Abstract
We study a type of double truncation where units are observed if and only if their death event occurs within a specific timespan. The resulting missing data mechanism is nonignorable and thus has to be reconsidered. Based on the density function of observed lifetimes and the random sample size, we derive a likelihood model that enables simultaneous estimation of the lifetime distribution and the parameters governing the birth process. In particular, knowledge of the population size is not required. We show that the model is identifiable under certain conditions by using results on exponential families. Bayesian estimators and corresponding standard errors for all involved parameters become available by using MCMC simulation. We describe how the simulation can be performed efficiently while maintaining sufficiently good mixing behaviour of the resulting chains. Both finite-sample and asymptotic properties of the investigated estimators are examined through a simulation study. The proposed method is applied to estimate the lifetime distribution of German companies.
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References
Benrabah O, Ould Saïd E, Tatachak A (2015) A kernel mode estimate under random left truncation and time series model: asymptotic normality. Stat Pap 56:887–910
Efron B, Petrosian V (1999) Nonparametric methods for doubly truncated data. J Am Stat Assoc 94:824–834
Emura T, Konno Y (2012) Multivariate normal distribution approaches for dependently truncated data. Stat Pap 53:133–149
Emura T, Hu Y-H, Konno Y (2015) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation. https://doi.org/10.1007/s00362-015-0730-y
Gelman A, Meng X-L, Stern HS (1996) Posterior predictive assessment of model fitness via realized discrepancies. Stat Sin 6:733–807
Gourieroux C, Monfort A (1995) Statistics and econometric models. Cambridge University Press, Cambridge
Hu Y-H, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation. Comput Stat 30:1199–1229. https://doi.org/10.1007/s00180-015-0564-z
Kalbfleisch JD, Lawless JF (1989) Inference based on retrospective ascertainment: an analysis of the data on transfusion-related aids. J Am Stat Assoc 84:360–372
Lawless J, Kalbfleisch JD (1992) Some issues in the collection and analysis of field reliability data. Kluwer Academic Publishers, Dordrecht, pp 141–152
Lee J, Berger JO (2001) Semiparametric Bayesian analysis of selection models. J Am Stat Assoc 96:1397–1409
Liang H-Y, Baek J-I (2016) Asymptotic normality of conditional density estimation with left-truncated and dependent data. Stat Pap 57:1–20
Shen P-S (2010) Nonparametric analysis of doubly truncated data. Ann Inst Stat Math 62:835–853
Shen P-S (2017) Pseudo maximum likelihood estimation for the Cox model with doubly truncated data. Stat Pap. https://doi.org/10.1007/s00362-016-0870-8
Tierney L (1994) Markov chains for exploring posterior distributions. Ann Stat 22:1701–1728
Wang M-C (1989) A semiparametric model for randomly truncated data. J Am Stat Assoc 84:742–748
Xu H-X, Chen Z-L, Wang J-F, Fan G-L (2017) Quantile regression and variable selection for partially linear model with randomly truncated data. Stat Pap. https://doi.org/10.1007/s00362-016-0867-3
Acknowledgements
The author is grateful to the two anonymous reviewers for their many constructive comments. Further, the author would like to thank Mr. W. Lohse for suggesting the website to collect data on insolvencies in Germany. The financial support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
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Dörre, A. Bayesian estimation of a lifetime distribution under double truncation caused by time-restricted data collection. Stat Papers 61, 945–965 (2020). https://doi.org/10.1007/s00362-017-0968-7
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DOI: https://doi.org/10.1007/s00362-017-0968-7