Nonparametric estimation of convex models via mixtures

It is well known that bootstrap bias-correction typically reduces bias and increases variance. It is generally anticipated that the resultant mean squared error will be reduced. We provide a real-life example where the mean squared error will either decrease or increase, depending on what is assumed for an underlying distribution. Using only concepts from first-year statistics graduate school curricula, the bias-corrected estimator and its mean squared error formula are developed in a simple closed form expression. Comparisons with the uncorrected estimator are made. The content of this example can be the basis for a classroom lecture, helping students vividly appreciate both what bootstrap bias-correction accomplishes and how modern statistics methodology contributes to solving a real problem.