Likelihood inference for lognormal data with left truncation and right censoring with an illustration.

*(English)*Zbl 1221.62038Summary: The lognormal distribution is quite commonly used as a life-time distribution. Data arising from life-testing and reliability studies are often left truncated and right censored. Here, the EM algorithm is used to estimate the parameters of the lognormal model based on left truncated and right censored data. The maximization step of the algorithm is carried out by two alternative methods, with one involving approximation using Taylor series expansion (leading to approximate maximum likelihood estimate) and the other based on the EM gradient algorithm [K. Lange, J. R. Stat. Soc., Ser. B 57, No. 2, 425–437 (1995; Zbl 0813.62021)]. These two methods are compared based on Monte Carlo simulations. The Fisher scoring method for obtaining the maximum likelihood estimates shows a problem of convergence under this setup, except when the truncation percentage is small. The asymptotic variance-covariance matrix of the MLEs is derived by using the missing information principle [T. A. Louis, ibid., Ser. B 44, 226–233 (1982; Zbl 0488.62018)], and then the asymptotic confidence intervals for scale and shape parameters are obtained and compared with corresponding bootstrap confidence intervals. Finally, some numerical examples are given to illustrate all the methods of inference developed here.

##### MSC:

62F10 | Point estimation |

62N01 | Censored data models |

62N05 | Reliability and life testing |

62F25 | Parametric tolerance and confidence regions |

65C05 | Monte Carlo methods |

##### Keywords:

maximum likelihood estimators; EM algorithm; lifetime data; left truncation; right censoring; missing information principle; asymptotic variances; parametric bootstrap confidence intervals##### Software:

SPLIDA
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\textit{N. Balakrishnan} and \textit{D. Mitra}, J. Stat. Plann. Inference 141, No. 11, 3536--3553 (2011; Zbl 1221.62038)

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