# 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¶

**simple sample estimate**

**leave-one-out measurement**

**pseudovalues**

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.

## 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

And of course the estimate of is given by

### Another Option¶

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