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## 5.2 The Grenander estimator 

Another prominent example of a cube root consistent estimator is the isotonic density estima- tor of Grenander (1956). The asymptotic properties of Grenander estimator have been studied by Prakasa Rao (1969), Groeneboom (1985), and Kim and Pollard (1990), among others. More recently, inconsistency of standard bootstrap-based approximations to the distribution of Grenan- der estimator has been pointed out by Kosorok (2008) and Sen, Banerjee, and Woodroofe (2010), among others. 

To describe the Grenander estimator, suppose { z$_{i}$ : 1 ≤ i ≤ n } is a random sample of a continu- ously distributed nonnegative random variable z, whose Lebesgue density f is continuous and non- increasing on [0 , ∞ ) . For a given evaluation point x 0 ∈ (0 , ∞ ) , the Grenander estimator ˆ f$_{n}$ ( x 0 ) of f ( x 0 ) is the left derivative at x 0 of the least concave majorant (LCM) of ˆ F$_{n}$ ( · ) = n − 1 ∑ n i $_{=1}$1( x$_{i}$ ≤ · ) , the empirical distribution function. 

Although not an M -estimator of the form considered in this note, the Grenander estimator can be handled by adapting some of the ideas used to obtain the results for M -estimators. Assuming f is differentiable at x 0 with strictly negative derivative f $^{′}$( x 0 ) and letting W denote a two-sided Wiener process with W (0) = 0 , it is well known that 

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a result that can be obtained by using empirical process methods similar to those used when deriving (2); for a textbook treatment, see van der Vaart and Wellner (1996, Example 3.2.14). 

A natural bootstrap analogue of ˆ f$_{n}$ ( x 0 ) is given by ˆ f ∗ $_{n}$( x 0 ) , the left derivative at x 0 of the LCM of ˆ F ∗ $_{n}$( · ) = n − 1 ∑ n i $_{=1}$1( x ∗ i ≤ · ) , where { x ∗ i : 1 ≤ i ≤ n } denotes a random sample from the empirical distribution of { x$_{i}$ : 1 ≤ i ≤ n } . Kosorok (2008) and Sen, Banerjee, and Woodroofe (2010) proved, among other things, that the distributional approximation 

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is inconsistent. Once again, this inconsistency can be attributed to the fact that the bootstrap approximation uses an estimator based on a process whose mean function, under the bootstrap 

