On the population median estimation using quartile double ranked set sampling

In this article, quartile double ranked set sampling (QDRSS) method is considered for estimating the population median. The sample median based on QDRSS is suggested as an estimator of the population median. The QDRSS is compared with the simple random sampling (SRS), ranked set sampling (RSS) and quartile ranked set sampling (QRSS) methods for estimating the population median. To verify this method a real data example is applied. It turns out that for the symmetric distributions considered in this study, the QDRSS estimators are unbiased estimators of the population median and are larger than their counterparts using SRS, RSS and QRSS based on the same sample size of measured units. For asymmetric distributions, QDRSS is biased. It is more efficient than the SRS and the QRSS for all samples of size m while it is more efficient than RSS if


Introduction
Ranked set sampling was first suggested by McIntyre (1952) as a cost efficient sampling procedure when compared to the commonly used simple random sampling in situations where visual ordering of set units can be done easily, but the exact measurement of the units is difficult and expensive. McIntyre (1952) found that the RSS is more efficient than SRS for estimating the population mean.
Let X be a random variable with a probability density function (pdf) () fx, and a cumulative distribution function (cdf) () Fx with mean  and variance 2  . Also, let ( : ) () im fx be the pdf of the ith order statistic of a random sample of size m, 12   Muttlak (1997) suggested median ranked set sampling for estimating the population mean. Al-Saleh and Al-Kadiri (2000) considered double ranked set sampling (DRSS) method for estimating the population mean, and they showed that the ranking at the second stage is easier than the ranking at the first stage.
The double ranked set sampling method can be described as follows: Randomly identify 3 m units from the target population and divide them randomly into m sets each of size 2 m . The procedure of ranked set sampling is applied to these sets to obtain m ranked set samples each of size m, again reapply the ranked set sampling procedure on the m ranked set samples to obtain a DRSS of size m.
Al-Saleh and Al-Omari (2002) generalized the DRSS to multistage ranked set sampling to increase the efficiency of the estimators for specific value of the sample size. Muttlak (2003) proposed quartile ranked set sampling (QRSS) for estimating the population mean. Al-Omari and Al-Saleh (2009) suggested quartile double ranked set sampling (QDRSS) for estimating the population mean. Al-Omari (2010) suggested an estimator of the population median using double robust extreme ranked set sampling. Entropy estimation and goodnessof-fit tests for the inverse Gaussian and Laplace distributions using paired ranked set sampling method is suggested by Al-Omari and Haq (2015). Biradar and Santosha (2015) proposed estimation of the population mean using paired ranked set sampling. Santos and Barrios (2015) considered predictive accuracy of logistic regression model using ranked set samples. Confidence intervals and hypothesis tests for a population mean using ranked set sampling are considered by Stella et al. (2015). For more about RSS and its modifications see Sinha et al. (2006)

Using SRS
and if m is even and

Using RSS
The RSS (McIntyre, 1952) involves randomly selecting 2 m units from the population. These units are randomly allocated into m sets, each of size m. The m units of each sample are ranked visually or by any inexpensive method with respect to a variable of interest. From the first set of m units, the smallest ranked unit is measured. From the second set of m units, the second smallest ranked unit is measured. The process is continued until from the mth set of m units the largest ranked unit is measured. The process can be repeated n times to get a sample of size mn from the initial m 2 n units.

. Using QDRSS
The quartile double ranked set sampling method (Al-Omari and Al-Saleh, 2009) can be carried out as follows: Step 1: Randomly select 3 m units from the target population and allocate them into m sets each of size 2 m units.

Simulation Study
In this section, a simulation study is considered to compare the proposed estimators for the population median using QDRSS, QRSS, RSS relative to SRS. Six probability distribution functions were considered for the populations: Uniform, Normal, Logistic, Exponential, Gamma and Weibull. 60,000 samples were generated and the averages of these samples were compared.
If the distribution is symmetric the efficiency of RSS, QRSS and QDRSS relative to SRS, is defined as, respectively, Results of simulation in terms of the efficiency and bias values for RSS, QRSS and DQRSS are summarized for 4,5 m  in Table 1, for 6,7 m  in Table 2, for 10,11 m  in Table 3 and for 12 m  in Table 4.    , while with even sample size we identify the first or the third quartile of the ith sample.

Real Data Application
In this section, to evaluate the performance of QDRSS in estimating the population median of a real data, a study is conducted to estimate the median weight of 342 students. Balanced ranked set sampling is considered and all samples were done without replacement. The skewness of the 342 observations is 1.244, which means that these data are asymmetrically distributed, and so the QDRSS estimators will be biased. Hence, the bias and mean squared error (MSE) of the estimators were computed. The efficiency of RSS, QRSS and QDRSS with respect to SRS are obtained using Equations (8), (9), and (10). The simulated median, bias, MSE and the efficiency values are summarized in Table 5.  Table 5 shows that there is a small difference between the true and the estimated median. This difference is due to skewness of the data used in this example. For 4 m  , RSS is more efficient than QDRSS. While, QDRSS is more efficient than RSS. In addition, it can be noted that QDRSS is more efficient than QRSS for all sample sizes considered in Table 5. Furthermore, the results of real data example are agreed with the results of the simulation study conducted in Section 4.

Conclusion
In estimating the population median, a good achievement is gained in efficiency using QDRSS, QRSS, RSS regardless the underlying distribution whether it is symmetric or asymmetric. QDRSS estimators are unbiased estimators of the population median when distributions are symmetric. In addition, it is found that QDRSS is more efficient than RSS if 4 m  and more efficient than QRSS in all cases considered in this study. However, the QDRSS is recommended for estimating the population median of symmetric distributions.