The art of fitting p-mode spectra: Part I. Maximum Likelihood Estimation

In this article we present our state of the art of fitting helioseismic p-mode spectra. We give a step by step recipe for fitting the spectra: statistics of the spectra both for spatially unresolved and resolved data, the use of Maximum Likelihood estimates, the statistics of the p-mode parameters, the use of Monte-Carlo simulation and the significance of fitted parameters. The recipe is applied to synthetic low-resolution data, similar to those of the LOI, using Monte-Carlo simulations. For such spatially resolved data, the statistics of the Fourier spectrum is assumed to be a multi-normal distribution; the statistics of the power spectrum is \emph{not} a $\chi^{2}$ with 2 degrees of freedom. Results for $l=1$ shows that all parameters describing the p modes can be obtained without bias and with minimum variance provided that the leakage matrix is known. Systematic errors due to an imperfect knowledge of the leakage matrix are derived for all the p-mode parameters.