Interpolating the peak of a cross-correlation using a centroid Forgive me if this question is poorly worded, I'm not sure if centroid is the word to use here.

Say I want to interpolate the peak of a cross-correlation function in order to get sub-sample delays. I can fit a parabola with three points, or simply upsample my signal through zero-padding/sinc-interpolation. However, I haven't seen any comments (or found any literature) on using some sort of weighted average on the main peak.

In case it's unclear, i mean performing the following:


Finding the maximum of the cross-correlation in order to get a coarse estimate.
$$
\hat{D} = argmax(R_{12})
$$
Taking the P nearest neighbors (where [-P,P] corresponds to the central peak area) and doing the following fine estimate 


$$
\hat{d} = \frac{\sum_{k =-P}^P kR_{12}(k)}{\sum_{k=-P}^PR_{12}(k)}
$$

I've tested it for a gaussian distributed, infinite snr signal, critically sampled (so peak is only three samples. Centroid, (yellow) versus some sinc interpolation (purple) and parabolic fitting (orange), and it seems to offer good performance.



It appears to have less maximum bias, despite the discontinuity. 
I also tested it with noise and the results were also okay. I get that this requires knowledge of the signal A.C.F. beforehand whereas fitting a parabola doesn't, but is there any other reason why this method is not seen in the literature?

Either way, I'm finding it somewhat advantageous in my application (also not sure wh), so I would like to read some more formal studies or descriptions of the technique. My background is not signal processing, so I'm afraid I'm overlooking some severe shortcoming. 

NOTE: I also tested the mean-square error. It seems to perform much worse there,
 but as I said, for my use case it seems to perform well. . (The black line is the cramer-rao lower bound. Not sure why the parabola goes below it, probably because it's a biased estimator...)

EDIT: Signal example of the application (this picture has quite a lot of distortion of the delayed signal, but it's what I have at hand currently).
There are a lot of papers on bin interpolation and some of them have obvious connections to interpolating a parabola and others aren't' so obvious. Given what you presented, I'm sure that you can make the connection between cross correlation and spectral analysis. I suggest  you look at:

M. D. Macleod, "Fast nearly ML estimation of the parameters of real or complex single tones or resolved multiple tones," in IEEE Transactions on Signal Processing, vol. 46, no. 1, pp. 141-148, Jan 1998.
doi: 10.1109/78.651200
Abstract: This paper presents new computationally efficient algorithms for estimating the parameters (frequency, amplitude, and phase) of one or more real tones (sinusoids) or complex tones (cisoids) in noise from a block of N uniformly spaced samples. The first algorithm is an interpolator that uses the peak sample in the discrete Fourier spectrum (DFS) of the data and its two neighbors. We derive Cramer-Rao bounds (CRBs) for such interpolators and show that they are very close to the CRB's for the maximum likelihood (ML) estimator. The new algorithm almost reaches these bounds. A second algorithm uses the five DFS samples centered on the peak to produce estimates even closer to ML. Enhancements are presented that maintain nearly ML performance for small values of N. For multiple complex tones with frequency separations of at least 4π/N rad/sample, unbiased estimates are obtained by incorporating the new single-tone estimators into an iterative “cyclic descent” algorithm, which is a computationally cheap nonlinear optimization. Single or multiple real tones are handled in the same way. The new algorithms are immune to nonzero mean signals and (provided N is large) remain near-optimal in colored and non-Gaussian noise
keywords: {Fourier analysis;Gaussian noise;amplitude estimation;frequency estimation;harmonic analysis;interpolation;iterative methods;maximum likelihood estimation;optimisation;phase estimation;signal sampling;spectral analysis;white noise;AWGN;Cramer-Rao bounds;FFT;amplitude;cisoids;colored noise;complex single tones;discrete Fourier spectrum;fast nearly ML estimation;frequency;frequency separations;interpolator;iterative cyclic descent algorithm;noise;nonGaussian noise;nonlinear optimization;nonzero mean signals;parameter estimation;peak sample;phase;real single tones;resolved multiple tones;sinusoids;unbiased estimates;uniformly spaced samples;Amplitude estimation;Computational efficiency;Frequency estimation;Iterative algorithms;Maximum likelihood estimation;Noise level;Parameter estimation;Phase estimation;Phase noise;Signal processing algorithms},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;arnumber=651200&amp;isnumber=14197

and if you can get into IEEE Explore, look at the papers that cite it. These are the State-OF-The-Art bin interpolation schemes, which you should be making your comparisons to.  

The only comment that I have about you centroid technique, would be that the accuracy would seem to depend on there being only a single delay in your data.  My experience is that where there is one delay, there are often many.  Having too wide a window would seem to be more susceptible to additional delays.