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Right-censored models by the expectile method

Gabriela Ciuperca

Abstract

Based on the expectile loss function and the adaptive LASSO penalty, the paper proposes and studies the estimation methods for the accelerated failure time (AFT) model. In this approach, we need to estimate the survival function of the censoring variable by the Kaplan-Meier estimator. The AFT model parameters are first estimated by the expectile method and afterwards, when the number of explanatory variables can be large, by the adaptive LASSO expectile method which directly carries out the automatic selection of variables. We also obtain the convergence rate and asymptotic normality for the two estimators, while showing the sparsity property for the censored adaptive LASSO expectile estimator. A numerical study using Monte Carlo simulations confirms the theoretical results and demonstrates the competitive performance of the two proposed estimators. The usefulness of these estimators is illustrated by applying them to three survival data sets.

Right-censored models by the expectile method

Abstract

Based on the expectile loss function and the adaptive LASSO penalty, the paper proposes and studies the estimation methods for the accelerated failure time (AFT) model. In this approach, we need to estimate the survival function of the censoring variable by the Kaplan-Meier estimator. The AFT model parameters are first estimated by the expectile method and afterwards, when the number of explanatory variables can be large, by the adaptive LASSO expectile method which directly carries out the automatic selection of variables. We also obtain the convergence rate and asymptotic normality for the two estimators, while showing the sparsity property for the censored adaptive LASSO expectile estimator. A numerical study using Monte Carlo simulations confirms the theoretical results and demonstrates the competitive performance of the two proposed estimators. The usefulness of these estimators is illustrated by applying them to three survival data sets.
Paper Structure (15 sections, 3 theorems, 58 equations, 8 figures, 2 tables)

This paper contains 15 sections, 3 theorems, 58 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model and assumptions
  3. Estimators and asymptotic behavior
  4. Simulation studies
  5. Numerical study on the choice of the tuning parameter $\lambda_n$
  6. Numerical study with respect to model error distributions
  7. Comparative numerical study by varying $p$, censoring rate, $\beta^0_0$, $n$
  8. Numerical study of $\widetilde{\textrm{$\mathbf{\beta}$}}_n$
  9. Conclusion of simulations
  10. Applications on real data
  11. Primary Biliary Cholangitis Data
  12. Myeloma
  13. Myeloma data of the R package "emplik"
  14. Myeloma data of the R package "survminer"
  15. Proofs

Key Result

Theorem 1

Under assumptions (A1)-(A7) we have: (i) $\widetilde{\textrm{$\mathbf{\beta}$}}_n -\textrm{$\mathbf{\beta}^0$}=O_\mathbb{P}(n^{-1/2})$. (ii) $n^{1/2}(\widetilde{\textrm{$\mathbf{\beta}$}}_n - \textrm{$\mathbf{\beta}^0$}) \overset{\cal L} {\underset{n \rightarrow \infty}{\longrightarrow}} {\cal N}_

Figures (8)

  • Figure 1: Percentage evolution of the true and false zeros with respect to $n$ for two sequences $\lambda_n$ ($\circ$ for $\lambda_n=n^{1/2}$, $\square$ for $\lambda_n =n^{1/2- 0.01}$) by censored adaptive LASSO expectile method, for model without intercept ($\beta^0_0=0$), when $p=50$, censoring rate $25\%$, $\varepsilon \sim{\cal G}(0,1)$.
  • Figure 2: Percentage evolution of the true and false zeros with respect to $n$ for two sequences $\lambda_n$ by three censored adaptive LASSO estimation methods, when $\varepsilon \sim{\cal G}(0,1)$, $p=50$, model with intercept ($\beta_0^0=2$) and censoring rate $25\%$.
  • Figure 3: Percentage evolution of the false zeros by three censored adaptive LASSO estimation methods, for model without intercept ($\beta^0_0=0$), when $p=50$ and censoring rate is $25\%$.
  • Figure 4: Percentage evolution of the false zeros by three censored adaptive LASSO estimation methods, for model without intercept ($\beta^0_0=0$), when $\varepsilon \sim{\cal G}(0,1)$ and censoring rate is $10\%$.
  • Figure 5: Percentage evolution of the true and false zeros by three censored adaptive LASSO estimation methods, for model without intercept ($\beta^0_0=0$), supposition with intercept, when $\varepsilon \sim{\cal G}(0,1)$ and $p=10$.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 2