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 Dq are nonfluctuating quantities, contrary to a recent claim in the literature. A recently proposed relation between D2 and the spectral compressibility χ is violated in the regime of strong multifractality, with χ→1 in the limit D2→0.