In what follows, we will illustrate the

applicability of the suggested CDF estimators and the proposed test by using

real data set known as body fat data set. This data set of size 252

observations is providedby Carnegie Mellon’s statistics library and can be

found at http://lib.stat.cmu.edu/datasets/bodyfat. Also, it has been used by

many authors.( e.g. Wang et al. (2008) and Zamanzade and Al-Omari (2016)). We

will consider this data set as the target population with overall and in-stratum

CDF are respectively given by

Similar to Zamanzade and Al-Omari (2016), we will select two variables among

the fifteen variables included in this data set. The first one is the abdomen

circumferencedenoted as the concomitant variable

, whereas the second variable is the percentage of body fatdenoted as

the interested variable

. These two variables are highly correlated as their correlation coefficient

is 83%, thus one can expect that the ranking process will be nearly prefect.Further,it

should be emphasizedthat

has a semi-symmetric shape, thuswe

do not need to transform the data priorimplementing the kernel-based estimators.

For the same values of

determined in Section

,

samples with replacement were

selected from the target population. For each sample, all the CDF estimators

studied in Section

were calculated and their REs to

were also obtained and reported

in the Table

. It should be mentioned that the IMSE are approximated over the

interval

with interval width .5, where

is the population quantile at

point

.

One can easily deduce that the results exhibited in

Table 5 are consistent with those shown in Table 1. As, one can easily order

the three estimators.

is the best overall CDF estimator, then

and

. Whereas for estimating the in-stratumCDF,

outperforms

which in turnoutperforms

.