# Smoothing Spline Semiparametric Density Models

## TL;DR

This paper introduces a unified framework for semiparametric density estimation using reproducing kernel Hilbert spaces, providing new models, estimation methods, and theoretical guarantees validated through simulations and real data.

## Contribution

It develops a systematic, kernel-based approach to semiparametric density modeling, estimation, and theory, unifying existing methods and extending their applicability.

## Key findings

- Proposed general semiparametric density models encompassing existing ones.
- Established joint consistency and convergence rates for estimators.
- Validated methods through simulations and real data application.

## Abstract

Density estimation plays a fundamental role in many areas of statistics and machine learning. Parametric, nonparametric and semiparametric density estimation methods have been proposed in the literature. Semiparametric density models are flexible in incorporating domain knowledge and uncertainty regarding the shape of the density function. Existing literature on semiparametric density models is scattered and lacks a systematic framework. In this paper, we consider a unified framework based on the reproducing kernel Hilbert space for modeling, estimation, computation and theory. We propose general semiparametric density models for both a single sample and multiple samples which include many existing semiparametric density models as special cases. We develop penalized likelihood based estimation methods and computational methods under different situations. We establish joint consistency and derive convergence rates of the proposed estimators for both the finite dimensional Euclidean parameters and an infinite-dimensional functional parameter. We validate our estimation methods empirically through simulations and an application.