In this part, we will illustrate how theconcomitant information can be useful for estimating the under Bohn-Wolfe (BW) model. Actually, there are two different ways toestimate introduced respectively by Ozturk(2008 and 2010). Since the second method is computationally intensive, the first onewill only be considered as done in Sgambellone (2013).
The BW model could beexpressed as where is the population in-stratum CDF and is the element of indicates the probability that order statistics is assigned to order statistics. The interesting characteristic associated toBW model is that reflects the ranking mechanismquality of the RSS. As if the rankings are done perfectly, this leads equals the identity matrix.Whereas if the ranking process is exact imperfect, all the elements of will equal . Ozturk (2008) showed that can be useful for diversestatistical applications such as constructing non parametric confidenceinterval and computing Mann-Whitney-Wilcoxon test.
Here, we will also add a newuseful usage for as illustrated in the next section.Ozturk (2008) decidedto estimate by minimizing the difference between the sampling and the expected in-stratumCDF under BW model. He considered as a sampling in-stratum CDF. Hementioned that the minimization process must be implemented in the light of thefollowing constraints: 1- Each must be within the interval . 2- The sum of each row and each column must be equal one.
3- for . It is obvious that the first constrain guarantees that is a probability matrix, whilethe second implies that satisfies the doubly stochasticcondition and third constrain refers to,for simplicity, the symmetry of in order to reduce the number of unknown items from into . Accordingly, the optimization problem can be formulated as Ozturk (2008) used the function Solve.QP in theR-library QUADPROG to obtain the solution of the optimization expressed in (2)denotedby . Computational details for getting can be found in Sgambellone (2013).
Generally speaking, it isreasonable to think if onewould like to obtain another estimate for such that be much better than , itis sufficient to replace with another more efficientestimator in (2). Since suffers from a seriousdisadvantage that depends on which does not always agree withthe SO constrain, it is good choice to interchange in (2) with , , and leading to new estimators , and respectively. It is observantthat and have the advantage ofincorporating the concomitant information turning most likely to more accuracythan this obtained from and .Toexamine the effect of using different in-stratum CDF estimators on estimating , we did a small simulation study. Since the BWmodel cannot be applicable for getting concomitant-based RSS, we used againDell and Clutter (1972) model to generate concomitant-based RSS under perfect ( ) and imperfect ( ) ranking with and and .
Foreach combination of and , iterations have been done. For each iteration,the estimators , and are computed.