Jack-knifing a Multitaper SDF estimator¶
Assume there is a parameter that parameterizes a distribution, and that the set of random variables are i.i.d. according to that distribution.
The basic jackknifed estimator of some parameter is found through forming pseudovalues based on the original set of samples. With n samples, there are n pseudovalues based on n “leave-one-out” sample sets.
General JN definitions¶
Now the jackknifed esimator is computed as
This estimator is known (?) to be distributed about the true parameter theta approximately as a Student’s t distribution, with standard error defined as
General Multitaper definition¶
The general multitaper spectral density function (sdf) estimator, using n orthonormal tapers, combines the n sdf estimators, and takes the form
For instance, using discrete prolate spheroidal sequences (DPSS) windows, the set, in their simplest form, are the eigenvalues of the spectral concentration system.
A natural choice for a leave-one-out measurement is (leaving out the dependence on argument f)
where is the vector of weights with the ith element set to zero. The natural log has been taken so that the estimate is distributed below and above more evenly.
I’m not quite clear on the form of the pseudovalues for multitaper combinations.
The simple option is to weight the different leave-one-out measurements equally, which leads to
And of course the estimate of is given by
Another approach seems obvious which weights the leave-one-out measurements according to the length of . It would look something like this
Then the pseudovalues are
and the jackknifed estimator is
and I would wager, the standard error is estimated as