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

estimate

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

where

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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-

for

. 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

into

. 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

,

,

and

leading to new estimators

,

and

respectively. It is observant

that

and

have the advantage of

incorporating the concomitant information turning most likely to more accuracy

than this obtained from

and

.

To

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

and

and

.For

each combination of

and

,

iterations have been done. For each iteration,

the estimators

,

and

are computed.