Interpreting a paper on spectral analysis

[boneh3ad]

I do a fair bit of spectral analysis of time series in my research, but to date my experience in the topic is almost exclusively from an engineering perspective rather than the more statistical approach. Of course I am aware that ultimately they are equivalent, but it means that my familiarity with the terminology and general language surrounding the statistical flavor of time series analysis is somewhat lacking.

With this in mind, there is a paper [1] I've been exploring that has be a bit confused and I was hoping there was someone around here that might have a bit of experience in this area that can guide me. In particular, I am a bit confused with section 4, where they lay out the proposed normalization method. I don't know if there is an abuse of notation somewhere or if it is my own unfamiliarity with the statistical version of this topic, but I am having a really tough time deciphering this and subsequently implementing it.

Any help would be appreciated.

Thanks.

[1] Hinich MJ, Wolinsky M, 2005. Normalizing bispectra. J. Statistical Planning and Inference (130) 1-2.

 

[olivermsun]

boneh3ad,

Are you trying to implement the "standard" normalization method advocated by Hinich and Wolinsky (2005)? (The method is also discussed in, e.g., Hinich and Clay (1968)).

 

[boneh3ad]

No, the standard "normalization" is not really a normalization, but simply calculating the skewness function using FFTs. I am looking to implement the chi-squared normalization discussed at the bottom in section 4.

 

[boneh3ad]

Now that I have a few moments to spare, I can expand on this a bit to be a bit more specific in my questions.

Calculating the skewness function estimate, $\hat{\Gamma}(f_{k_1},f_{k_2})$, is straightforward and really just boils down to using something akin to Welch's method of windowing and averaging the signal to calculate the spectra and bispectra and dividing accordingly. That part is clear to me. There are several other portions from the paper that require clarification for me. I have several more fundamental questions and one regarding the practical aspects of implementing section 4.

Issue 1
Near the end of section 2, the discuss an assumption that $L=N^e$ and that $0<e<0.5$. It is not clear to me why this condition must be satisfied.

Issue 2
Extensive use is made of $\gamma_e$, which is the skewness of $e(t)$ (which is a different $e$ from the one above, which is rather confusing). Given my relative unfamiliarity with the statistical approach to time series analysis, I am not, at this point, entirely clear on how to calculate $\gamma_e$ when all I know at this point is $x(t)$. If anyone knows of a solid reference on this I'd appreciate it. I have checked out a book or two from the library and am slowing getting myself up to speed, but it is slow going while I juggle this with other job responsibilities.

Issue 3
In section 4, they discuss the statistical normalization based on a non-central $\chi^2$ distribution. They suggest it has two degrees of freedom but I am struggling to determine why. It seems to me that, for any pair $(k_1,k_2)$, there is only one squared value here, which is $\hat{\Gamma}(f_{k_1},f_{k_2})$. Is it because $k_1$ and $k_2$ are allowed to freely vary? If so, then it seems like my understanding of $\chi^2$ variables has taken a nosedive since I took the course a decade-plus ago. On a related note, why is the $N^{2e-1}$ term necessary here?

Issue 4
I suspect that the noncentrality parameter has the "unhatted" version of $\Gamma$ because it should be based on the expected value of $\Gamma$. I obviously don't know the true value there, but that seems to be why they use $\gamma_e^2$ since that should be a known quantity and $|\Gamma| = |\gamma_e|$ under the null hypothesis that the time series is linear. What I don't understand, then, is that if $\lambda$ is the same for all bifrequencies under this null hypothesis, how can they suggest then using $\hat{\lambda}$ based on multiple presumably different $\lambda$ values? I feel like there is missing information here and potentially typos. In short, I suppose it is the calculation of $\hat{\lambda}$ that has me confused. It is supposed to be based on $\gamma_e$, which is the subject of Issue 2 above, but is it based on that value calculated for each signal window and then averaged? That would make some sense based on how all of the other hatted quantities are calculated.

Thanks again.