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Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation

Author

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  • Takeshi Emura

    (National Central University)

  • Ya-Hsuan Hu

    (National Central University)

  • Yoshihiko Konno

    (Japan Women’s University)

Abstract

Biased sampling affects the inference for population parameters of interest if the sampling mechanism is not appropriately handled. This paper considers doubly-truncated data arising in lifetime data analysis in which samples are subject to both left- and right-truncations. To correct for the sampling bias with doubly-truncated data, maximum likelihood estimator (MLE) has been proposed under a parametric family called the special exponential family (Efron and Petrosian, in J Am Stat Assoc 94:824–834, 1999). However, there is still a lack of justifying the fundamental properties for the MLE, including consistency and asymptotic normality. In this paper, we point out that the classical asymptotic theory for the independent and identically distributed data is not suitable for studying the MLE under double-truncation due to the non-identical truncation intervals. Alternatively, we formalize the asymptotic results under independent but not identically distributed data that suitably takes into account for the between-sample heterogeneity of truncation intervals. We establish the consistency and asymptotic normality of the MLE under a reasonably simple set of regularity conditions. Then, we give asymptotically valid techniques to estimate standard errors and to construct confidence intervals. Simulations are conducted to verify the suggested techniques, and childhood cancer data are used for illustration.

Suggested Citation

  • Takeshi Emura & Ya-Hsuan Hu & Yoshihiko Konno, 2017. "Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation," Statistical Papers, Springer, vol. 58(3), pages 877-909, September.
    • Handle: RePEc:spr:stpapr:v:58:y:2017:i:3:d:10.1007_s00362-015-0730-y
    DOI: 10.1007/s00362-015-0730-y
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    References listed on IDEAS

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    1. Moreira, Carla & Van Keilegom, Ingrid, 2013. "Bandwidth selection for kernel density estimation with doubly truncated data," LIDAM Reprints ISBA 2013018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Takeshi Emura & Yoshihiko Konno, 2012. "Multivariate normal distribution approaches for dependently truncated data," Statistical Papers, Springer, vol. 53(1), pages 133-149, February.
    3. Pao-sheng Shen, 2010. "Nonparametric analysis of doubly truncated data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(5), pages 835-853, October.
    4. Henry T Robertson & David B Allison, 2012. "A Novel Generalized Normal Distribution for Human Longevity and other Negatively Skewed Data," PLOS ONE, Public Library of Science, vol. 7(5), pages 1-7, May.
    5. Carla Moreira & Jacobo Uña-Álvarez & Ingrid Keilegom, 2014. "Goodness-of-fit tests for a semiparametric model under random double truncation," Computational Statistics, Springer, vol. 29(5), pages 1365-1379, October.
    6. Moreira, C. & Van Keilegom, I., 2013. "Bandwidth selection for kernel density estimation with doubly truncated data," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 107-123.
    7. P. Sankaran & S. Sunoj, 2004. "Identification of models using failure rate and mean residual life of doubly truncated random variables," Statistical Papers, Springer, vol. 45(1), pages 97-109, January.
    8. Pao-Sheng Shen, 2011. "Testing quasi-independence for doubly truncated data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(3), pages 753-761.
    9. T. Emura & K. Murotani, 2015. "An algorithm for estimating survival under a copula-based dependent truncation model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 734-751, December.
    10. Joan Castillo, 1994. "The singly truncated normal distribution: A non-steep exponential family," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(1), pages 57-66, March.
    11. A. Philippou & G. Roussas, 1975. "Asymptotic normality of the maximum likelihood estimate in the independent not identically distributed case," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 27(1), pages 45-55, December.
    12. Ya-Hsuan Hu & Takeshi Emura, 2015. "Maximum likelihood estimation for a special exponential family under random double-truncation," Computational Statistics, Springer, vol. 30(4), pages 1199-1229, December.
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    Cited by:

    1. Rafael Weißbach & Dominik Wied, 2022. "Truncating the exponential with a uniform distribution," Statistical Papers, Springer, vol. 63(4), pages 1247-1270, August.
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    3. Achim Dörre & Chung-Yan Huang & Yi-Kuan Tseng & Takeshi Emura, 2021. "Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model," Computational Statistics, Springer, vol. 36(1), pages 375-408, March.

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