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.
Which of the following best describes the Akaike Information Criterion (AIC) used in model selection?
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Correct Answer: D
Solution
D is correct.
The Akaike Information Criterion is defined as: AIC=−2ℓ^+2k where ℓ^ is the maximized log-likelihood and k is the number of estimated parameters. The criterion balances fit (via −2ℓ^) against complexity (via 2k). 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?
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Correct Answer: E
Solution
E is correct.
AIC =−2ℓ^+2p penalizes each parameter by 2. BIC =−2ℓ^+plnn penalizes each by lnn. The BIC penalty exceeds AIC's when lnn>2, i.e., n>e2≈7.4. For any dataset with n≥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 n complete observations is θ^=Xˉ. Using the delta method, find the asymptotic variance of the MLE of S(t)=e−t/θ, evaluated at t=500 and θ^=1000.
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Correct Answer: D
Solution
D is correct.
The survival function is S(t)=e−t/θ. By the delta method: Var(S^(t))≈(dθdS)2Var(θ^)dθdS=θ2te−t/θVar(θ^)=nθ2Var(S^(t))=(θ2te−t/θ)2⋅nθ2=nθ2t2e−2t/θ At t=500, θ=1000: Var=n×1065002e−1=106n250000e−1=4ne−1.
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