As mentioned above, the resulting of exploiting the

concomitant information is raising the precision of the inferential procedures.

This result motivates, recently, Zamanzade and Mahdizadeh (2017) to modify

for getting another overall CDF

estimator expected to be better than

. Their idea is basically based on the two well-known facts expressed as

where

is a proper function. Taking

as

in (1) yielding anotheroverall CDF

estimator given by

It is clear that

efficiently uses the auxiliary information provided by

and it is still unbiased

estimator for

with less variance than

, in view of (1), which implies that sampling estimator for

expected to outperform

. In order to

obtain the sampling estimator for

denoted by

,it is required that

and

have the same numberof

observations which does not agree with the RSS strategy. Hence, Zamanzade and Mahdizadeh (2017) decided to

overcome this dilemma by interpolating the unmeasured values of

using their corresponding values of

. Of course, these interpolated values affect negatively on the

efficiency of

. It is also logically to expect that if the relation between

and

is not strong, the interpolated

values will no longer be accurate. Fortunately, Zamanzade and Mahdizadeh (2017) investigated

numerically that

is still valid even when the

rankings are done completely random, i.e.

and

are independent variables, as the

efficiency-reduction isnegligible.Without loss of generality, we will assume

positive relation between

and

. Hence, the steps of calculating

can be summarized in the

following steps.

1- Combining

and their corresponding values of

into two new variables

respectively.

2- Sorting ascending

according to

values yielding

.

3- Obtain the isotonized

values for

and save these values in

.

4- For each

, obtain the corresponding

by utilizing the linear

interpolation formula given by

.

5. Lastly,

can be directly get as

Likewise, we can easily propose a new

in-stratum CDF estimator based on the information supported by the concomitant

variable. Our suggested estimator for in-stratum CDF can be defined as

Remark

:It is of interest to notethat the suggested estimator

enjoys with some attractive

properties similar to

:1-It incorporates efficiently the concomitant information. 2-It is also

unbiased estimator for

with a smallervariance than

,according to the

identity given in (1), under the perfect ranking.3- It satisfies the SO condition due to step 3mentioned above.