Estimation, Evaluation, and Selection of Actuarial Models

50 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
For the product-limit estimator,
" jY
Var[S n (y j )] = Var
i=1
jY
· S(yi ) 2
=
=
i=1
i=1
µ #
ri − S i
r i
S(y i−1 ) 2 + [S(y i−1) − S(y i )]S(y i )
r i S(y i−1 ) 2
¸
− S(y j) 2
S(y 0 ) 2
jY
·ri S(y i ) 2 ¸
+[S(y i−1 ) − S(y i )]S(y i )
r i S(y i−1 ) 2
− S(y j) 2
S(y 0 ) 2
jY
· S(yi ) 2
¸
r i S(y i )+[S(y i−1 ) − S(y i )]
=
S(y
i=1 i−1 ) 2 − S(y j) 2
r i S(y i )
S(y 0 ) 2
( jY
= S(y j) 2 ·
S(y 0 ) 2 1+ S(y ¸ )
i−1) − S(y i )
− 1 .
r
i=1
i S(y i )
This formula is unpleasant to work with, so the following approximation is often used. It is based on
the fact that for any set of small numbers a 1 ,...,a n , the product (1+a 1 ) ···(1+a n ) is approximately
1+a 1 + ···+ a n . This follows because the missing terms are all products of two or more of the
a i s. If they are all small to begin with, the products will be even smaller, and so can be ignored.
Applying this produces the approximation
Var[S n (y j )] . =
¸2 ·S(yj )
S(y 0 )
jX
i=1
S(y i−1 ) − S(y i )
.
r i S(y i )
Because it is unlikely that the survival function is known, an estimated value needs to be inserted.
Recall that the estimated value of S(y j ) is actually conditional on being alive at age y 0 . Also,
(r i − s i )/r i is an estimate of S(y i )/S(y i−1 ). Then,
dVar[S n (y j )] . = S n (y j ) 2
jX
i=1
s i
r i (r i − s i ) .
This formula is known as Greenwood’s approximation. It is the only version that will be used in
this Note.
Example 3.25 Using Data Set D1, estimate the variance of S 30 (3) both directly and using Greenwood’s
formula. Do the same for 2ˆq 3 .
Because there is no censoring or truncation, the empirical formula can be used to directly
estimate this variance. There were three deaths out of 30 individuals, and therefore
dVar[S 30 (3)] = (3/30)(27/30) = 81
30 30 3 .
For Greenwood’s approximation, r 1 =30, s 1 =1, r 2 =29,ands 2 =2. The approximation is
µ 27 2 µ 1
30 30(29) + 2
29(27)
= 81
30 3 .