Dwarf spheroidal satellite galaxies (dSphs) of the Milky Way are considered ideal targets for particle Dark Matter (DM) identification. Indirect detection strategies entail examining dSphs in search for signals of annihilating or decaying DM, in the form of excess electrons or gamma- and X-ray photons above the astrophysical background. To robustly compare model predictions with the observed fluxes of such product particles, most analyses of astrophysical data - which are generally frequentist - rely on estimating the abundance of DM by calculating the so-called J-factor. This quantity is usually inferred from the kinematic properties of the stellar population of a dSph using Jeans equation, commonly by means of Bayesian techniques. Previous works have, therefore, combined different statistical methods when analysing astrophysical data from dSphs. In this thesis, I describe the development of a new, fully-frequentist approach for constructing profile likelihood curves for the J-factor of dSphs. I then use kinematic data from 20 dSphs to derive estimates of their maximum likelihood J-factor and its confidence intervals. The obtained J-factors and their uncertainties are in good agreement with previous, Bayesian-derived values. This technique is validated using a publicly available simulation suite, released by Gaia Challenge, by evaluating its coverage and bias. The results of these tests indicate that the method possess good statistical properties. The implications of these findings for DM searches are discussed, together with future improvements and extensions of this technique.