  Indirect detection strategies of particle Dark Matter (DM) in Dwarf
spheroidal satellite galaxies (dSphs) typically entail searching for
annihilation signals above the astrophysical background. To robustly compare
model predictions with the observed fluxes of product particles, most analyses
of astrophysical data -- which are generally frequentist -- rely on estimating
the abundance of DM by calculating the so-called $\textit{J-factor}$. 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 which entail the presence (and additional systematic uncertainty) of
prior choice. Here, extending earlier work, we develop a scheme to derive the
profile likelihood for $J$-factors of dwarf spheroidals for models with five or
more free parameters. We validate our method on a publicly available simulation
suite, released by the Gaia Challenge, finding satisfactory statistical
properties for coverage and bias. We present the profile likelihood function
and maximum likelihood estimates for the $J$-factor of ten dSphs . As an
illustration, we apply these profiles likelihood to recently published analyses
of gamma-ray data with the Fermi Large Area Telescope to derive new, consistent
upper limits on the DM annihilation cross-section. We do this for a subset of
systems, generally referred to as $\textit{classical dwarfs}$. The implications
of these findings for DM searches are discussed, together with future
improvements and extensions of this technique.
