## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 11, Number 2 (2017), 2800-2831.

### Converting high-dimensional regression to high-dimensional conditional density estimation

#### Abstract

There is a growing demand for nonparametric conditional density estimators (CDEs) in fields such as astronomy and economics. In astronomy, for example, one can dramatically improve estimates of the parameters that dictate the evolution of the Universe by working with full conditional densities instead of regression (i.e., conditional mean) estimates. More generally, standard regression falls short in any prediction problem where the distribution of the response is more complex with multi-modality, asymmetry or heteroscedastic noise. Nevertheless, much of the work on high-dimensional inference concerns regression and classification only, whereas research on density estimation has lagged behind. Here we propose *FlexCode*, a fully nonparametric approach to conditional density estimation that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities and not just expectations in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with different structures in the data (e.g., a large number of irrelevant components and nonlinear manifold structure) as well as different data types (e.g., functional data, mixed data types and sample sets). We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on simulated as well as real-world data, such as photometric galaxy data, Twitter data, and line-of-sight velocities in a galaxy cluster.

#### Article information

**Source**

Electron. J. Statist., Volume 11, Number 2 (2017), 2800-2831.

**Dates**

Received: July 2016

First available in Project Euclid: 4 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1499133755

**Digital Object Identifier**

doi:10.1214/17-EJS1302

**Mathematical Reviews number (MathSciNet)**

MR3679910

**Zentralblatt MATH identifier**

1366.62078

**Subjects**

Primary: 62G07: Density estimation 62G15: Tolerance and confidence regions

Secondary: 62G08: Nonparametric regression

**Keywords**

Nonparametric inference conditional density high-dimensional data prediction intervals functional conditional density estimation

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Izbicki, Rafael; B. Lee, Ann. Converting high-dimensional regression to high-dimensional conditional density estimation. Electron. J. Statist. 11 (2017), no. 2, 2800--2831. doi:10.1214/17-EJS1302. https://projecteuclid.org/euclid.ejs/1499133755