance function $V(\mu)=\mu^p$. For $1<p<2$ it is a compound Poisson-Gamma: a positive probability mass at zero (no claims) plus a continuous positive part. This lets a single GLM model aggregate loss cost (pure premium), which is exactly zero for policies with no claims and continuous otherwise.

How do you interpret a GLM coefficient $\beta_j$ under a log link versus a logit link?

Log link: $e^{\beta_j}$ is the multiplicative effect on the mean for a one-unit increase in $x_j$ (a relativity); a percent change of $(e^{\beta_j}-1)\times100\%$. Logit link: $e^{\beta_j}$ is the odds ratio — the multiplicative effect on the odds of the event for a one-unit increase in $x_j$.

Define deviance in a GLM and state its role in model comparison.

Deviance is $D = 2\big[\ell(\text{saturated}) - \ell(\text{fitted})\big]$ scaled appropriately, measuring lack of fit relative to the saturated model (smaller is better). Differences in deviance between nested GLMs follow an approximate $\chi^2$ distribution (with df = difference in parameters), enabling likelihood-ratio model comparison.

What are deviance residuals and Pearson residuals in a GLM?

Pearson residual: $$r_i^P = \frac{y_i - \hat\mu_i}{\sqrt{V(\hat\mu_i)}}$$ Deviance residual: $r_i^D = \text{sign}(y_i-\hat\mu_i)\sqrt{d_i}$, where $d_i$ is observation $i$'s contribution to the deviance. Both assess fit; deviance residuals are usually closer to normal for non-normal responses.

What is overdispersion in a Poisson GLM, how is it detected, and how can it be addressed?

Overdispersion occurs when observed variance exceeds the mean (the Poisson assumption $\text{Var}=\mu$ fails). Detected when the deviance or Pearson statistic divided by its degrees of freedom $\gg 1$. Remedies: use a quasi-Poisson model (estimate dispersion $\phi$), or a Negative Binomial distribution.

Define strict stationarity and weak (covariance) stationarity for a time series.

Strict stationarity: the joint distribution of $(X_{t_1},\dots,X_{t_k})$ is invariant to time shifts. Weak/covariance stationarity: constant mean $E[X_t]=\mu$, constant variance, and autocovariance $\text{Cov}(X_t,X_{t+h})=\gamma(h)$ depending only on lag $h$, not on $t$.

Write the AR(1) model and state the stationarity condition and its theoretical ACF.

AR(1): $X_t = \phi X_{t-1} + \varepsilon_t$, $\varepsilon_t$ white noise. Stationary if $|\phi|<1$. Its autocorrelation function decays geometrically: $\rho(h)=\phi^{|h|}$.

Write the MA(1) model and give its autocorrelation function.

MA(1): $X_t = \varepsilon_t + \theta\varepsilon_{t-1}$. Its ACF cuts off after lag 1: $$\rho(1)=\frac{\theta}{1+\theta^2}, \qquad \rho(h)=0 \text{ for } h\geq 2$$ An MA($q$) process is always stationary.

How do the ACF and PACF patterns help identify AR($p$) vs MA($q$) models?

AR($p$): ACF tails off (decays gradually), PACF cuts off after lag $p$. MA($q$): ACF cuts off after lag $q$, PACF tails off. ARMA($p,q$): both ACF and PACF tail off gradually.

What does the ARIMA($p,d,q$) notation mean, and what is the role of $d$?

ARIMA combines AutoRegressive (order $p$), Integrated (differencing order $d$), and Moving Average (order $q$) components. $d$ is the number of times the series is differenced ($\nabla X_t = X_t - X_{t-1}$) to remove trend and achieve stationarity before fitting the ARMA($p,q$) part.

Why is differencing used on nonstationary series, and what does the random walk model look like?

Differencing $\nabla X_t = X_t - X_{t-1}$ removes trends and stabilizes the mean, converting many nonstationary series to stationary. The random walk $X_t = X_{t-1} + \varepsilon_t$ is nonstationary (variance grows with $t$), but its first difference $\nabla X_t = \varepsilon_t$ is white noise (stationary).

What is the purpose of a unit root test such as the Augmented Dickey-Fuller (ADF) test?

It tests whether a series has a unit root (i.e., is nonstationary / integrated). The null hypothesis is the presence of a unit root (nonstationary); rejecting it supports stationarity. This guides whether differencing is needed before ARMA modeling.

What is a seasonal ARIMA (SARIMA) model and how is its notation written?

SARIMA extends ARIMA with seasonal AR, differencing, and MA terms at the seasonal period $s$. Notation: $\text{ARIMA}(p,d,q)\times(P,D,Q)_s$, where lowercase terms are the non-seasonal orders and uppercase $(P,D,Q)$ are the seasonal orders applied at lag $s$ (e.g., $s=12$ for monthly data).

Define white noise and explain why residuals of a good time series model should resemble it.

White noise is a sequence with mean zero, constant variance, and zero autocorrelation at all nonzero lags. If a fitted model has captured all structure, the residuals should be white noise — any remaining autocorrelation indicates the model is missing signal. The Ljung-Box test checks for residual autocorrelation.

What is exponential smoothing (simple) and how does it forecast?

Simple exponential smoothing forecasts with a weighted average giving exponentially decreasing weights to older observations: $$\hat{X}_{t+1} = \alpha X_t + (1-\alpha)\hat{X}_t, \quad 0<\alpha<1$$ Larger $\alpha$ weights recent data more. It suits series with no trend or seasonality (Holt and Holt-Winters extend it to trend and seasonality).

Distinguish a point forecast from a prediction interval, and explain why forecast intervals widen with horizon.

A point forecast is a single best-estimate value (e.g., the conditional mean) for a future period; a prediction interval gives a range expected to contain the actual value with stated probability, accounting for forecast uncertainty. Intervals widen with horizon because accumulated future shocks add variance the further ahead you forecast.

How are a regression tree split and prediction determined under CART?

At each node CART chooses the predictor and split point that minimizes the residual sum of squares (RSS) of the two resulting child nodes (recursive binary splitting). The prediction in each terminal (leaf) node is the mean of the training responses falling in that node.

What impurity measures are used to choose splits in a classification tree?

The Gini index $G=\sum_k \hat{p}_k(1-\hat{p}_k)$ and cross-entropy/deviance $-\sum_k \hat{p}_k\ln\hat{p}_k$, where $\hat{p}_k$ is the proportion of class $k$ in a node. Both are minimized by node purity and are preferred over classification error rate for growing trees because they are more sensitive.

Why and how are decision trees pruned?

A fully grown tree overfits. Cost-complexity (weakest-link) pruning minimizes $$\sum_{\text{leaves}} \text{RSS} + \alpha|T|$$ where $|T|$ is the number of terminal nodes and $\alpha\geq 0$ tunes the penalty (chosen by cross-validation). Larger $\alpha$ yields smaller trees, trading fit for simplicity and better generalization.

Contrast bagging, random forests, and boosting as ensemble tree methods.

Bagging fits trees on bootstrap samples and averages them, reducing variance. Random forests add randomness by considering only a random subset of predictors at each split, decorrelating the trees for further variance reduction. Boosting grows trees sequentially, each fitting the residuals/errors of the prior ensemble, slowly reducing bias; it uses small trees and a learning rate.

What is the out-of-bag (OOB) error in bagging and random forests?

Each bootstrap sample omits about $1/3$ of observations (the out-of-bag set). Each observation is predicted using only the trees that did not train on it, and the averaged error over all observations is the OOB error — a built-in estimate of test error that avoids separate cross-validation.

State the K-means clustering objective and algorithm steps.

Objective: minimize within-cluster sum of squares $$\sum_{k=1}^{K}\sum_{i\in C_k}\|x_i-\mu_k\|^2.$$ Steps: (1) initialize $K$ centroids; (2) assign each point to its nearest centroid; (3) recompute centroids as cluster means; (4) repeat 2-3 until assignments stop changing. It converges to a local optimum, so multiple random starts are used.

Compare K-means and hierarchical clustering.

K-means requires choosing $K$ in advance, produces a single flat partition, and is sensitive to initialization. Hierarchical (agglomerative) clustering builds a nested tree (dendrogram) without pre-specifying $K$ — clusters are obtained by cutting the dendrogram at a chosen height — using a linkage rule (single, complete, average, Ward) to measure inter-cluster distance.

What is Principal Component Analysis (PCA) and what does the first principal component maximize?

PCA is an unsupervised dimension-reduction technique that finds orthogonal linear combinations (principal components) of the variables. The first principal component is the unit-norm direction maximizing the variance of the projected data; subsequent components maximize remaining variance subject to being orthogonal to earlier ones. Components are the eigenvectors of the covariance/correlation matrix.

Why should variables be standardized before PCA, and what does a scree plot show?

PCA is scale-sensitive: variables with larger units dominate the variance, so standardizing (zero mean, unit variance) prevents arbitrary units from driving the components. A scree plot shows the proportion of variance explained by each component; the 'elbow' helps decide how many components to retain.

In binary classification, define the confusion matrix terms and compute sensitivity, specificity, and precision.

From TP, FP, TN, FN: Sensitivity (recall, TPR) $=\frac{TP}{TP+FN}$; Specificity (TNR) $=\frac{TN}{TN+FP}$; Precision $=\frac{TP}{TP+FP}$. Accuracy $=\frac{TP+TN}{\text{total}}$.

What does an ROC curve plot, and how is AUC interpreted?

The ROC curve plots true positive rate (sensitivity) against false positive rate ($1-$specificity) as the classification threshold varies. The Area Under the Curve (AUC) summarizes performance: $1.0$ is perfect, $0.5$ is no better than random; it equals the probability that the model ranks a random positive above a random negative.

Define precision and recall and explain the F1 score.

Precision $=\frac{TP}{TP+FP}$ (of predicted positives, how many are correct); Recall $=\frac{TP}{TP+FN}$ (of actual positives, how many are found). The F1 score is their harmonic mean: $$F_1 = \frac{2\cdot\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{Recall}}$$ used when a single balanced measure is needed, especially with class imbalance.

What is a lift chart (gains chart) and how is it used to evaluate predictive models in insurance?

Observations are sorted by predicted value and grouped (e.g., deciles); the chart compares actual outcomes across the ranked groups. Good models show a steep, monotonic increase in actual losses/frequency from lowest to highest predicted bins, demonstrating the model's ability to discriminate (segment) risk. The lift ratio compares top-bin to bottom-bin or to the overall average.

Distinguish supervised from unsupervised learning and give an example of each.

Supervised learning has a labeled response/target and learns to predict it (e.g., regression, classification, GLMs, trees). Unsupervised learning has no target and finds structure within the predictors (e.g., K-means clustering, hierarchical clustering, PCA).

What is the difference between the training, validation, and test sets, and what is each used for?

Training set: fits model parameters. Validation set: tunes hyperparameters and selects among models. Test set: provides a final, unbiased estimate of generalization performance and must not influence model building. Cross-validation can substitute for a separate validation set.

Explain overfitting versus underfitting and how each appears in training vs test error.

Overfitting: model too flexible — low training error but high test error (high variance, fits noise). Underfitting: model too simple — high training and high test error (high bias, misses structure). The goal is the flexibility level minimizing test error, where the two errors are balanced.