# Non-Smooth Backfitting for Excess Risk Additive Regression Model with Two Survival Time-Scales

## TL;DR

This paper introduces a novel backfitting algorithm for a complex non-parametric survival model with two time-scales, avoiding smoothing and providing large sample properties and confidence bands.

## Contribution

It proposes a smoothing-free backfitting method for a two time-scale survival model with additive covariate effects, extending existing estimation techniques.

## Key findings

- Successfully estimates cumulative intensities without smoothing.
- Provides large sample properties and confidence bands.
- Applied to myocardial infarction data to distinguish effects.

## Abstract

We present a new backfitting algorithm estimating the complex structured non-parametric survival model of Scheike (2001) without having to use smoothing. The considered model is a non-parametric survival model with two time-scales that are equivalent up to a constant that varies over the subjects. Covariate effects are modelled linearly on each time scale by additive Aalen models. Estimators of the cumulative intensities on the two time-scales are suggested by solving local estimating equations jointly on the two time-scales. We are able to estimate the cumulative intensities solving backfitting estimating equations without using smoothing methods and we provide large sample properties and simultaneous confidence bands. The model is applied to data on myocardial infarction providing a separation of the two effects stemming from time since diagnosis and age.