Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Parametric Estimation Practice Questions

Parametric estimation on SOA Exam FAM covers maximum likelihood estimation (MLE), method of moments, and goodness-of-fit testing for insurance loss distributions. These techniques are foundational to actuarial modeling.

48 Questions

24 Easy

19 Medium

5 Hard

2026 Syllabus

Sample Questions

Question 1 Easy

Which of the following is a property of maximum likelihood estimators for large samples?

Solution

D is correct.

Key asymptotic properties of MLEs:

1. **Consistency**:

\hat{\theta}_{\text{MLE}} \xrightarrow{p} \theta

n \to \infty

.

2. **Asymptotic normality**:

\sqrt{n}(\hat{\theta} - \theta) \xrightarrow{d} N(0, 1/I(\theta))

where

I(\theta)

is the Fisher information.

3. **Asymptotic efficiency**: The MLE achieves the Cramér-Rao lower bound asymptotically, meaning it has the smallest asymptotic variance among consistent estimators.

4. **Invariance**: If

\hat{\theta}

is the MLE of

\theta

, then

g(\hat{\theta})

is the MLE of

g(\theta)

.

Important caveats:
For example, the MLE of

\sigma^2

in the normal model divides by

n

, not n-1.

Question 2 Medium

A sample of 10 losses from an exponential distribution has 8 exact observations summing to 2,400 and 2 right-censored observations at 500 each. Calculate the MLE of

\theta

Solution

D is correct.

For right-censored exponential data, the MLE of θ is:

\hat{θ} = \frac{\text{total exposure}}{\text{number uncensored}} = \frac{2{,}400 + 2(500)}{8} = \frac{3{,}400}{8} = 425

Total exposure includes both exact observations and censored exposure times.

Question 3 Hard

Loss amounts follow a Pareto distribution with pdf

f(x) = \frac{\alpha\theta^\alpha}{(x+\theta)^{\alpha+1}}

for

x > 0

. Both

\alpha

and

\theta

are unknown. A sample of 3 observations is: 100, 400, 500. Write the system of equations that the MLEs must satisfy.

Solution

E is correct.

The log-likelihood is:

\ell(\alpha, \theta) = n\ln\alpha + n\alpha\ln\theta - (\alpha+1)\sum\ln(x_i + \theta)

. Taking partial derivatives:

\frac{\partial\ell}{\partial\alpha} = \frac{n}{\alpha} + n\ln\theta - \sum\ln(x_i + \theta) = 0

\frac{\partial\ell}{\partial\theta} = \frac{n\alpha}{\theta} - (\alpha+1)\sum\frac{1}{x_i + \theta} = 0

. With

n = 3

and the given data: Equation 1:

\frac{3}{\alpha} + 3\ln\theta - [\ln(100+\theta) + \ln(400+\theta) + \ln(500+\theta)] = 0

. Equation 2:

\frac{3\alpha}{\theta} - (\alpha+1)\left[\frac{1}{100+\theta} + \frac{1}{400+\theta} + \frac{1}{500+\theta}\right] = 0

. This system must be solved simultaneously using numerical methods (e.g., Newton-Raphson). From Equation 1, we can express

\alpha

as a function of

\theta

and substitute into Equation 2. From Equation 1:

\alpha = \frac{3}{\sum\ln(x_i+\theta) - 3\ln\theta}

(profile likelihood approach).

Create a Free Account to Access All 48 Questions →

Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Parametric Estimation Practice Questions

Sample Questions

Guides & Articles

About FreeFellow