A comparison of quasar emission reconstruction techniques for $z\geq5.0$
  Lyman-$\alpha$ and Lyman-$\beta$ transmission
  Reconstruction techniques for intrinsic quasar continua are crucial for the
precision study of Lyman-$\alpha$ (Ly-$\alpha$) and Lyman-$\beta$ (Ly-$\beta$)
transmission at $z>5.0$, where the $\lambda<1215 A$ emission of quasars is
nearly completely absorbed. While the number and quality of spectroscopic
observations has become theoretically sufficient to quantify Ly-$\alpha$
transmission at $5.0<z<6.0$ to better than $1\%$, the biases and uncertainties
arising from predicting the unabsorbed continuum are not known to the same
level. In this paper, we systematically evaluate eight reconstruction
techniques on a unified testing sample of $2.7<z<3.5$ quasars drawn from eBOSS.
The methods include power-law extrapolation, stacking of neighbours, and six
variants of Principal Component Analysis (PCA) using direct projection, fitting
of components, or neural networks to perform weight mapping. We find that
power-law reconstructions and the PCA with fewest components and smallest
training sample display the largest biases in the Ly-$\alpha$ forest
($-9.58\%/+8.22\%$ respectively). Power-law extrapolations have larger scatters
than previously assumed of $+13.1\%/-13.2\%$ over Ly-$\alpha$ and
$+19.9\%/-20.1\%$ over Ly-$\beta$. We present two new PCAs which achieve the
best current accuracies of $9\%$ for Ly-$\alpha$ and $17\%$ for Ly-$\beta$. We
apply the eight techniques after accounting for wavelength-dependent biases and
scatter to a sample $19$ quasars at $z>5.7$ with IR X-Shooter spectroscopy,
obtaining well-characterised measurements for the mean flux transmission at
$4.7<z<6.3$. Our results demonstrate the importance of testing and, when
relevant, training, continuum reconstruction techniques in a systematic way.
