Suppose we have observed L time series, all of which can be regarded as independent realizations from the same stationary process with SDF S(·). Suppose that the lth such series has a unique sample...

Suppose we have observed L time series, all of which can be regarded as independent realizations from the same stationary process with SDF S(·). Suppose that the lth such series has a unique sample size Nl and that the series are ordered such that
  Suppose we use the lth series to form a direct spectral estimate
 at some frequency f such that 0 <><>   is an unbiased estimator of S(f). We then combine the L different direct spectral estimates together to form

where αl > 0 for all l.

(a) What condition do we need to impose on the weights αl so that Sˆ (f) is also an unbiased estimator of S(f)?

(b) Assuming that (i) the αl ’s are chosen so that
  is unbiased and (ii) the usual approach to approximating the distribution of direct spectral estimates holds, determine the EDOFs ν for

(c) For an unbiased estimator, how should the weights αl be set so that ν is maximized?

(d) Does the fact that the time series all have different sample sizes Nl influence the EDOFs ν?