In this part, we will illustrate how the
concomitant information can be useful for estimating the

 under Bohn-Wolfe (BW) model. Actually, there are two different ways to

 introduced respectively by Ozturk
(2008 and 2010). Since the second method is computationally intensive, the first one
will only be considered as done in Sgambellone (2013). The BW model could be
expressed as 


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 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 to
BW model is that

 reflects the ranking mechanism
quality 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 diverse
statistical applications such as constructing non parametric confidence
interval and computing Mann-Whitney-Wilcoxon test. Here, we will also add a new
useful usage for

 as illustrated in the next section.

Ozturk (2008) decided
to estimate

by minimizing the difference between the sampling and the expected in-stratum
CDF under BW model. He considered

 as a sampling in-stratum CDF. He
mentioned that the minimization process must be implemented in the light of the
following constraints: 1- Each

 must be within the interval

. 2- The sum of each row and each column must be equal one. 3-


. It is obvious that the first constrain guarantees that

 is a probability matrix, while
the second implies that

 satisfies the doubly stochastic
condition and third constrain refers to,for simplicity, the symmetry of

in order to reduce the number of unknown items from


. Accordingly, the optimization problem can be formulated as

Ozturk (2008) used the function Solve.QP in the
R-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 one
would like to obtain another estimate for

 such that be much better than

,  itis sufficient to replace

 with another more efficient
estimator in (2). Since

 suffers from a serious
disadvantage that depends on

 which does not always agree with
the SO constrain, it is good choice to interchange

 in (2) with 




 leading to new estimators



 respectively. It is observant


 have the advantage of
incorporating the concomitant information turning most likely to more accuracy
than this obtained from



examine the effect of using different in-stratum CDF estimators on estimating

, we did a small simulation study. Since the BW
model cannot be applicable for getting concomitant-based RSS, we used again
Dell and Clutter (1972) model to generate concomitant-based RSS under perfect (

) and imperfect (

) ranking with



each combination of



 iterations have been done. For each iteration,
the estimators



 are computed. 


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