Retrospective sampling of survival data based on a Poisson birth process: conditional maximum likelihood

Abstract

Truncated survival data are observed retrospectively, if the death event falls into the study period. We find that estimating the survival distribution requires a model for the birth event. With births generated by a parametric Poisson process, we analyse the likelihood of parametric survival, stochastically independent of birth. Conditioning on the number of observed units reduces the number of parameters by one, and to two in our application. The compact support of an observation simplifies the proof of consistency. Furthermore, that we only need to show identification separately for the birth and the death distribution is helpful for demonstrating asymptotic normality. For identification of the survival parameters, a stronger criterion is needed than for a simple random sample, but is fulfilled in our application. In a simulation study, we find that the variance inflation by truncation can be substantial, and apparently is indeed so in our application. From 55,000 German companies that went insolvent between 2014 and 2016, we infer an average time to insolvency of six years and a negative linear trend of corporate foundations after the German reunification in 1990. Companies in Northern Germany survive longer than in the south.

Keywords:

• Double truncation
• conditional maximum likelihood
• nonhomogeneous births
• asymptotics

Acknowledgments

For support in the process of the data acquisition, we thank W. Lohse and D. Ollrogge, and for general advice, W. Krämer. The linguistic and idiomatic advice of Brian Bloch is also gratefully acknowledged. We are also indebted to A. Meister and two anonymous reviewers for valuable advice.

Disclosure statement

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

Additional information

Funding

The financial support from the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged (Grant 386913674 ‘Multi-state, multi-time, multi-level analysis of health-related demographic events: Statistical aspects and applications’).