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On the construction of unbiased estimators for the group testing problem. (English) Zbl 1442.62047
The paper deals with the debiased estimation in the group testing procedure. The research in the group testing led to the development of several estimators with the goal of bias minimization. An unbiased estimator based on sequential binomial sampling (see [J. P. Burman, J. R. Stat. Soc., Suppl. 8, 98–103 (1946; Zbl 0063.00666); M. A. Girshick et al., Ann. Math. Stat. 17, 13–23 (1946; Zbl 0063.01635)]) has been used recently by W. K. Kremers [Sequential Anal. 9, No. 1, 43–58 (1990; Zbl 0693.62064)], K.-i. Koike [Sequential Anal. 12, No. 3–4, 253–259 (1993; Zbl 0792.62071)], the authors and P. S. Albert [Sequential Anal. 37, No. 1, 1–17 (2018; Zbl 1390.62163)]. Previous research has focused heavily on the simple case where no misclassification is assumed and only one trait is to be tested. The present paper treats the problem of unbiased estimation in these broader areas, giving constructions of such estimators for several cases. The necessary and sufficient conditions for unbiased estimation of a function of the parameter vector of an inverse multinomial model is given. The result generalize Theorem 4.1 formulated in [M. H. DeGroot, Ann. Math. Stat. 30, 80–101 (1959; Zbl 0091.30901)]. It allows to construct unbiased estimators under group testing models for one and two traits. The results presented here are refereed to testing for single or multiple diseases throughout. It is shown that it is impossible to find any proper unbiased estimator (i.e. an estimator giving only values in the parameter space). This is shown to hold generally under any binomial or multinomial sampling plans.
MSC:
62F10 Point estimation
62L12 Sequential estimation
62P10 Applications of statistics to biology and medical sciences; meta analysis
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