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Hazard Rate for Associated Data in Deconvolution Problems: Asymptotic Normality

Benjrada Mohammed Essalih

TL;DR

The paper addresses hazard-rate estimation under a contaminated-associated data model, using a deconvolution kernel approach to estimate the latent density and distribution and form the hazard estimator $\lambda_n(x)=\dfrac{f_n(x)}{1-F_n(x)}$. It establishes quadratic-mean convergence and asymptotic normality under ordinary-smooth errors and positive association, providing explicit expressions for the asymptotic variance $\sigma^2(x)$ and a bias expansion, along with a plug-in variance estimator for inference. The results enable principled statistical inference for hazard rates in reliability and survival analyses when data are weakened by measurement errors and dependence, and are complemented by simulation studies demonstrating finite-sample performance. The paper also outlines open problems for supersmooth and non-standard errors and for unknown error distributions, guiding future extensions of deconvolution-based hazard-rate estimation.

Abstract

In reliability theory and survival analysis, observed data are often weakly dependent and subject to additive measurement errors. Such contamination arises when the underlying data are neither independent nor strongly mixed but instead exhibit association. This paper focuses on estimating the hazard rate by deconvolving the density function and constructing an estimator of the distribution function. We assume that the data originate from a strictly stationary sequence satisfying association conditions. Under appropriate smoothness assumptions on the error distribution, we establish the quadratic-mean convergence and asymptotic normality of the proposed estimators. The finite-sample performance of both the hazard rate and distribution function estimators is evaluated through a simulation study. We conclude with a discussion of open problems and potential future research directions.

Hazard Rate for Associated Data in Deconvolution Problems: Asymptotic Normality

TL;DR

The paper addresses hazard-rate estimation under a contaminated-associated data model, using a deconvolution kernel approach to estimate the latent density and distribution and form the hazard estimator . It establishes quadratic-mean convergence and asymptotic normality under ordinary-smooth errors and positive association, providing explicit expressions for the asymptotic variance and a bias expansion, along with a plug-in variance estimator for inference. The results enable principled statistical inference for hazard rates in reliability and survival analyses when data are weakened by measurement errors and dependence, and are complemented by simulation studies demonstrating finite-sample performance. The paper also outlines open problems for supersmooth and non-standard errors and for unknown error distributions, guiding future extensions of deconvolution-based hazard-rate estimation.

Abstract

In reliability theory and survival analysis, observed data are often weakly dependent and subject to additive measurement errors. Such contamination arises when the underlying data are neither independent nor strongly mixed but instead exhibit association. This paper focuses on estimating the hazard rate by deconvolving the density function and constructing an estimator of the distribution function. We assume that the data originate from a strictly stationary sequence satisfying association conditions. Under appropriate smoothness assumptions on the error distribution, we establish the quadratic-mean convergence and asymptotic normality of the proposed estimators. The finite-sample performance of both the hazard rate and distribution function estimators is evaluated through a simulation study. We conclude with a discussion of open problems and potential future research directions.

Paper Structure

This paper contains 16 sections, 13 theorems, 170 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Estimates and Assumptions
  3. Estimation
  4. Assumptions and Notation
  5. Comments on hypotheses
  6. Main Results
  7. The Quadratic Mean Convergence
  8. The Asymptotic Normality
  9. Future Directions
  10. The Case of Supersmooth Error
  11. The Case of Non-Standard Error
  12. Simulation study
  13. Designing the Simulated Data
  14. Estimation with Large Sample Sizes
  15. Asymptotic Normality and Confidence Interval
  16. ...and 1 more sections

Key Result

Proposition 5

1) For all $x\in \mathbb{R}$, we have 2) Under conditions (H2)-1, (H2)-2, and supposing that $f$ and $F$ are in $C_{2}\left( \mathbb{R} \right)$, we find

Figures (3)

  • Figure 1: The true hazard rate function (corresponding to NSR = 0) is compared against its estimates obtained under different NSR.
  • Figure 2: Hazard rate estimator computed from large sample size observations contaminated by additive noise with $\text{NSR} = 0.1$, $0.2$, and $0.5$. The solid line represents the true hazard rate function, the dashed line corresponds to the hazard rate estimator for $n = 10000$, and the dotted line represents the hazard rate estimator for $n = 15000$.
  • Figure 3: Normal-probability plots of $\lambda _{n}(0.5)$ based on Laplacian errors with $\text{NSR} = 0.1$.

Theorems & Definitions (28)

  • Definition 1
  • Remark 2
  • Example 3
  • Example 4
  • Proposition 5
  • Theorem 6
  • Corollary 7
  • Remark 8
  • Proposition 9
  • Proposition 10
  • ...and 18 more