Exam MAS-I Practice Question 64
Study guide: Exam MAS-I
Claim sizes are exponential with unknown mean \theta. From a sample of n = 10 claims, an actuary considers three estimators of \theta:
| Estimator | Definition | Bias | Variance |
|---|---|---|---|
T_1 |
\bar{X} |
0 | \theta^2/10 |
T_2 |
\sum X_i / 11 |
-\theta/11 |
\tfrac{10}{121}\theta^2 |
T_3 |
n \cdot \min(X_1, \ldots, X_{10}) |
0 | \theta^2 |
The actuary will use whichever estimator has the smallest mean square error. A colleague points out that the variance of T_2 is below the Cramér–Rao lower bound for this model.
Which choice, with its justification, is correct?
- A)
T_1— it attains the Cramér–Rao bound, so no estimator can have a smaller MSE - B)
T_1— a variance below the bound shows thatT_2's figures are in error - C)
T_2— it has the smallest variance, and a smaller variance means a smaller MSE - D)
T_2— the bound limits unbiased estimators only, andT_2has the smallest MSE - E)
T_3— it is unbiased, and an estimator must be unbiased before its MSE counts
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