Jack-knifing a Multitaper SDF estimator

Assume there is a parameter \theta that parameterizes a distribution, and that the set of random variables \lbrace Y_1, Y_2, ..., Y_n \rbrace are i.i.d. according to that distribution.

The basic jackknifed estimator \tilde{\theta} of some parameter \theta is found through forming pseudovalues \hat{\theta}_i 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

simple sample estimate
\hat{\theta} = \dfrac{1}{n}\sum_i Y_i
leave-one-out measurement
\hat{\theta}_{-i} = \dfrac{1}{n-1}\sum_{k \neq i}Y_k
pseudovalues
\hat{\theta}_i = n\hat{\theta} - (n-1)\hat{\theta}_{-i}

Now the jackknifed esimator is computed as

\tilde{\theta} = \dfrac{1}{n}\sum_i \hat{\theta}_i = n\hat{\theta} - \dfrac{n-1}{n}\sum_i \hat{\theta}_{-i}

This estimator is known (?) to be distributed about the true parameter theta approximately as a Student’s t distribution, with standard error defined as

s^{2} = \dfrac{n-1}{n}\sum_i \left(\hat{\theta}_i - \tilde{\theta}\right)^{2}

General Multitaper definition

The general multitaper spectral density function (sdf) estimator, using n orthonormal tapers, combines the n \lbrace \hat{S}_i^{mt} \rbrace sdf estimators, and takes the form

\hat{S}^{mt}(f) = \dfrac{\sum_{k} w_k(f)^2S^{mt}_k(f)}{\sum_{k} |w_k(f)|^2} = \dfrac{\sum_{k} w_k(f)^2S^{mt}_k(f)}{\lVert \vec{w}(f) \rVert^2}

For instance, using discrete prolate spheroidal sequences (DPSS) windows, the \rbrace w_i \lbrace 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)

\ln\hat{S}_{-i}^{mt} = \ln\dfrac{\sum_{k \neq i} w_k^2S^{mt}_k}{\lVert \vec{w}_{-i} \rVert^2} = \ln\sum_{k \neq i} w_k^2S^{mt}_k - \ln\lVert \vec{w}_{-i} \rVert^2

where \vec{w}_{-i} 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 S(f) more evenly.

Multitaper Pseudovalues

I’m not quite clear on the form of the pseudovalues for multitaper combinations.

One Option

The simple option is to weight the different leave-one-out measurements equally, which leads to

\ln\hat{S}_{i}^{mt} = n\ln\hat{S}^{mt} - (n-1)\ln\hat{S}_{-i}^{mt}

And of course the estimate of S(f) is given by

\ln\tilde{S}^{mt} (f) = \dfrac{1}{n}\sum_i \ln\hat{S}_i^{mt}(f)

Another Option

Another approach seems obvious which weights the leave-one-out measurements according to the length of \vec{w}_{-i}. It would look something like this

let
g = {\lVert \vec{w} \rVert^2}
g_i = {\lVert \vec{w}_{-i} \rVert^2}

Then the pseudovalues are

\ln\hat{S}_i^{mt} = \left(\ln\hat{S}^{mt} + \ln g\right) - \left(\ln\hat{S}_{-i}^{mt} + \ln g_i\right)

and the jackknifed estimator is

\ln\tilde{S}^{mt} = \sum_i \ln\hat{S}_i^{mt} - \ln g

and I would wager, the standard error is estimated as

s^2 = \dfrac{1}{n}\sum_i \left(\ln\hat{S}_i^{mt} - \ln\tilde{S}^{mt}\right)^2

Consensus in Literature (??)

From what I can tell from a couple of sources from Thompson [#f1]_, [#f2]_, this is the approach for estimating the variance.