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Any help on either/both of these would be greatly appreciated. --------------------------------------------------- Suppose that P ( Y = i ) = p ( i ) for
Any help on either/both of these would be greatly appreciated.
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Suppose that P(Y=i)=p(i) for i=1,2,...,m. Assume that you can simulate X(i) which corresponds to the value of X given that Y=i. We are interested in estimating =E(X) using optimal stratified sampling. Consider the estimator
(n1,...,nm)=i=1mnip(i)j=1niXj(i).
where the Xj(i)'s are i.i.d. copies of X(i) and all of the random variables involved are independent.
- Use calculus to find an expression for the values (n1,...,nm) that minimizes the variance of (n1,...,nm) subject to the constraint that n1++nm=N. Feel free to treat the ni's as continuous variables for this exercise.
- Discuss briefly how to implement an estimator based on the optimal ni's given that such estimator depends on Var(X(i)), which is unknown.
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