Estimation, Evaluation, and Selection of Actuarial Models

48 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
Constructing confidence intervals is usually very difficult. However, there is a method for
constructing approximate confidence intervals that is often accessible. Suppose we have an estimator
ˆθ of the parameter θ such that E(ˆθ) = . θ, Var(ˆθ) = . v(θ), andˆθ has approximately a normal
distribution. With all these approximations, we have
Ã
1 − α =Pr
. −z α/2 ≤ ˆθ
!
− θ
p ≤ z α/2 (3.2)
v(θ)
and solving for θ produces the desired interval. Sometimes this is difficult to do (due to the
appearance of θ in the denominator) and so we may replace v(θ) in (3.2) with v(ˆθ) to obtain a
further approximation
1 − α =Pr
. q
µˆθ − zα/2
qv(ˆθ) ≤ θ ≤ ˆθ + z α/2 v(ˆθ)
(3.3)
where z α is the 100(1 − α)th percentile of the standard normal distribution.
Example 3.24 Use (3.2) and (3.3) to construct approximate 95% confidence intervals for p(2)
using Data Set A.
From (3.2),
⎛
⎞
0.95 = Pr ⎝−1.96 ≤ p n(2) − p(2)
q ≤ 1.96⎠ .
p(2)[1−p(2)]
n
Solve this by making the inequality an equality and then squaring both sides to obtain (dropping
the argument of (2) for simplicity),
(p n − p) 2 n
p(1 − p)
= 1.96 2
np 2 n − 2npp n + np 2 = 1.96 2 p − 1.96 2 p 2
0 = (n +1.96 2 )p 2 − (2np n +1.96 2 )p + np 2 n
p = 2np n +1.96 2 ± p (2np n +1.96 2 ) 2 − 4(n +1.96 2 )np 2 n
2(n +1.96 2 )
which provides the two endpoints of the confidence interval. Inserting the numbers from Data Set
A(p n =0.017043, n =94,935) produces a confidence interval of (0.016239, 0.017886).
Equation (3.3) provides the confidence interval directly as
r
pn (1 − p n )
p n ± 1.96
.
n
Inserting the numbers from Data Set A gives 0.017043±0.000823 for an interval of (0.016220, 0.017866).
The answers for the two methods are very similar, whichisthecasewhenthesamplesizeislarge.
The results are reasonable, because it is well known that the normal distribution is a reasonable
approximation to the binomial. ¤