SOA SRMStatistics for Risk Modeling

A regression with $n=50$ observations and $p=4$ predictors has $SSE=40$ and $SST=200$. Calculate the adjusted $R^2$.

A logistic model has $\text{logit}(p)=-1+0.5x$. Calculate the predicted probability at $x=2$.

A PCA on four standardized variables yields eigenvalues 3.0, 1.0, 0.5 and 0.5. What proportion of total variance is explained by the first principal component?

Compared with a single deep decision tree, a random forest typically has:

Explain the bias–variance decomposition of expected test error and use it to compare (a) k-nearest neighbours with small vs large $k$, and (b) lasso with small vs large $\lambda$.

An AR(1) model is $y_t=5+0.6y_{t-1}+\varepsilon_t$. Given $y_{100}=20$, calculate the 2-step-ahead point forecast $\hat y_{102}$.

K-means with $k=2$ is run on points $\{1,2,3,9,10,11\}$, correctly separating $\{1,2,3\}$ and $\{9,10,11\}$. Calculate the total within-cluster sum of squares (WCSS).

Regressing predictor $x_3$ on the other predictors gives $R^2=0.80$. Calculate the variance inflation factor for $x_3$.

Compare the Gini index and entropy as splitting criteria for classification trees, and explain why classification error rate is rarely used for growing (as opposed to pruning) a tree.

Relative to 10-fold cross-validation, leave-one-out cross-validation (LOOCV) of the test error typically has: