# Large Deviations of the Estimated Cumulative Hazard Rate

## TL;DR

This paper investigates the large deviation properties of the Nelson-Aalen estimator for the cumulative hazard rate, revealing asymmetries that cause overestimation especially in small samples and rare event scenarios.

## Contribution

It derives large deviation principles for the Nelson-Aalen estimator, providing new insights into its small sample behavior and bias tendencies.

## Key findings

- Rate functions are asymmetric, favoring overestimation.
- Overestimation bias is strongest with small samples.
- Implications for risk assessment of rare events.

## Abstract

Survivorship analysis allows to statistically analyze situations that can be modeled as waiting times to an event. These waiting times are characterized by the cumulative hazard rate, which can be estimated by the Nelson-Aalen estimator or diverse confidence estimators based on asymptotic statistics. To better understand the small sample properties of these estimators, the speed of convergence of the estimate to the exact value is examined. This is done by deriving large deviation principles and their rate functions for the estimators and examining their properties. It is shown that these rate functions are asymmetric, leading to a tendency of the estimated cumulative hazard rate to overestimate the true cumulative hazard rate. This tendency is strongest in the cases of (1) small sample sizes and (2) low tail probabilities. Taking this tendency into account can improve risk assessments of rare events and of cases where only little data is available.