You are given the following estimated survival properties and risk sets determined using the Kaplan-Meier method.

[math]j[/math]	[math]t_{(j)}[/math]	[math]r_{j}[/math]	[math]\hat{p}_{j}[/math]
1	17.2	29	0.9655
2	22.1	27	0.9259
3	32.7	24	0.9583
4	45.0	20	0.9500

Calculate the standard deviation of [math]\hat{S}(25)[/math] using Greenwood's formula.

• 0.0543
• 0.0556
• 0.0579
• 0.0604
• 0.0626

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Answer: C

[math]s d[\hat{S}(25)] \approx \hat{S}(25) \sqrt{\sum_{t_{(j)} \leq 25} \frac{d_{j}}{r_{j}\left(r_{j}-d_{j}\right)}}[/math]

[math]\hat{S}(25)=\hat{p}_{1} \times \hat{p}_{2}=0.8940[/math]

[math]d_{1}=r_{1}\left(1-\hat{p}_{1}\right)=1 ; \quad d_{2}=r_{2}\left(1-\hat{p}_{2}\right)=2[/math]

[math]\Rightarrow s d[\hat{S}(25)] \approx 0.8940 \sqrt{\left(\frac{1}{29 \times 28}+\frac{2}{27 \times 25}\right)}=0.0579[/math]

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.