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Block Empirical Likelihood Inference for Longitudinal Generalized Partially Linear Single-Index Models

Tianni Zhang, Yuyao Wang, Yu Lu, Mengfei Ran

Abstract

Generalized partially linear single-index models (GPLSIMs) provide a flexible and interpretable semiparametric framework for longitudinal outcomes by combining a low-dimensional parametric component with a nonparametric index component. For repeated measurements, valid inference is challenging because within-subject correlation induces nuisance parameters and variance estimation can be unstable in semiparametric settings. We propose a profile estimating-equation approach based on spline approximation of the unknown link function and construct a subject-level block empirical likelihood (BEL) for joint inference on the parametric coefficients and the single-index direction. The resulting BEL ratio statistic enjoys a Wilks-type chi-square limit, yielding likelihood-free confidence regions without explicit sandwich variance estimation. We also discuss practical implementation, including constrained optimization for the index parameter, working-correlation choices, and bootstrap-based confidence bands for the nonparametric component. Simulation studies and an application to the epilepsy longitudinal study illustrate the finite-sample performance.

Block Empirical Likelihood Inference for Longitudinal Generalized Partially Linear Single-Index Models

Abstract

Generalized partially linear single-index models (GPLSIMs) provide a flexible and interpretable semiparametric framework for longitudinal outcomes by combining a low-dimensional parametric component with a nonparametric index component. For repeated measurements, valid inference is challenging because within-subject correlation induces nuisance parameters and variance estimation can be unstable in semiparametric settings. We propose a profile estimating-equation approach based on spline approximation of the unknown link function and construct a subject-level block empirical likelihood (BEL) for joint inference on the parametric coefficients and the single-index direction. The resulting BEL ratio statistic enjoys a Wilks-type chi-square limit, yielding likelihood-free confidence regions without explicit sandwich variance estimation. We also discuss practical implementation, including constrained optimization for the index parameter, working-correlation choices, and bootstrap-based confidence bands for the nonparametric component. Simulation studies and an application to the epilepsy longitudinal study illustrate the finite-sample performance.
Paper Structure (32 sections, 8 theorems, 93 equations, 3 figures, 4 tables, 1 algorithm)

This paper contains 32 sections, 8 theorems, 93 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Longitudinal GPLSIM
  4. Sieve Approximation
  5. Profile Estimating Equations
  6. Block Empirical Likelihood for $\pmb{\theta}$
  7. Bootstrap Inference for $\eta_0(\cdot)$
  8. Algorithm
  9. Inputs
  10. Profile Algorithm for $\widehat{\pmb{\theta}}$ and $\widehat{\eta}$
  11. BEL Statistic
  12. Asymptotic Theory
  13. Assumptions
  14. Main Results
  15. Simulation Studies
  16. ...and 17 more sections

Key Result

Theorem 1

Under Assumptions ass:sampling--ass:H, there exists a sequence of roots $\widehat{\pmb{\theta}}$ of eq:Utheta such that Moreover, with $\widehat{\eta}$ defined by the spline sieve at $\widehat{\pmb{\theta}}$,

Figures (3)

  • Figure 1: Representative fit of $\eta_0(u)$ and $\widehat{\eta}(u)$ with $95\%$ bootstrap bands under Bernoulli case. ($a$) Top Left: $n=200$, $\rho=0.0$; ($b$) Top Right: $n=200$, $\rho=0.3$; ($c$) Bottom: $n=200$, $\rho=0.6$.
  • Figure 2: Representative fit of $\eta_0(u)$ and $\widehat{\eta}(u)$ with $95\%$ bootstrap bands under Gaussian case. ($a$) Left: $n=100$, $\rho=0.0$; ($b$) Right: $n=200$, $\rho=0.0$.
  • Figure 3: Estimated time effect $\widehat{\eta}(t)$ on the epilepsy dataset. All $\eta$ estimates are post-processed to satisfy the same identifiability constraint.

Theorems & Definitions (20)

  • Theorem 1: Consistency and rates
  • Remark 1
  • Theorem 2: Asymptotic normality
  • Remark 2
  • Lemma 1: Quadratic expansion of BEL
  • Remark 3
  • Theorem 3: Wilks phenomenon for BEL
  • Remark 4
  • Lemma 2: Spline approximation
  • proof
  • ...and 10 more