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

.