This is from a discussion of analyzing a time series with a sinusoid + noise model. The troublesome statement is:
...data values near the beginning and end of a record are most important for frequency estimation, in agreement with intuitive common sense.
Intuitive? If there were any variation in importance, I would expect the values near the middle of the record to be most important, as reflected by our standard window functions (Hamming, Hann, Blackman, etc.), which all have peaks in the middle. What gives?
The following passage should provide enough context.
$$(\omega)_{\mathrm{est}} = \hat{\omega} \pm \frac{\sigma}{|\hat{B_1}|}\sqrt{48/N^3}\qquad(2.10)$$ blah blah blah...
The frequency estimate becomes $$f_{\mathrm{est}} = \hat{f}\pm\delta f \; \mathrm{Hz}$$ where now, not distinguishing between $N$ and $(N-1)$, $$\delta f = \frac{\sigma}{2\pi\hat{B_1}T}\sqrt{48/N} = \frac{1.1\sigma}{\hat{B_1}T\sqrt{N}} \; \mathrm{Hz} \qquad (2.11)$$ Comparing this with (2.10) we now see that to improve the accuracy of the estimate the two most important factors are how long we sample (the $T$ dependence) and the signal-to-noise ratio. We could double the number of data values in one of two ways, by doubling the total sampling time or by doubling the sampling rate. However, (2.11) clearly indicates that doubling the sampling time is to be preferred. This indicates that data values near the beginning and end of a record are most important for frequency estimation, in agreement with intuitive common sense [emphasis mine].
For intuition, consider how you'd measure the frequency by eye, if you don't have a lot of computational resources. Simply counting waves and troughs gives you a fairly good initial estimate of the frequency. In order to make it more precise, what you'd want to do is find two zero crossings and divide the number of waves between them with their distance.
You known the numerator exactly (that's just a matter of counting), and the uncertainty in locating the zero crossings is the same no matter which crossing you select. So the relative error will be minimized if you measure the zero crossings as far from each other as you can -- that is at one close to the beginning of the record and one close to the end. So the precise samples there contribute more importantly to your estimate than the one in the middle do.
(Less simplified, one should probably speak about estimating the phase at the beginning and end of the record and use the phase difference as the fractional part of the number of waves in the entire record -- but that's more or less the same thing. In this view, the samples in the middle matter less, because they are farthest from the two points where we need to know the phase precisely).