12.2 Means, variances, and interval estimation

When all of the information is available, working with the empirical estimate of the survival function is straightforward. To see that with complete data the empirical estimator of the survival function is unbiased and consistent, recall that the empirical estimate of S(x) is Sn (x) = Y/n, where Y is the number of observations in the sample that are greater than x. Then Y must have a binomial distribution with parameters n and S(x) and

equation

demonstrating that the estimator is unbiased. The variance is

equation

which has a limit of zero, thus verifying consistency.

To make use of the result, the best we can do for the variance is estimate it. It is unlikely we know the value of S(x) because that is the quantity we are trying to estimate. The estimated variance is given by

equation

The same results hold for empirically estimated probabilities. Let p = Pr(a < X ≤ b). The empirical estimate of p is = Sn(a) − Sn(b). Arguments similar to those used for Sn(x) verify that is unbiased and consistent, with Var() = p(1 − p)/n.

When doing mortality studies or evaluating ...

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