Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local differential privacy in survival analysis using private failure indicators

Egea Maxime, Escobar-Bach Mikael

TL;DR

This work considers the estimation of the cumulative hazard function, and equivalently the distribution function, with censored data under a setup that preserves the privacy of the survival database and proposes a non-parametric kernel estimator that remains consistent under the privatization.

Abstract

We consider the estimation of the cumulative hazard function, and equivalently the distribution function, with censored data under a setup that preserves the privacy of the survival database. This is done through a $α$-locally differentially private mechanism for the failure indicators and by proposing a non-parametric kernel estimator for the cumulative hazard function that remains consistent under the privatization. Under mild conditions, we also prove lowers bounds for the minimax rates of convergence and show that estimator is minimax optimal under a well-chosen bandwidth.

Local differential privacy in survival analysis using private failure indicators

TL;DR

This work considers the estimation of the cumulative hazard function, and equivalently the distribution function, with censored data under a setup that preserves the privacy of the survival database and proposes a non-parametric kernel estimator that remains consistent under the privatization.

Abstract

We consider the estimation of the cumulative hazard function, and equivalently the distribution function, with censored data under a setup that preserves the privacy of the survival database. This is done through a -locally differentially private mechanism for the failure indicators and by proposing a non-parametric kernel estimator for the cumulative hazard function that remains consistent under the privatization. Under mild conditions, we also prove lowers bounds for the minimax rates of convergence and show that estimator is minimax optimal under a well-chosen bandwidth.
Paper Structure (13 sections, 8 theorems, 145 equations, 1 figure)

This paper contains 13 sections, 8 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model
  3. Notations and estimators
  4. Assumptions
  5. Local differential privacy
  6. Risk optimization
  7. Simulations
  8. Proof
  9. Technical lemmas
  10. Proof of the Theorem \ref{['Theo:Upperbound']}
  11. Risk analysis for $\Lambda_n-\Lambda$
  12. Risk analysis for $\widehat{\Lambda}_n-\Lambda_n$
  13. Proof of Theorem \ref{['theorem:minimax']}

Key Result

Lemma 3.1

Let $\widetilde{Q}$ defines the privacy mechanism that takes $(Y,\delta,X)$ and returns $(Y,Z,X)$ as described above. Then $\widetilde{Q}\in\widetilde{\mathcal{Q}}_\alpha$.

Figures (1)

  • Figure 1: Graph of Mean Squared Error (MSE) as a function of time with $50 \%$ of censoring for the first line and $25\%$ for the second.

Theorems & Definitions (13)

  • Lemma 3.1
  • Definition 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof
  • Lemma 6.3
  • ...and 3 more