# A generic approach to nonparametric function estimation with mixed data

## TL;DR

This paper demonstrates that adding noise to discrete variables allows existing nonparametric estimators designed for continuous data to be effectively extended to mixed data, simplifying implementation and preserving asymptotic properties.

## Contribution

It provides a theoretical justification for using noise addition to handle discrete variables in nonparametric estimation, enabling straightforward extension of continuous estimators to mixed data.

## Key findings

- Adding noise from a specific class justifies continuous convolution estimators for mixed data.
- Asymptotic properties of estimators transfer from continuous to mixed data settings.
- The approach simplifies implementation of nonparametric methods with mixed data.

## Abstract

In practice, data often contain discrete variables. But most of the popular nonparametric estimation methods have been developed in a purely continuous framework. A common trick among practitioners is to make discrete variables continuous by adding a small amount of noise. We show that this approach is justified if the noise distribution belongs to a certain class. In this case, any estimator developed in a purely continuous framework extends naturally to the mixed data setting. Estimators defined that way will be called continuous convolution estimators. They are extremely easy to implement and their asymptotic properties transfer directly from the continuous to the mixed data setting.