Abstract
Debiased estimation has long been an area of research in the group testing literature. This has led to the development of several estimators with the goal of bias minimization and, recently, an unbiased estimator based on sequential binomial sampling. Previous research, however, has focused heavily on the simple case where no misclassification is assumed and only one trait is to be tested. In this paper, we consider the problem of unbiased estimation in these broader areas, giving constructions of such estimators for several cases. We show that, outside of the standard case addressed previously in the literature, it is impossible to find any proper unbiased estimator, that is, an estimator giving only values in the parameter space. This is shown to hold generally under any binomial or multinomial sampling plans.
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References
Bilder, C.R. and Tebbs, J.M. (2005). Empirical Bayes estimation of the disease transmission probability in multiple-vector-transfer designs. Biom. J.47, 502–516.
Burrows, P.M. (1987). Improved estimation of pathogen transmission rates by group testing. Phytopathology77, 363–365.
Chiang, C.L. and Reeves, W.C. (1962). Statistical estimation of virus infection rates in mosquito vector populations. Am. J. Hyg.75, 377–391.
Ding, J. and Xiong, W. (2015). Robust group testing for multiple traits with misclassification. J. Appl. Stat.42, 2115–2125.
Ding, J. and Xiong, W. (2016). A new estimator for a population proportion using group testing. Communications in Statistics – Simulation and Computation45, 101–114.
DeGroot, M.H. (1959). Unbiased sequential estimation for binomial populations. Ann. Math. Stat.30, 80–101.
Dorfman, R. (1943). The detection of defective members of large populations. Ann. Math. Stat.14, 436–440.
Gibbs, A.J. and Gower, J.C. (1960). The use of a multiple–transfer method in plant virus transmission studies—some statistical points arising in the analysis of results. Annals of Applied Biology48, 75–83.
Girshick, M.A., Mosteller, F. and Savage, L. (1946). J Unbiased estimates for certain binomial sampling problems with applications. Ann. Math. Stat.17, 13–23.
Haber, G. and Malinovsky, Y. (2017). Random walk designs for selecting pool sizes in group testing estimation with small samples. Biom. J.59, 1382–1398.
Haber, G., Malinovsky, Y. and Albert, P.S. (2018). Sequential estimation in the group testing problem. Seq. Anal.37, 1–17.
Hall, W.J. (1963). Estimators with minimum bias. University of California Press, Berkeley, Bellman, R. (ed.), p. 167–199.
Hepworth, G. and Biggerstaff, B. (2017). Bias correction in estimating proportions by pooled testing. J. Agric. Biol. Environ. Stat.22, 602–614.
Hepworth, G. and Watson, R. (2009). Debiased estimation of proportions in group testing. J. R. Stat. Soc. Ser. C58, 105–121.
Huang, S., Huang, M.L., Shedden, K. and Wong, W.K. (2017). Optimal group testing designs for estimating prevalence with uncertain testing errors. J. R. Stat. Soc. Ser. B79, 1547–1563.
Hughes-Oliver, J.M. and Rosenberger, W. (2000). Efficient estimation of the prevalence of multiple rare traits. Biometrika87, 315–327.
Hughes-Oliver, J.M. and Swallow, W.H. (1994). A two-stage adaptive group testing procedure for estimating small proportions. J. Am. Stat. Assoc.89, 982–993.
Hung, M. and Swallow, W.H. (1999). Robustness of group testing in the estimation of proportions. Biometrics55, 231–237.
Koike, K. (1993). Unbiased estimation for sequential multinomial sampling plans. Seq. Anal.12, 253–259.
Kremers, W. (1990). Completeness and unbiased estimation in sequential multinomial sampling. Seq. Anal.9, 43–58.
Lehmann, E.L. and Casella, G. (1998). Theory of point estimation, 2nd edn. Springer, New York.
Li, Q., Liu, A. and Xiong, W. (2017). D-optimality of group testing for joint estimation of correlated rare diseases with misclassification. Stat. Sin.27, 823–838.
Liu, A., Liu, C., Zhang, Z. and Albert, P.S. (2012). Optimality of group testing in the presence of misclassification. Biometrika99, 245–251.
McMahan, C.S., Tebbs, J.M. and Bilder, C.R. (2013). Regression models for group testing data with pool dilution effects. Biostatistics14, 284–298.
Pfeiffer, R.M., Rutter, J.L., Gail, M.H., Struewing, J. and Gastwirth, J.L. (2002). Efficiency of DNA pooling to estimate joint allele frequencies and measure linkage disequilibrium. Genet. Epidemiol.22, 94–102.
Santos, J.D. and Dorgman, D. (2016). An approximate likelihood estimator for the prevalence of infections in vectors using pools of varying sizes. Biom. J.58, 1248–1256.
Sirazhdinov, S. (1956). On estimators with minimum bias for a binomial distribution. Theory of Probability and its Applications1, 150–156.
Swallow, W.H. (1985). Group testing for estimating infection rates and probabilities of disease transmission. Phytopathology75, 882–889.
Tebbs, J.M., Bilder, C.R. and Koser, B.K. (2003). An empirical Bayes group-testing approach to estimating small proportions. Communications in Statistics – Theory and Methods32, 983–995.
Tebbs, J.M., McMahan, C.S. and Bilder, C.R. (2013). Two-stage hierarchical group testing for multiple infections with application to the Infertility Prevention Project. Biometrics69, 1064–1073.
Thompson, K.H. (1962). Estimation of the proportion of vectors in a natural population of insects. Biometrics18, 568–578.
Tu, X.M., Litvak, E. and Pagano, M. (1995). On the informativeness and accuracy of pooled testing in estimating prevalence of a rare disease: application to HIV screening. Biometrika82, 287–297.
Warasi, M.S., Tebbs, J.M., McMahan, C.S. and Bilder, C.R. (2016). Estimating the prevalence of multiple diseases from two-stage hierarchical pooling. Statistics In Medicine35, 3851–3864.
Zhang, Z., Liu, C., Kim, S. and Liu, A. (2014). Prevalence estimation subject to misclassification: the mis-substitution bias and some remedies. Stat. Med.33, 4482–4500.
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The authors would like to thank the Editor, Associate Editor, and two anonymous referees whose input greatly improved the presentation of this paper.
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Haber, G., Malinovsky, Y. On the Construction of Unbiased Estimators for the Group Testing Problem. Sankhya A 82, 220–241 (2020). https://doi.org/10.1007/s13171-018-0156-4
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DOI: https://doi.org/10.1007/s13171-018-0156-4
Keywords and phrases.
- Binomial sampling plans
- Group testing
- Multinomial sampling plans
- Sequential estimation
- Unbiased estimation