Estimation, Evaluation, and Selection of Actuarial Models

46 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
Suppose the value of x is between the boundaries c j−1 and c j .LetY be the number of observations
at or below c j−1 and let Z be the number of observations above c j−1 and at or below c j .
Then
S n (x) =1− Y (c j − c j−1 )+Z(x − c j−1 )
n(c j − c j−1 )
and
E[S n (x)] = 1 − n[1 − S(c j−1)](c j − c j−1 )+n[S(c j−1 ) − S(c j )](x − c j−1 )
n(c j − c j−1 )
= S(c j−1 ) c j − x
+ S(c j ) x − c j−1
.
c j − c j−1 c j − c j−1
This estimator is biased (although it is an unbiased estimator of the true interpolated value). The
variance is
Var[S n (x)] = (c j − c j−1 ) 2 Var(Y )+(x − c j−1 ) 2 Var(Z)+2(c j − c j−1 )(x − c j−1 )Cov(Y,Z)
[n(c j − c j−1 )] 2
where Var(Y )=nS(c j−1 )[1 − S(c j−1 )], Var(Z) =n[S(c j−1 ) − S(c j )][1 − S(c j−1 )+S(c j )], and
Cov(Y,Z) =−n[1 − S(c j−1 )][S(c j−1 ) − S(c j )]. For the density estimate,
f n (x) =
Z
n(c j − c j−1 )
and
E[f n (x)] = S(c j−1) − S(c j )
c j − c j−1
which is biased for the true density function. The variance is
Var[f n (x)] = [S(c j−1) − S(c j )][1 − S(c j−1 )+S(c j )]
n(c j − c j−1 ) 2 .
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Example 3.20 For Data Set C estimate S(10,000), f(10,000) and the variance of your estimators.
The point estimates are
99(17,500 − 7,500) + 42(10,000 − 7,500)
S 227 (10, 000) = 1 − =0.51762
227(17,500 − 7,500)
42
f 227 (10,000) =
227(17,500 − 7,500) =0.000018502.
The estimated variances are
and
dVar[S 227 (10,000)] =
99
10,0002
227 227 128
42
+2,5002
227 227 185
99
− 2(10,000)(2,500)
227 227
42
227(10,000) 2 =0.00094713
42 185
227 227
dVar[f 227 (10,000)] =
227(10,000) 2 =6.6427 × 10−12 .