For the general linear CV method with m control variates, the variance of the controlled e...

For the general linear CV method with m control variates, the variance of the controlled estimator was given in Eq. (11.3), where the weights a_l must be specified.

(a) Find the optimal (i.e., variance-minimizing) weights for the cases m = 2 and 3.

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(b) Assuming that the control variates are uncorrelated with each other, find the optimal weights for any m.

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(c) For both cases (a) and (b) above, give a method for estimating the optimal weights. If we know the variances of the control variates (as we might if they are just averages of input variates), could your estimators be improved? In (a), what if we also know the covariances between the control variates?