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Minimax estimation for Varying Coefficient Model via Laguerre Series

Rida Benhaddou, Khalid Chokri, Jackson Pinschenat

Abstract

We delve into the estimation of the functional coefficients and inference for varying coefficient model. Applying Laguerre series, we develop an estimator for the vector of functional coefficients that attains asymptotically optimal convergence rates in the minimax sense. These rates are derived for functional coefficients that belong to Laguerre-Sobolev space. The method is based on approximating the functional coefficients using truncated Laguerre series and choosing empirical Laguerre coefficients that minimize the least squares criterion. In addition, we establish the asymptotic normality of the estimator for the functional coefficients, construct their confidence intervals, and establish point-wise hypothesis tests about their true values. A simulations study is carried out to examine the finite-sample properties of the proposed methodology. A real data set is considered as well, and results based on the proposed methodology are compared to those based on selected existing approaches.

Minimax estimation for Varying Coefficient Model via Laguerre Series

Abstract

We delve into the estimation of the functional coefficients and inference for varying coefficient model. Applying Laguerre series, we develop an estimator for the vector of functional coefficients that attains asymptotically optimal convergence rates in the minimax sense. These rates are derived for functional coefficients that belong to Laguerre-Sobolev space. The method is based on approximating the functional coefficients using truncated Laguerre series and choosing empirical Laguerre coefficients that minimize the least squares criterion. In addition, we establish the asymptotic normality of the estimator for the functional coefficients, construct their confidence intervals, and establish point-wise hypothesis tests about their true values. A simulations study is carried out to examine the finite-sample properties of the proposed methodology. A real data set is considered as well, and results based on the proposed methodology are compared to those based on selected existing approaches.
Paper Structure (12 sections, 6 theorems, 75 equations, 4 figures, 2 tables)

This paper contains 12 sections, 6 theorems, 75 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Estimation Algorithm
  3. Estimator of a Single Functional Coefficient, $\beta_l(t)$, and Asymptotic Analysis
  4. Estimator of the Vector of Functional Coefficients, ${\boldsymbol{\beta}}(t)$, and Asymptotic Analysis
  5. Minimax Lower Bounds for Estimating Vector of Functional Coefficients, $\widehat{\boldsymbol{\beta}}(t)$
  6. Asymptotic Normality of the Estimator for $\beta_l(t)$
  7. Confidence intervals for $\beta_l(t)$
  8. Hypothesis Test about the true $\beta_l(t)$
  9. Simulations Study
  10. Analysis of Real Data
  11. Residual Analysis
  12. Proofs

Key Result

Lemma 1

Let AssumptionsA.2- A.4 hold. Then, as $n\rightarrow \infty$ where ${\bf I}_l$ is the $\sum^{\bf r}_{d=1}M_d$-dimensional diagonal matrix whose $l^{th}$ block diagonal matrix of size $(M_l)\times(M_l)$ is the identity matrix and the rest of the diagonal elements are all zero.

Figures (4)

  • Figure 1: Actual versus estimated $\beta_l$ based on the three different methods
  • Figure 2: Plots of $\beta_l(t)$, $l=0, 1, 2, 3$, and their confidence bands versus age
  • Figure 3:
  • Figure 4:

Theorems & Definitions (12)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 4
  • Remark 5
  • ...and 2 more