1. How can the MFHB bootstrap procedure be applied to estimate the distribution of statistics that are functions of integrated periodograms?
The MFHB bootstrap procedure can be applied to estimate the distribution of statistics that are functions of integrated periodograms by appropriately modifying the procedure. For example, if g is a smooth function and the statistic of interest is n(Rn - R), where Rn = g(Mn) and R = g(M), the bootstrap procedure can be adapted to approximate the distribution of n(Rn - R). This involves imposing smoothness assumptions on the function g, splitting it into real and imaginary parts, and applying the delta method to obtain the limiting result. The algorithm consists of steps such as applying the MFHB algorithm, calculating the pseudo periodogram matrices, and approximating the distribution of the sample cross-correlation. The algorithm works on real and imaginary parts separately, allowing for real differentiability of g instead of complex differentiability. This approach has been proven to be asymptotically valid for the sample cross-correlation and successfully imitates the expression for the limiting variance.
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2. How to choose MFHB parameters?
The choice of MFHB parameters, f and b, is crucial for the consistency of the procedure. Assumption 5 and 6 provide general requirements for these parameters. For f, kernel estimators can be used, with h controlling the number of periodogram ordinates. Cross-validation approaches can help select h. For b, a practical rule suggests selecting the smallest integer larger or equal to 3*n^0.30. This satisfies Assumption 6 and ensures good performance in practice. Different combinations of h and b are tested to assess sensitivity of bootstrap estimates.
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3. How does the MFHB procedure compare to the MBB in terms of performance for estimating the standard deviation of cross-correlation estimates?
The MFHB procedure outperforms the MBB procedure in terms of performance for estimating the standard deviation of cross-correlation estimates. The results presented in Table 1 show that the MFHB mean square errors are consistently lower than those of the MBB procedure, regardless of the choice of bandwidth h. Additionally, the MFHB estimates are less sensitive to the choice of subsampling parameter b compared to the MBB procedure's sensitivity to the choice of block size b. This indicates that the MFHB procedure is a more reliable and efficient method for estimating cross-correlation estimates in the given context.
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