# Unbiased estimator for the following survival function

I have the following data, where $$+$$ denotes right censoring.

$$2, 6, 7^+, 8^+, 10, 11, 14^+, 16$$ I would like to find an unbiased estimator for $$S(6)S(10)$$, where $$S(\cdot)$$ mean survival function.

My try

KM estimate for $$\hat{S}(6) = \frac{7}{8} \times \frac{6}{7} = \frac{3}{4}$$, $$\hat{S}(10) = \frac{7}{8} \times \frac{6}{7} \times \frac{3}{4} = \frac{9}{16}$$, where $$\hat{S}(\cdot)$$ denotes Kaplan-Meier estimator.

But $$\hat{S}(6)\hat{S}(10)$$ would never work, without their independence, and I have heard that KM estimator is biased, so I'm stuck at how I should proceed.

• We estimate the parameters, not the statistics, such as $\hat S(6)\hat S(10). Oct 26, 2018 at 1:36 • @a_statistician, thx for the comment, edited. Oct 26, 2018 at 1:40 • What is the meaning of$S(6)S(10)$? One person survivals to 6 and 10, or two persons, one survivals to 6 another survivals to 10? Oct 26, 2018 at 1:47 • @a_statistician it's just$P(X > 6) P(X > 10)$, where$X\$ is the lifetime. Oct 26, 2018 at 1:50
• Need two conditions: 1)KM is unbiased. 2)person 1's survival is independent from person 2's survival. 2) should be true. I do not know where you get the idea that 1) is not true. Oct 26, 2018 at 2:03