  A recent experiment on s-wave neutron scattering from $^{192,194,196}$Pt
found that the reduced neutron width distributions deviate significantly from
the expected Porter-Thomas distribution (PTD), and several explanations have
been proposed within the statistical model of compound nucleus reactions. Here,
we study the statistics of reduced neutron widths in the reaction $n+ ^{194}$Pt
within a model that combines the standard statistical model with a realistic
treatment of the neutron channel. We find that, if the correct secular energy
dependence of the average neutron widths is used, then the reduced neutron
width distribution is in excellent agreement with the PTD for a reasonable
range of the neutron-nucleus coupling strength and depth of the neutron channel
potential. Within our parameter range, there can be a near-threshold bound or
virtual state of the neutron channel potential that modifies the energy
dependence of the average width from the $\sqrt{E}$ dependence, commonly
assumed in experimental analysis, in agreement with the proposal of H. A.
Weidenm\"uller [1]. In these cases, the reduced neutron width distributions
extracted using the $\sqrt{E}$ dependence are significantly broader than the
PTD. We identify a relatively narrow range of parameters where this effect is
significant.
