Free SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) Construction and Selection of Parametric Models Practice Questions

Parametric model construction on SOA Exam ASTAM tests maximum likelihood estimation (MLE), minimum distance estimation, hypothesis testing, model selection criteria (AIC, BIC, likelihood ratio), and simulation methods.

204 questions 51 easy 108 medium 45 hard 2026 syllabus

Sample Questions

Question 1 Easy
Which of the following best describes the Akaike Information Criterion (AIC) used in model selection?
Solution
D is correct.

The Akaike Information Criterion is defined as: AIC=−2ℓ^+2k\text{AIC} = -2\hat{\ell} + 2k where ℓ^\hat{\ell} is the maximized log-likelihood and kk is the number of estimated parameters. The criterion balances fit (via −2ℓ^-2\hat{\ell}) against complexity (via 2k2k). Lower AIC is preferred.
Question 2 Medium
Which of the following correctly describes the relationship between the AIC and BIC model selection criteria and their preference for model complexity?
Solution
E is correct.

AIC =−2ℓ^+2p= -2\hat{\ell} + 2p penalizes each parameter by 22. BIC =−2ℓ^+pln⁡n= -2\hat{\ell} + p\ln n penalizes each by ln⁡n\ln n. The BIC penalty exceeds AIC's when ln⁡n>2\ln n > 2, i.e., n>e2≈7.4n > e^2 \approx 7.4. For any dataset with n≥8n \ge 8 — which includes virtually every actuarial application — BIC penalizes additional parameters more heavily than AIC and therefore selects simpler (fewer-parameter) models.
Question 3 Hard
The MLE of the exponential mean based on nn complete observations is θ^=Xˉ\hat{\theta} = \bar{X}. Using the delta method, find the asymptotic variance of the MLE of S(t)=e−t/θS(t) = e^{-t/\theta}, evaluated at t=500t = 500 and θ^=1000\hat{\theta} = 1000.
Solution
D is correct.

The survival function is S(t)=e−t/θS(t) = e^{-t/\theta}. By the delta method: Var(S^(t))≈(dSdθ)2Var(θ^)\text{Var}(\hat{S}(t)) \approx \left(\frac{dS}{d\theta}\right)^2 \text{Var}(\hat{\theta}) dSdθ=tθ2e−t/θ\frac{dS}{d\theta} = \frac{t}{\theta^2}e^{-t/\theta} Var(θ^)=θ2n\text{Var}(\hat{\theta}) = \frac{\theta^2}{n} Var(S^(t))=(tθ2e−t/θ)2⋅θ2n=t2e−2t/θnθ2\text{Var}(\hat{S}(t)) = \left(\frac{t}{\theta^2}e^{-t/\theta}\right)^2 \cdot \frac{\theta^2}{n} = \frac{t^2 e^{-2t/\theta}}{n\theta^2} At t=500t = 500, θ=1000\theta = 1000: Var=5002e−1n×106=250000e−1106n=e−14n\text{Var} = \frac{500^2 e^{-1}}{n \times 10^6} = \frac{250000 e^{-1}}{10^6 n} = \frac{e^{-1}}{4n}.
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