  Statistics of the inverse participation ratio (IPR) at the critical point of
the localization transition is studied numerically for the power-law random
banded matrix model. It is shown that the IPR distribution function is
scale-invariant, with a power-law asymptotic ``tail''. This scale invariance
implies that the fractal dimensions $D_q$ are non-fluctuating quantities,
contrary to a recent claim in the literature. A recently proposed relation
between $D_2$ and the spectral compressibility $\chi$ is violated in the regime
of strong multifractality, with $\chi\to 1$ in the limit $D_2\to 0$.
