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Smoothing spline density estimation from doubly truncated data

David Bamio, Jacobo de Uña-Álvarez

TL;DR

This work tackles nonparametric density estimation when data are doubly truncated, introducing a bias-corrected smoothing spline density estimator derived from a penalized likelihood with a logistic transform. By extending the smoothing spline framework to a two-dimensional truncation mechanism, the authors establish existence and convergence properties and contrast with kernel density estimation, which relies on the NPMLE and can suffer from nonexistence or boundary issues. Through extensive simulations, the corrected spline generally achieves lower integrated squared error than KDE, while providing robust performance and avoiding NPMLE degeneracy. Real-data applications to Quasars and Parkinson's disease demonstrate the method's practical bias correction and uncertainty quantification via bootstrap, highlighting advantages in boundary behavior and interpretability in truncated settings.

Abstract

In Astronomy, Survival Analysis and Epidemiology, among many other fields, doubly truncated data often appear. Double truncation generally induces a sampling bias, so ordinary estimators may be inconsistent. In this paper, smoothing spline density estimation from doubly truncated data is investigated. For this purpose, an appropriate correction of the penalized likelihood that accounts for the sampling bias is considered. The theoretical properties of the estimator are discussed, and its practical performance is evaluated through simulations. Two real datasets are analyzed using the proposed method for illustrative purposes. Comparison to kernel density smoothing is included.

Smoothing spline density estimation from doubly truncated data

TL;DR

This work tackles nonparametric density estimation when data are doubly truncated, introducing a bias-corrected smoothing spline density estimator derived from a penalized likelihood with a logistic transform. By extending the smoothing spline framework to a two-dimensional truncation mechanism, the authors establish existence and convergence properties and contrast with kernel density estimation, which relies on the NPMLE and can suffer from nonexistence or boundary issues. Through extensive simulations, the corrected spline generally achieves lower integrated squared error than KDE, while providing robust performance and avoiding NPMLE degeneracy. Real-data applications to Quasars and Parkinson's disease demonstrate the method's practical bias correction and uncertainty quantification via bootstrap, highlighting advantages in boundary behavior and interpretability in truncated settings.

Abstract

In Astronomy, Survival Analysis and Epidemiology, among many other fields, doubly truncated data often appear. Double truncation generally induces a sampling bias, so ordinary estimators may be inconsistent. In this paper, smoothing spline density estimation from doubly truncated data is investigated. For this purpose, an appropriate correction of the penalized likelihood that accounts for the sampling bias is considered. The theoretical properties of the estimator are discussed, and its practical performance is evaluated through simulations. Two real datasets are analyzed using the proposed method for illustrative purposes. Comparison to kernel density smoothing is included.
Paper Structure (11 sections, 3 theorems, 7 equations, 4 figures, 3 tables)

This paper contains 11 sections, 3 theorems, 7 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methods and theoretical properties
  3. Smoothing spline density estimator
  4. Kernel density estimator
  5. Existence of the estimators
  6. Convergence of the estimators
  7. Simulation study
  8. Real data illustrations
  9. Quasars data
  10. Parkinson's disease data
  11. Main conclusions and discussion

Key Result

Theorem 1

Suppose $L(\eta)$ is a continuous and strictly convex functional in a Hilbert space $\mathcal{H}$ and $J(\eta)$ is a square seminorm in $\mathcal{H}$ with a finite dimensional null space $\mathcal{N}_J\subset\mathcal{H}$. If $L(\eta)$ has a minimizer in $\mathcal{N}_J$, then $L(\eta)+J(\eta)$ has a

Figures (4)

  • Figure 1: Density estimation results after 250 Montecarlo simulations of sample size $n=200$ for the the interval sampling settings (constant $\tau$). Represented in blue is the mean of the 250 density estimates obtained for each point in the support grid; in red, the 2.5% top and bottom percentiles. The theoretical density is represented in a black dashed line.
  • Figure 2: Density estimation results after 250 Montecarlo simulations of sample size $n=200$ for settings with random observation interval length $\tau$. Represented in blue is the mean of the 250 density estimates obtained for each point in the support grid; in red, the 2.5% top and bottom percentiles. The theoretical density for each setting is represented in a black dashed line.
  • Figure 3: Density estimation results for the Quasars data obtained by three methods: ordinary smoothing spline (left), corrected smoothing spline (middle) and kernel density estimation (right). Represented in blue is the density estimate of the real data; in red, the 2.5% top and bottom percentiles from 250 bootstrap estimates.
  • Figure 4: Density estimation results of the PDlate data obtained by the three methods: ordinary smoothing spline (left), corrected smoothing spline (middle) and kernel density estimation (right). Represented in blue is the density estimate of the real data; in red, the 2.5% top and bottom percentiles from 250 bootstrap estimates.

Theorems & Definitions (3)

  • Theorem 1: guqiu
  • Theorem 2: xiao
  • Theorem 3: xiao