Estimation, Evaluation, and Selection of Actuarial Models

3.2. MEASURES OF QUALITY 39
1 2 3 5 2 3 4 6
3 4 5 7 5 6 7 9
Combining the common values, the sample mean, usually denoted ¯X, has the following probability
distribution.
x p ¯X(x)
1 1/16
2 2/16
3 3/16
4 2/16
5 3/16
6 2/16
7 2/16
9 1/16
The expected value of the estimator is
E( ¯X) = [1(1) + 2(2) + 3(3) + 4(2) + 5(3) + 6(2) + 7(2) + 9(1)]/16 = 4.5
and so the sample mean is an unbiased estimator of the population mean for this example. ¤
Example 3.5 For Example 3.2 determine the bias of the sample mean and the sample median as
estimators of the population mean.
Thesamplemeanis ¯X =(X 1 + X 2 + X 3 )/3 where each X j represents one of the observations
from the exponential population. Its expected value is
µ
E( ¯X)
X1 + X 2 + X 3
=E
= 1 3
3 [E(X 1)+E(X 2 )+E(X 3 )] = 1 (θ + θ + θ) =θ
3
and therefore the sample mean is an unbiased estimator of the population mean.
Investigating the sample median is a bit more difficult. The distribution function of the middle
of three observations can be found as follows, using Y as the random variable of interest and X as
the random variable for an observation from the population.
F Y (y) = Pr(Y ≤ y) =Pr(X 1 ,X 2 ,X 3 ≤ y)+Pr(X 1 ,X 2 ≤ y,X 3 >y)
+Pr(X 1 ,X 3 ≤ y,X 2 >y)+Pr(X 2 ,X 3 ≤ y, X 1 >y)
= F X (y) 3 +3F X (y) 2 [1 − F X (y)]
= [1− e −y/θ ] 3 +3[1− e −y/θ ] 2 e −y/θ .
The density function is
f Y (y) =F 0 Y (y) = 6 θ
³e −2y/θ − e −3y/θ´
.
The expected value of this estimator is
E(Y |θ) =
Z ∞
0
= 5θ
6 .
y 6 θ
³e −2y/θ − e −3y/θ´
dy