Behavior near zero of the distribution of GCV smoothing parameter estimates

It has been noticed by several authors that there is a small but non-zero probability that the GCV estimate [lambda] of the smoothing parameter in spline and related smoothing problems will be extremely small, leading to gross undersmoothing. We obtain an upper bound to the probability that the GCV function, whose minimizer provides [lambda], has a (possibly local) minimum at 0. This upper bound goes to 0 exponentially fast as the sample size gets large. For the medium- to small-sample case we study this probability both by Monte Carlo evaluation of a formula for the exact probability that the GCV function has a minimum at 0 as well as by replicated calculations of [lambda].