Model based inference using ranked set samples
Section 6. Example

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In this section we apply the proposed estimators to a data set which contains a sheep population in a research farm at Ataturk University, Erzurum, Turkey. Data set contains birth weights, mothers’ weights at mating and the weights at the 7^th month after birth for 224 lambs. The entire data set is given in Hollander, Wolfe and Chicken (2014, page 709). Variable of interest is the weights $\left(Y\right)$ at the 7^th month after birth for 224 lambs. We use birth weights $\left(X_1\right)$ and mothers’ weights $\left(X_2\right)$ at mating as auxiliary variables to perform ranking process. The ranking variables are positively correlated with the variable of interest $Y.$ The correlation coefficient $\left(\rho \mathrm{<mo>=</mo>}\text{corr}\left(X\mathrm{,}Y\right)\right)$ between $X_1\mathrm{,}Y$ and $X_2\mathrm{,}Y$ are 0.8425 and 0.5941, respectively. The histogram of the variable of interest, $Y,$ is roughly symmetric. Mean and variance of $Y$ are ${\overline{Y}}_N\mathrm{<mo>=</mo>}$ 28.125kg and $S_N^2\mathrm{<mo>=</mo>}$ 15.23kg², respectively, where $S_n^2={\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^{224}}{\left(Y_i-{\overline{Y}}_N\right)}^2/223.$ We treated these 224 lambs as a realization from a super population having finite mean $\mu$ and variance ${\sigma }^2\text{}.$ We constructed samples based RSS sampling design using this finite population. Samples are generated for sample and set size combinations, $\left(n\mathrm{,}H\right),$ (10, 2), (15, 3), (20, 4), (25, 5). Simulation size is taken to be 50,000.

In this example, we incorporate the sampling cost to RSS and SRS sampling designs with concomitant ranking in RSS. We first need to determine reasonable costs associated with various aspects of RSS. Weight measurement is obtained from seven-month-old lambs. These animals are very active and measurement cost is substantial. The measurement process usually require three people for separating the lamb from the flock, bringing it to scale, holding it firm during the measurement. Suppose that the farm employs the workers in an annual salary of $50,000. This corresponds to a rate of approximatley $25 per hour per person. Assume that the measurement of a lamb takes about 5 minutes. The measurement cost for a lamb then would be about $c_{qy}\mathrm{<mo>=</mo>\; 3}\left(25/12\right)\approx 6$ for three workers. Ranking will be performed using auxilairy variables $X_1$ and $X_2\text{}.$ These variables are maintained in the data base for some other purposes. Only cost to sampling would be due to personal cost for ranking. Ranking will be performed in the office by selecting sets at random from the data base and ranking them based on auxiliary variables. Suppose that ranking a set of size $H$ takes about 1/2 minute. This leads to ranking cost of $H{c}_{qx}\mathrm{<mo>=</mo>\; <mi>\$</mi>\; 0.21.}$ We may assume that cost related to identification of a unit in the population is negligible $\left(c_i\mathrm{<mo>=</mo>\; 0}\right).$ Under these stipulations, the cost ratio of selecting and measuring a unit in RSS and SRS is given by $\text{ratio}\mathrm{<mo>=</mo>}\left(H{c}_i+H{c}_{qx}+c_{qy}\right)/\left(c_i+c_{qy}\right)\mathrm{<mo>=</mo>}\left(6+0.21\right)/6\mathrm{<mo>=</mo>\; 1.035.}$ Since this ratio is less than all entries in Table 4.1, we anticipate that ${\overline{Y}}_{\text{RC}}$ provides higher efficiency than ${\overline{Y}}_{\text{SRS}}\text{}.$

Table 6.1 presents the estimated MSPE and relative efficiency of RSS esimator as well as the coverage probability of the confidence interval of $\mu$ for different $\rho$ and sample size combinations. It is clear that the RSS estimator outperform the SRS estimator for all simulation parameter combinations. Estimated MSPEs and coverage probabilities also show similar behaviors as in Section 3. The estimated MSPE values are very close to the simulated MSPE values. The coverage probabilities of the confidence intervals based on $t-$ approximation appear to be very close to the nominal coverage probabiliy, 0.950.

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