Multiplicative deconvolution in survival analysis under dependency

Abstract

We study the non-parametric estimation of an unknown survival function S with support on ${\mathbb{R}}_{+}$ based on a sample with multiplicative measurement errors. The proposed fully data-driven procedure is based on the estimation of the Mellin transform of the survival function and a regularization of the inverse of the Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is handled by a data-driven choice of the cut-off parameter. In order to discuss the bias term, we consider the Mellin–Sobolev spaces which characterize the regularity of the unknown survival function S through the decay of its Mellin transform. For the analysis of the variance term, we consider the independent and identically distributed case and incorporate dependent observations in form of Bernoulli shift processes and β-mixing sequences. Additionally, we show minimax optimality over Mellin–Sobolev spaces of the spectral cut-off estimator.

Keywords:

• Survival analysis
• non-parametric statistics
• multiplicative measurement errors
• functional dependence measure
• adaptivity

2020 Mathematics Subject Classifications:

• Primary 62G05
• Secondary 62N02
• 62C20

Acknowledgments

We thank the Editor, Associate Editor and the referees for their helpful comments and valuable suggestions. Furthermore, we want to thank Rainer Dahlhaus, Fabienne Comte, Jan Johannes, and Stefan Richter for their helpful advice and their support of our work.

Disclosure statement

No potential conflict of interest was reported by the author(s).