Statistical inference based on the nonparametric maximum likelihood estimator under double-truncation

Takeshi Emura*, Yoshihiko Konno, Hirofumi Michimae

*Corresponding author for this work

Research output: Contribution to journalJournal Article peer-review

19 Scopus citations


Doubly truncated data consist of samples whose observed values fall between the right- and left- truncation limits. With such samples, the distribution function of interest is estimated using the nonparametric maximum likelihood estimator (NPMLE) that is obtained through a self-consistency algorithm. Owing to the complicated asymptotic distribution of the NPMLE, the bootstrap method has been suggested for statistical inference. This paper proposes a closed-form estimator for the asymptotic covariance function of the NPMLE, which is computationally attractive alternative to bootstrapping. Furthermore, we develop various statistical inference procedures, such as confidence interval, goodness-of-fit tests, and confidence bands to demonstrate the usefulness of the proposed covariance estimator. Simulations are performed to compare the proposed method with both the bootstrap and jackknife methods. The methods are illustrated using the childhood cancer dataset.

Original languageEnglish
Pages (from-to)397-418
Number of pages22
JournalLifetime Data Analysis
Issue number3
StatePublished - 15 07 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2014, Springer Science+Business Media New York.


  • Asymptotic variance
  • Bootstrap
  • Confidence band
  • Goodness-of-fit test
  • Survival analysis


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