Estimation, Evaluation, and Selection of Actuarial Models

40 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
This estimator is clearly biased, 1 with bias Y (θ) = 5θ
6 − θ = − θ 6
. On the average, this estimator
underestimates the true value. It is also easy to see that the sample median can be turned into an
unbiased estimator by multiplying it by 1.2. ¤
For Example 3.2 we have two estimators (the sample mean and 1.2 times the sample median)
that are both unbiased. We will need additional criteria to decide which one we prefer. Some
estimators exhibit a small amount of bias, which vanishes as the sample size goes to infinity.
Definition 3.6 Let ˆθ n be an estimator of θ based on a sample size of n. The estimator is asymptotically
unbiased if
lim E(ˆθ n |θ) =θ
n→∞
for all θ.
Example 3.7 Suppose a random variable has the uniform distribution on the interval (0, θ). Consider
the estimator ˆθ n =max(X 1 ,...,X n ). Show that this estimator is asymptotically unbiased.
Let Y n be the maximum from a sample of size n. Then
The expected value is
F Yn (y) = Pr(Y n ≤ y) =Pr(X 1 ≤ y,...,X n ≤ y)
= F X (y) n
= (y/θ) n
f Yn (y) = ny n−1 /θ n , 0