Abstract
Randomized nomination sampling (RNS) is a rank-based sampling technique which has been shown to be effective in several nonparametric studies involving environmental, agricultural, medical and ecological applications. In this paper, we investigate parametric inference using RNS design for estimating an unknown vector of parameters θ in some parametric families of distributions. We examine both maximum likelihood (ML) and method of moments (MM) approaches. We introduce four types of RNS-based data as well as necessary EM algorithms for the ML estimation under each data type, and evaluate the performance of corresponding estimators in estimating θ compared with those based on simple random sampling (SRS). Our results can address many parametric inference problems in reliability theory, sport analytics, fisheries, etc. Theoretical results are augmented with numerical evaluations, where we also study inference based on imperfect ranking. We apply our methods to a real data problem in order to study the distribution of the mercury contamination in fish body using RNS designs.
Similar content being viewed by others
References
Al-Odat, M. and Al-Saleh, M.F. (2001). A variation of ranked set sampling. Journal of Applied Statistical Science10, 2, 137–146.
Bhattacharya, D. and Samaniego, F.J. (2010). Estimating component characteristics from system failure-time data. Naval Research Logistics (NRL)57, 4, 380–389.
Bhavsar, S.P., Gewurtz, S.B., McGoldrick, D.J., Keir, M.J. and Backus, S.M. (2010). Changes in mercury levels in great lakes fish between 1970s and 2007. Environmental science & technology44, 9, 3273–3279.
Boyles, R.A. and Samaniego, F.J. (1986). Estimating a distribution function based on nomination sampling. Journal of the American Statistical Association81, 396, 1039–1045.
Gemayel, N.M., Stasny, E.A. and Wolfe, D.A. (2010). Optimal ranked set sampling estimation based on medians from multiple set sizes. Journal of Nonparametric Statistics22, 4, 517–527.
Ghosh, K. and Tiwari, R.C. (2009). A unified approach to variations of ranked set sampling with applications. Journal of Nonparametric Statistics21, 4, 471–485.
Jafari Jozani, M. and Johnson, B.C. (2012). Randomized nomination sampling for finite populations. Journal of Statistical Planning and Inference142, 7, 2103–2115.
Jafari Jozani, M. and Mirkamali, S.J. (2010). Improved attribute acceptance sampling plans based on maxima nomination sampling. Journal of Statistical Planning and Inference140, 9, 2448–2460.
Jafari Jozani, M. and Mirkamali, S.J. (2011). Control charts for attributes with maxima nominated samples. Journal of Statistical Planning and Inference141, 7, 2386–2398.
Kvam, P.H. and Samaniego, F.J. (1993). On estimating distribution functions using nomination samples. Journal of the American Statistical Association88, 424, 1317–1322.
McGoldrick, D.J., Clark, M.G., Keir, M.J., Backus, S.M. and Malecki, M.M. (2010). Canada’s national aquatic biological specimen bank and database. Journal of Great Lakes Research36, 2, 393–398.
Mehrotra, K. and Nanda, P. (1974). Unbiased estimation of parameters by order statistics in the case of censored samples. Biometrika61, 3, 601–606.
Nourmohammadi, M., Jafari Jozani, M. and Johnson, B.C. (2014). Confidence intervals for quantiles in finite populations with randomized nomination sampling. Computational Statistics & Data Analysis73, 112–128.
Nourmohammadi, M., Jafari Jozani, M. and Johnson, B.C. (2015a). Distribution-free tolerance intervals with nomination samples: Applications to mercury contamination in fish. Statistical Methodology26, 16–33.
Nourmohammadi, M., Jafari Jozani, M. and Johnson, B.C. (2015b). Nonparametric confidence intervals for quantiles with randomized nomination sampling. Sankhya A77, 2, 408–432.
Samawi, H.M., Ahmed, M.S. and Abu-Dayyeh, W. (1996). Estimating the population mean using extreme ranked set sampling. Biometrical Journal38, 5, 577–586.
Tiwari, R.C. (1988). Nonparametric bayes estimation of a distribution under nomination sampling. Reliability, IEEE Transactions on37, 5, 558–561.
Tiwari, R.C. and Wells, M.T. (1989). Quantile estimation based on nomination sampling. Reliability, IEEE Transactions on38, 5, 612–614.
Wells, M.T., Tiwari, R.C. et al. (1990). Estimating a distribution function based on minima-nomination sampling,.
Willemain, T.R. (1980). Estimating the population median by nomination sampling. Journal of the American Statistical Association75, 372, 908–911.
Acknowledgements
The authors gratefully acknowledge the partial support of the Natural Sciences and Engineering Research Council of Canada (NSERC). We would like to thank two anonymous referees for their useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Nourmohammadi, M., Jafari Jozani, M. & Johnson, B.C. Parametric Inference Using Nomination Sampling with an Application to Mercury Contamination in Fish. Sankhya A 82, 115–146 (2020). https://doi.org/10.1007/s13171-018-00159-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-018-00159-8
Keywords
- Randomized nomination sampling
- Method of moments
- Maximum likelihood
- Modified maximum likelihood
- EM algorithm