Inference for the extreme value distribution under progressive type-II censoring.

*(English)*Zbl 1048.62090Summary: The extreme value distribution has been extensively used to model natural phenomena such as rainfall and floods, and also in modeling lifetimes and material strengths. Maximum likelihood estimation (MLE) for the parameters of the extreme value distribution leads to likelihood equations that have to be solved numerically, even when the complete sample is available. We discuss point and interval estimation based on progressively Type-II censored samples. Through an approximation in the likelihood equations, we obtain explicit estimators which are approximations to the MLEs. Using these approximate estimators as starting values we obtain the MLEs using an iterative method and examine numerically their bias and mean squared error.

The approximate estimators compare quite favorably to the MLEs in terms of both bias and efficiency. Results of the simulation study, however, show that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory for both these estimators and particularly so when the effective sample size is small. We, therefore, suggest the use of unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. The results are presented for a wide range of sample sizes and different progressive censoring schemes. We conclude with an illustrative example.

The approximate estimators compare quite favorably to the MLEs in terms of both bias and efficiency. Results of the simulation study, however, show that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory for both these estimators and particularly so when the effective sample size is small. We, therefore, suggest the use of unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. The results are presented for a wide range of sample sizes and different progressive censoring schemes. We conclude with an illustrative example.

##### MSC:

62N01 | Censored data models |

65C60 | Computational problems in statistics (MSC2010) |

62G32 | Statistics of extreme values; tail inference |

65C05 | Monte Carlo methods |

62Q05 | Statistical tables |

62N02 | Estimation in survival analysis and censored data |

##### Keywords:

maximum likelihood estimator; Monte Carlo simulation; pivotal quantity; progressive type-II censoring
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\textit{N. Balakrishnan} et al., J. Stat. Comput. Simulation 74, No. 1, 25--45 (2004; Zbl 1048.62090)

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