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We show how to use properties of the vectors which are iterated in thetransfer-matrix approach to Anderson localization, in order to generate thestatistical distribution of electronic wavefunction amplitudes at arbitarydistances from the origin of $L^{d-1} \times \infty$ disordered systems. For$d=1$ our approach is shown to reproduce exact diagonalization resultsavailable in the literature. In $d=2$, where strips of width $ L \leq 64$ siteswere used, attempted fits of gaussian (log-normal) forms to the wavefunctionamplitude distributions result in effective localization lengths growing withdistance, contrary to the prediction from single-parameter scaling theory. Wealso show that the distributions possess a negative skewness $S$, which isinvariant under the usual histogram-collapse rescaling, and whose absolutevalue increases with distance. We find $0.15 \lesssim -S \lesssim 0.30$ for therange of parameters used in our study, .Comment: RevTeX 4, 6 pages, 4 eps figures. Phys. Rev. B (final version, to be  published

Failure of single-parameter scaling of wave functions in Anderson localization