As Zamanzade and Mahdizadeh (2017) to modify for

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

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. 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.

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