  Estimation of the Saupe tensor is central to the determination of molecular
structures from residual dipolar couplings (RDC) or chemical shift
anisotropies. Assuming a given template structure, the singular value
decomposition (SVD) method proposed in Losonczi et al. 1999 has been used
traditionally to estimate the Saupe tensor. Despite its simplicity, whenever
the template structure has large structural noise, the eigenvalues of the
estimated tensor have a magnitude systematically smaller than their actual
values. This leads to systematic error when calculating the eigenvalue
dependent parameters, magnitude and rhombicity. We propose here a Monte Carlo
simulation method to remove such bias. We further demonstrate the effectiveness
of our method in the setting when the eigenvalue estimates from multiple
template protein fragments are available and their average is used as an
improved eigenvalue estimator. For both synthetic and experimental RDC datasets
of ubiquitin, when using template fragments corrupted by large noise, the
magnitude of our proposed bias-reduced estimator generally reaches at least 90%
of the actual value, whereas the magnitude of SVD estimator can be shrunk below
80% of the true value.
