Density estimation using cellular binary trees and an application to monotone densities
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910916635787264 |
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| author | Devroye, Luc Hamdan, Jad |
| author_facet | Devroye, Luc Hamdan, Jad |
| contents | Consider a density $f$ on $[0,1]$ that must be estimated from an i.i.d. sample $X_1,...,X_n$ drawn from $f$. In this note, we study binary-tree-based histogram estimates that use recursive splitting of intervals. If the decision to split an interval is a (possibly randomized) function of the number of data points in the interval only, then we speak of an estimate of complexity one. We exhibit a universally consistent estimate of complexity one. If the decision to split is a function of the cardinalities of k equal-length sub-intervals, then we speak of an estimate of complexity k. We propose an estimate of complexity two that can estimate any bounded monotone density on $[0,1]$ with optimal expected total variation error $O(n^{-1/3})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_08006 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Density estimation using cellular binary trees and an application to monotone densities Devroye, Luc Hamdan, Jad Statistics Theory Probability Consider a density $f$ on $[0,1]$ that must be estimated from an i.i.d. sample $X_1,...,X_n$ drawn from $f$. In this note, we study binary-tree-based histogram estimates that use recursive splitting of intervals. If the decision to split an interval is a (possibly randomized) function of the number of data points in the interval only, then we speak of an estimate of complexity one. We exhibit a universally consistent estimate of complexity one. If the decision to split is a function of the cardinalities of k equal-length sub-intervals, then we speak of an estimate of complexity k. We propose an estimate of complexity two that can estimate any bounded monotone density on $[0,1]$ with optimal expected total variation error $O(n^{-1/3})$. |
| title | Density estimation using cellular binary trees and an application to monotone densities |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2203.08006 |