On distribution function estimation with partially rank-ordered set samples: estimating mercury level in fish using length frequency data

ABSTRACT

We study the non-parametric estimation of a continuous distribution function F based on the partially rank-ordered set (PROS) sampling design. A PROS sampling design first selects a random sample from the underlying population and uses judgement ranking to rank them into partially ordered sets, without measuring the variable of interest. The final measurements are then obtained from one of the partially ordered sets. Considering an imperfect PROS sampling procedure, we first develop the empirical distribution function (EDF) estimator of F and study its theoretical properties. Then, we consider the problem of estimating F, where the underlying distribution is assumed to be symmetric. We also find a unique admissible estimator of F within the class of nondecreasing step functions with jumps at observed values and show the inadmissibility of the EDF. In addition, we introduce a smooth estimator of F and discuss its theoretical properties. Finally, we expand on various numerical illustrations of our results via several simulation studies and a real data application and show the advantages of PROS estimates over their counterparts under the simple random and ranked set sampling designs.

KEYWORDS:

• imperfect subsetting
• kernel function
• mean integrated square error
• optimal bandwidth
• ranked set sampling

Acknowledgments

The authors would like to thank the referees for their constructive comments. Parts of this research were done when Sahar Nazari visited the University of Manitoba.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Mohammad Jafari Jozani gratefully acknowledges the research support of the Natural Sciences and Engineering Research Council of Canada. Sahar Nazari and Mahmood Kharrati-Kopaei acknowledge the Research Council of Shiraz University. Sahar Nazari gratefully acknowledges the financial support provided by NSERC Canada.