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distribution, fails to replicate that of its population counterpart. To be specific, the empirical distribution function ˆ F$_{n}$, upon which ˆ f$_{n}$ ( x 0 ) is based, is an unbiased estimator of the cumulative distribution function F. Around x 0 , that function admits a quadratic approximation of the form 

<img src='content_image/312806.jpg'>

The bootstrap mean of the function function ˆ F ∗ n upon which ˆ f ∗ $_{n}$( x 0 ) is based is given by ˆ F$_{n}$. Unlike F, the function ˆ F$_{n}$ does not admit a quadratic approximation around x 0 , a fact to which Sen, Banerjee, and Woodroofe (2010, p. 1968) attribute the inconsistency of the bootstrap. Adapt- ing the logic used to motivate the functional form of ˜ m$_{n}$, a reshaped version of ˆ F ∗ n is given by 

<img src='content_image/312807.jpg'>

where ˜ f ′ $_{n}$( x 0 ) is an estimator of f $^{′}$( x 0 ) . Letting ˜ f ∗ $_{n}$( x 0 ) denote the left derivative at x 0 of the LCM of ˜ F ∗ $_{n}$, the hope is of course that the approximation 

<img src='content_image/312808.jpg'>

is consistent under mild conditions on ˜ f ′ $_{n}$( x 0 ) . In perfect analogy with Theorem 1, it turns out that this is indeed the case. 

Theorem 3 Suppose that f is differentiable at x 0 with strictly negative derivative f $^{′}$( x 0 ) . If ˜ f ′ $_{n}$( x 0 ) → P f $^{′}$( x 0 ) , then 

<img src='content_image/312809.jpg'>

The consistency requirement ˜ f ′ $_{n}$( x 0 ) → P f $^{′}$( x 0 ) associated with the Grenander estimator is mild, being met by standard nonparametric estimators. One obvious choice, and one we have found to work well in simulations, is a standard kernel density estimator with its corresponding plug-in MSE-optimal bandwidth selector; for details, see the supplemental appendix. Another possibility, also discussed in the supplemental appendix, is to use an estimator based on numerical derivatives. 

