c) (12 points) (i) Describe the Nelson-Aalen estimate NA (t) of the cumulative hazard function H(t),...

Transcribed Image Text:

c) (12 points) (i) Describe the Nelson-Aalen estimate ÎNA (t) of the cumulative hazard function H(t), and the Kaplan-Meier (product-limit) estimate Skm(t) of the survival function S(t), explaining any notation that you use. (ii) Using the general approximation Var(In X) ≈ X-² Var(X) (applicable as long as the random variable X is 'close' to its expected value E(X)) and deploying the known formula for the variance of the Nelson-Aalen estimator HNA (t), de- rive an approximate expression for the variance of SNA(t) which is an estimator of the survival function S(t) based on the estimator of ĦNA(t). (iii) Let t(1) be the smallest (uncensored) death time in a sample. Compare the expression for Var{ SNA(t(1))} obtained in part (ii) with the corresponding value provided by Greenwood's formula for the variance of SKM (t(1)), and prove a suitable inequality. When are these two values close to each other? c) (12 points) (i) Describe the Nelson-Aalen estimate ÎNA (t) of the cumulative hazard function H(t), and the Kaplan-Meier (product-limit) estimate Skm(t) of the survival function S(t), explaining any notation that you use. (ii) Using the general approximation Var(In X) ≈ X-² Var(X) (applicable as long as the random variable X is 'close' to its expected value E(X)) and deploying the known formula for the variance of the Nelson-Aalen estimator HNA (t), de- rive an approximate expression for the variance of SNA(t) which is an estimator of the survival function S(t) based on the estimator of ĦNA(t). (iii) Let t(1) be the smallest (uncensored) death time in a sample. Compare the expression for Var{ SNA(t(1))} obtained in part (ii) with the corresponding value provided by Greenwood's formula for the variance of SKM (t(1)), and prove a suitable inequality. When are these two values close to each other?

Answer rating: 100% (QA)

ANSWER i The NelsonAalen estimate HNAt is a nonparametric estimator of the cumulative hazard function Ht in survival analysis The cumulative hazard fu... View the full answer