Estimation, Evaluation, and Selection of Actuarial Models

3.2. MEASURES OF QUALITY 41
3.2.3 Consistency
A second desirable property of an estimator is that it works well for extremely large samples.
Slightly more formally, as the sample size goes to infinity, the probability that the estimator is in
error by more than a small amount goes to zero. A formal definition is
Definition 3.8 An estimator is consistent (often called, in this context, weakly consistent) if
for all δ > 0 and any θ,
lim
n→∞ Pr(|ˆθ n − θ| > δ) =0.
Asufficient (although not necessary) condition for weak consistency is that the estimator be
asymptotically unbiased and Var(ˆθ n ) → 0.
Example 3.9 Prove that if the variance of a random variable is finite, the sample mean is a
consistent estimator of the population mean.
From Exercise 50, the sample mean is unbiased. In addition,
⎛
Var( ¯X) = Var⎝ 1 n
nX
j=1
X j
⎞
⎠
= 1 nX
n 2 Var(X j )
j=1
= Var(X)
n
Example 3.10 Show that the maximum observation from a uniform distribution on the interval
(0, θ) is a consistent estimator of θ.
→ 0.
From Example 3.7, the maximum is asymptotically unbiased. The second moment is
¤
E(Y 2 n )=
Z θ
0
ny n+1 θ −n dy =
n
n +2 yn+2 θ −n¯¯¯¯
θ
0
= nθ2
n +2
and then
Var(Y n )=
µ nθ2 nθ 2
n +2 − =
n +1
nθ 2
(n +2)(n +1) 2 → 0. ¤
Exercise 52 Explain why the sample mean may not be a consistent estimator of the population
mean for a Pareto distribution.