The distance between a naive cumulative estimator and its least concave majorant
We consider the process $\widehat\Lambda_n-\Lambda_n$, where $\Lambda_n$ is a cadlag step estimator for the primitive $\Lambda$ of a nonincreasing function $\lambda$ on $[0,1]$, and $\widehat\Lambda_n$ is the least concave majorant of $\Lambda_n$. We extend the results in Kulikov and Lopuha\"a...
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| Format | Journal Article |
| Language | English |
| Published |
16.06.2017
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| Abstract | We consider the process $\widehat\Lambda_n-\Lambda_n$, where $\Lambda_n$ is a
cadlag step estimator for the primitive $\Lambda$ of a nonincreasing function
$\lambda$ on $[0,1]$, and $\widehat\Lambda_n$ is the least concave majorant of
$\Lambda_n$. We extend the results in Kulikov and Lopuha\"a (2006, 2008) to the
general setting considered in Durot (2007). Under this setting we prove that a
suitably scaled version of $\widehat\Lambda_n-\Lambda_n$ converges in
distribution to the corresponding process for two-sided Brownian motion with
parabolic drift and we establish a central limit theorem for the $L_p$-distance
between $\widehat\Lambda_n$ and $\Lambda_n$. |
|---|---|
| AbstractList | We consider the process $\widehat\Lambda_n-\Lambda_n$, where $\Lambda_n$ is a
cadlag step estimator for the primitive $\Lambda$ of a nonincreasing function
$\lambda$ on $[0,1]$, and $\widehat\Lambda_n$ is the least concave majorant of
$\Lambda_n$. We extend the results in Kulikov and Lopuha\"a (2006, 2008) to the
general setting considered in Durot (2007). Under this setting we prove that a
suitably scaled version of $\widehat\Lambda_n-\Lambda_n$ converges in
distribution to the corresponding process for two-sided Brownian motion with
parabolic drift and we establish a central limit theorem for the $L_p$-distance
between $\widehat\Lambda_n$ and $\Lambda_n$. |
| Author | Lopuhaä, Hendrik P Musta, Eni |
| Author_xml | – sequence: 1 givenname: Hendrik P surname: Lopuhaä fullname: Lopuhaä, Hendrik P – sequence: 2 givenname: Eni surname: Musta fullname: Musta, Eni |
| BackLink | https://doi.org/10.48550/arXiv.1706.05173$$DView paper in arXiv |
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| Snippet | We consider the process $\widehat\Lambda_n-\Lambda_n$, where $\Lambda_n$ is a
cadlag step estimator for the primitive $\Lambda$ of a nonincreasing function... |
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| SubjectTerms | Mathematics - Statistics Theory Statistics - Theory |
| Title | The distance between a naive cumulative estimator and its least concave majorant |
| URI | https://arxiv.org/abs/1706.05173 |
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