  Published in 1999, Christodoulou proved that the naked singularities of a
self-gravitating scalar field are not stable in spherical symmetry and
therefore the cosmic censorship conjecture is true in this context. The
original proof is by contradiction and sharp estimates are obtained strictly
depending on spherical symmetry. In this paper, appropriate a priori estimates
for the solution are obtained. These estimates are more relaxed but sufficient
for giving another robust argument in proving the instability, in particular
not by contradiction. In another related paper, we are able to prove
instability theorems of the spherical symmetric naked singularities under
certain isotropic gravitational perturbations without symmetries. The argument
given in this paper plays a central role.
