Convergence rates of a class of multivariate density estimators based on adaptive partitioning
Abstract/Contents
- Abstract
- Density estimation is a fundamental problem in statistics. It is a building block for other statistical methods, such as classification, nonparametric testing, and data compression. In this dissertation, I focus on a non-parametric approach to multivarite density estimation, and study the asymptotic behavior of this approach under both frequentist and Bayesian settings. The estimated density function is obtained based on consideration of a sequence of approximating spaces to the space of densities. The approximating spaces consist of piecewise constant density functions supported by binary partitions with increasing complexity. The partition is learned by maximizing either the likelihood of the corresponding histogram on that partition, or the marginal posterior distribution of the partition under a suitable prior. We analyze the convergence rate of the maximum likelihood estimator and the posterior concentration rate of the Bayesian estimator, and conclude that for a relatively rich class of density functions the rate does not directly depend on the dimension. Another result has significant practical implications: We show that the Bayesian method can adapt to the unknown complexity of the density function, thus achieving an optimal convergence rate without any prior information. We also apply this method to several specific problems, including spatial adaptation, estimation of functions of bounded variation, and variable selection.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2016 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Liu, Linxi |
|---|---|
| Associated with | Stanford University, Department of Statistics. |
| Primary advisor | Wong, Wing Hung |
| Thesis advisor | Wong, Wing Hung |
| Thesis advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Thesis advisor | Diaconis, Persi |
| Advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Advisor | Diaconis, Persi |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Linxi Liu. |
|---|---|
| Note | Submitted to the Department of Statistics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2016. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2016 by Linxi Liu
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
Also listed in
Loading usage metrics...