Free SOA Exam SRM (Statistics for Risk Modeling) Linear Models Practice Questions

The Pearson residual for observation $i$ in a GLM is defined as:
$$r_{P,i} = \frac{y_i - \hat{\mu}_i}{\sqrt{V(\hat{\mu}_i)}}$$
where $V(\hat{\mu}_i)$ is the variance function of the assumed distribution evaluated at the fitted mean. This standardizes the raw residual by the expected standard deviation under the model.

Why each other option is incorrect:
- (D) Dividing by $\hat{\mu}_i$ rather than $\sqrt{V(\hat{\mu}_i)}$ is not the Pearson residual. For Poisson, $V(\mu) = \mu$, so the Pearson residual uses $\sqrt{\hat{\mu}_i}$.
- (C) This is the raw (response) residual, not Pearson.
- (A) This is the definition of the deviance residual, not the Pearson residual.
- (E) Dividing by a single estimated $\hat{\sigma}$ is the internally studentized residual in OLS, not the GLM Pearson residual.

Statement (C) is FALSE. Lasso does not always select the correct set of important variables. Its variable selection performance depends on conditions such as the irrepresentable condition, signal strength, and correlation among predictors. With highly correlated predictors, lasso may arbitrarily select one from a group of correlated variables.

Why each other option is actually true:
- (E) Lasso minimizes $\text{RSS} + \lambda \sum |\beta_j|$, which is the $L_1$ penalty.
- (D) The geometry of the $L_1$ constraint region (a diamond in 2D) means solutions often occur at corners where some coefficients are exactly zero.
- (A) Larger $\lambda$ imposes a tighter constraint, forcing more coefficients to zero.
- (B) Lasso is designed for high-dimensional settings and can handle $p > n$ scenarios.

Ln(8500) +/- 1.96*0.15 => exp => (6370, 11340). PI: Gamma(5,1700) quantiles => (3200, 22560).

Choice C is incorrect because it uses the wrong shape parameter alpha=2 instead of 5 for the PI.
Choice B is incorrect because the PI is constructible using the gamma distribution with known shape.
Choice D is incorrect because the CI is too narrow from a normal approximation on the original scale.
Choice E is incorrect because it reverses the CI and PI widths.