Abstract
Estimating equation approaches have been widely used in statistics inference. Important examples of estimating equations are the likelihood equations. Since its introduction by Sir R. A. Fisher almost a century ago, maximum likelihood estimation (MLE) is still the most popular estimation method used for fitting probability distribution to data, including fitting lifetime distributions with censored data. However, MLE may produce substantial bias and even fail to obtain valid confidence intervals when data size is not large enough or there is censoring data. In this paper, based on nonlinear combinations of order statistics, we propose new estimation equation approaches for a class of probability distributions, which are particularly effective for skewed distributions with small sample sizes and censored data. The proposed approaches may possess a number of attractive properties such as consistency, sufficiency and uniqueness. Asymptotic normality of these new estimators is derived. The construction of new estimation equations and their numerical performance under different censored schemes are detailed via Weibull distribution and generalized exponential distribution.
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Notes
The monotonicity property of \(t\mapsto W_N (t)\) is guaranteed for the examples provided in the paper, hence the approach in the textbook Maritz (1995) is sufficiently general in that cases. However, our theoretical arguments for proving asymptotic normality adapt rapidly to nonmonotonic maps \(t\mapsto W_N (t)\).
In this section, \(c\) and \(C\) are constants that may change from line to line.
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Acknowledgments
The authors thank the Editor, the Associate Editor and referees for constructive comments and suggestions that led to substantial improvements in the paper. Keming Yu is partially supported by NSF grant 11261048 of China. Bing Xing Wang is supported by Foundation of Ministry of Education of China under Grant No. 12YJA910005. Valentin Patilea gratefully acknowledges financial support from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project PN-II-ID-PCE-2011-3-0893.
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Yu, K., Wang, B.X. & Patilea, V. New estimating equation approaches with application in lifetime data analysis. Ann Inst Stat Math 65, 589–615 (2013). https://doi.org/10.1007/s10463-012-0385-6
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DOI: https://doi.org/10.1007/s10463-012-0385-6