Asymptotic normality of the Lk-error of the Grenander estimator

By V. Kulikov, Hendrik P. Lopuhaä. 2006

66 citations recorded by the source corpus.

We investigate the limit behavior of the $L_k$-distance between a decreasing density $f$ and its nonparametric maximum likelihood estimator $\hat{f}_n$ for $k\geq1$. Due to the inconsistency of $\hat{f}_n$ at zero, the case $k=2.5$ turns out to be a kind of transition point. We extend asymptotic normality of the $L_1$-distance to the $L_k$-distance for $1\leq k 1$, we show that the $L_k$-distance between $f$ and $\hat{f}_n$ is asymptotically equivalent to the $L_k$-distance between $U_n$ and $g$.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1e0d85bcc3ac544169695936997d5eed479c41a7

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Citations in the corpus

Divide and conquer in nonstandard problems and the super-efficiency phenomenon (2016, unknown): “rg condition is satisﬁed. A.16. Proof of Theorem 7.1 The (generalized) inverse Uˆ nof fˆntakes the form (8.6) with Λnreplaced by the empirical distribution function Fn. It follows from Theorem 2.1 in [15], which compares to our Lemma 8.4 in a diﬀerent setting, that there exists C >0 such that for”
The behavior of the NPMLE of a decreasing density near the boundaries of the support (2006, unknown): “the boundaries of [0,1] allow an adequate treatment of the L k-distance between fˆ n and f. It turns out that for k >2.5, the inconsistency of fˆ n starts to aﬀect the behavior of kfˆ n −fk k (see [10]). In Section 2 we give a brief outline of our approach for studying diﬀerences such as (1.2) an”
Testing equality of functions under monotonicity constraints (2012, methodology): “In Section 2, we describe the general setup, state our main results, and discuss the differences between the approach in this paper and the ones used in Groeneboom, Hooghiemstra, and Lopuhaä (1999), Kulikov and Lopuhaä (2005) and Durot (2007) to prove similar results.”