Sample Questions
Question 1
Easy
Which of the following statements is TRUE regarding the use of an offset variable in a generalized linear model?
Solution
C is correct. An offset is a known quantity that enters the linear predictor with a coefficient fixed at 1 (rather than being estimated from the data). For a Poisson GLM with log link, the linear predictor is , so the predicted count scales proportionally with exposure. This is the standard mechanism for handling unequal time-at-risk or other normalizing quantities in count and severity models.
Question 2
Medium
An actuary fitting a Poisson GLM to auto insurance claim count data observes that the Pearson chi-squared statistic divided by its degrees of freedom is 3.8.
Which of the following best describes this situation and the appropriate response?
Which of the following best describes this situation and the appropriate response?
Solution
(A) is correct.
For a well-fitting Poisson model, the dispersion parameter should be close to 1, since the Poisson distribution has (variance equals mean, i.e., ). (C) ratio of 3.8 is substantially greater than 1, indicating overdispersion: the observed variance is roughly 3.8 times the variance assumed by the Poisson model. The standard remedies are quasi-Poisson (which rescales standard errors by ) or negative binomial regression (which has an extra dispersion parameter).
Question 3
Hard
Determine which one of the following statements about Principal Component Regression (PCR) is FALSE.
Solution
C is correct.
Let us evaluate each statement:
A: TRUE. Standardizing predictors before PCA is recommended because PCA is sensitive to the scale of variables.
D: FALSE. PCR does NOT perform feature selection in the traditional sense. It selects principal components (linear combinations of all original features), but it does not select or exclude individual features. All original variables contribute to each principal component. Feature selection methods like LASSO actually zero out coefficients.
C: TRUE. This is a known limitation/assumption of PCR -- it assumes that the directions of maximum variance in the predictors are the directions most associated with the response.
(B): TRUE. PCR can reduce overfitting by using fewer principal components than original features, effectively reducing dimensionality.
E: TRUE. By definition, the first principal component is the direction of maximum variance in the data.
Let us evaluate each statement:
A: TRUE. Standardizing predictors before PCA is recommended because PCA is sensitive to the scale of variables.
D: FALSE. PCR does NOT perform feature selection in the traditional sense. It selects principal components (linear combinations of all original features), but it does not select or exclude individual features. All original variables contribute to each principal component. Feature selection methods like LASSO actually zero out coefficients.
C: TRUE. This is a known limitation/assumption of PCR -- it assumes that the directions of maximum variance in the predictors are the directions most associated with the response.
(B): TRUE. PCR can reduce overfitting by using fewer principal components than original features, effectively reducing dimensionality.
E: TRUE. By definition, the first principal component is the direction of maximum variance in the data.
