DESI DR1 Ly$\alpha$ 1D power spectrum: Validation of estimators

The Data Release 1 (DR1) of the Dark Energy Spectroscopic Instrument (DESI) is the largest sample to date for small-scale Ly$\alpha$ forest cosmology, accessed through its one-dimensional power spectrum ($P_{\mathrm{1D}}$). The Ly$\alpha$ forest $P_{\mathrm{1D}}$ is extracted from quasar spectra that are highly inhomogeneous (both in wavelength and between quasars) in noise properties due to intrinsic properties of the quasar, atmospheric and astrophysical contamination, and also sensitive to low-level details of the spectral extraction pipeline. We employ two estimators in DR1 analysis to measure $P_{\mathrm{1D}}$: the optimal estimator and the fast Fourier transform (FFT) estimator. To ensure robustness of our DR1 measurements, we validate these two power spectrum and covariance matrix estimation methodologies against the challenging aspects of the data. First, using a set of 20 synthetic 1D realizations of DR1, we derive the masking bias corrections needed for the FFT estimator and the continuum fitting bias needed for both estimators. We demonstrate that both estimators, including their covariances, are unbiased with these corrections using the Kolmogorov-Smirnov test. Second, we substantially extend our previous suite of CCD image simulations to include 675,000 quasars, allowing us to accurately quantify the pipeline's performance. This set of simulations reveals biases at the highest $k$ values, corresponding to a resolution error of a few percent. We base the resolution systematics error budget of DR1 $P_{\mathrm{1D}}$ on these values, but do not derive corrections from them since the simulation fidelity is insufficient for precise corrections.