Estimation, Evaluation, and Selection of Actuarial Models

42 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
3.2.4 Mean squared error
While consistency is nice, most estimators have this property. What would be truly impressive is
an estimator that is not only correct on average, but comes very close most of the time and, in
particular, comes closer than rival estimators. One measure, for a finite sample, is motivated by
the definition of consistency. The quality of an estimator could be measured by the probability
that it gets within δ of the true value–that is, by measuring Pr(|ˆθ n − θ| < δ). But the choice of δ
is arbitrary and we prefer measures that cannot be altered to suit the investigator’s whim. Then
we might consider E(|ˆθ n −θ|), the average absolute error. But we know that working with absolute
values often presents unpleasant mathematical challenges, and so the following has become widely
accepted as a measure of accuracy.
Definition 3.11 The mean squared error (MSE) of an estimator is
MSEˆθ(θ) =E[(ˆθ − θ) 2 |θ].
Note that the MSE is a function of the true value of the parameter. An estimator may perform
extremely well for some values of the parameter, but poorly for others.
Example 3.12 Consider the estimator ˆθ =5of an unknown parameter θ. The MSE is (5 − θ) 2 ,
which is very small when θ is near 5 but becomes poor for other values. Of course this estimate is
both biased and inconsistent unless θ is exactly equal to 5. ¤
A result that follows directly from the various definitions is
MSEˆθ(θ) =E{[ˆθ − E(ˆθ|θ)+E(ˆθ|θ) − θ] 2 |θ} = Var(ˆθ|θ)+[biasˆθ(θ)] 2 . (3.1)
If we restrict attention to only unbiased estimators, the best such could be defined as follows.
Definition 3.13 An estimator, ˆθ, iscalledauniformly minimum variance unbiased estimator
(UMVUE) if it is unbiased and for any true value of θ there is no other unbiased estimator
that has a smaller variance.
Because we are looking only at unbiased estimators it would have been equally effective to make
the definition in terms of MSE. We could also generalize the definition by looking for estimators
that are uniformly best with regard to MSE, but the previous example indicates why that is not
feasible. There are a few theorems that can assist with the determination of UMVUEs. However,
such estimators are difficult to determine. On the other hand, MSE is still a useful criterion for
comparing two alternative estimators.
Example 3.14 For Example 3.2 compare the MSEs of the sample mean and 1.2 times the sample
median.
The sample mean has variance
Var(X)
3
= θ2
3 .