Model based inference using ranked set samples
Section 5. Empirical results

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In this section, we conduct a simulation study to check the finite sample properties of the estimator for different values of simulation parameters. Data sets are generated from normal $\left(\mu \mathrm{<mo>=</mo>\; 10,}\sigma \mathrm{<mo>=</mo>\; 4}\right)$ and log normal $\left(\mu \mathrm{<mo>=</mo>\; 0,}\sigma \mathrm{<mo>=</mo>\; 1}\right)$ super populations. We consider two different finite populations with population sizes $N\mathrm{<mo>=</mo>}$ 150 and $N\mathrm{<mo>=</mo>}$ 1,000 to see the impact of population sizes on the estimators. Sample and set size combinations $\left(n\mathrm{,}H\right)$ are selected to be (10, 2), (15, 3), (20, 4), (25, 5). The quality of ranking information is modeled through a perceptual error model in Dell and Clutter (1972). The Dell and Clutter model considers two variables, the variable of interest $Y$ and a correlated ranking variable $X.$ The ranking variable is modeled through an additive model $X\mathrm{<mo>=</mo>}Y+ϵ,$ where $ϵ$ is a random noise generated independently with respect to $Y.$ To implement the perceptual error model, we generate a set (size $H)$ of simple random sample, $Y\mathrm{<mo>=</mo>}\left(Y_1\mathrm{,}{Y}_2\mathrm{,}\dots \mathrm{,}{Y}_H\right),$ from the true population of interest with mean $\mu$ and variance ${\sigma }^2\text{}.$ Another set (size $H)$ of random numbers are generated from a normal distribution with mean zero and variance, ${\sigma }_{ϵ}^2\text{},$ $\mathbf{ϵ}\mathrm{<mo>=</mo>}\left({ϵ}_1\text{}\mathrm{,}{ϵ}_2\text{}\mathrm{,}\dots \mathrm{,}{ϵ}_H\right).$ The perceptual error model is then defined by $X_i\mathrm{<mo>=</mo>}{Y}_i+{ϵ}_i$ $i\mathrm{<mo>=</mo>\; 1,}\mathrm{2,}\dots \mathrm{,}H.$ The random numbers $\left(X_i\mathrm{,}{Y}_i\right)$ are ranked with respect to the first components $\left(X_{\left(i\right)}\right)$ and the second components are taken to be the judgment ranked order statistics $\left(Y_{\left[i\right]}\right).$ The quality of the ranking information is controlled by the correlation coefficient between $Y$ and $X,$ $\rho \mathrm{<mtext>\hspace{0.17em}</mtext>\; <mo>=</mo>\; <mtext>\hspace{0.17em}</mtext>}\text{corr}\left(Y\mathrm{,}X\right)\mathrm{<mo>=</mo>}{\left(\frac{{\sigma }^2}{{\sigma }^2+{\sigma }_{ϵ}^2}\right)}^{1/2}\text{}.$ Since the units are ranked based on concomitant variable $X,$ the ranking model is equivalent to concomitant ranking in Section 3. In the simulation study, we used $\rho \mathrm{<mo>=</mo>\; 1}$ for perfect ranking and $\rho \mathrm{<mo>=</mo>}$ 0.75, 0.50 for imperfect ranking.

In each replication of the simulation, a finite population of size $P^N$ is generated from the normal super population with specified mean $\mu$ and standard deviation $\sigma ,$ $P^N\mathrm{<mo>=</mo>}\left\{y_1\mathrm{,}\dots \mathrm{,}{y}_N\right\}.$ A ranked set sample is then constructed from this finite population, a realization from normal super population, with specified set and cycle sizes. The quality of ranking information in each RSS sample is controlled generating random noise vector $\mathbf{ϵ}$ with specified $\rho$ (or equivalently ${\sigma }_{ϵ})$ in the perceptual error model. The simulation size is taken to be 50,000.

Simulation results are presented in Tables 5.1, 5.2, 5.3 and 5.4. There are several features that need to be discussed in these tables. For different $\rho$ and sample size combinations $\left(n\mathrm{,}H\right),$ the relative efficiencies of the RSS estimator with respect to the SRS estimator are given by

$${\mathrm{RE}}_{RC}\mathrm{<mo>=</mo>}\frac{V\left({\overline{Y}}_{\text{SRS}}\right)}{V\left({\overline{Y}}_{\text{RC}}\right)}(5.1)$$

where $V\left({\overline{Y}}_{\text{SRS}}\right)$ and $V\left({\overline{Y}}_{\text{RC}}\right)$ are the MSPE of SRS and RSS sample means from the simulation study under a super population model in equation (2.1), respectively. In equation (5.1), the relative efficiency values $\left({\text{RE}}_{\text{RC}}\right)$ greater than one indicate that the RSS estimator is more efficient than the SRS estimator. In all these tables, the RSS sample mean estimator performs better than the SRS estimator. Its efficiency is an increasing function of set size $H$ and correlation coefficient $\rho$ as expected. Under the concomitant cost model, if the cost ratio of obtaining a unit in RSS and a unit in SRS is less than the RP values in Table 4.1, the RSS sample mean has higher efficiency than the SRS sample mean.

The impact of the finite population size $N$ can be observed by comparing the efficiency results in Tables 5.1 and 5.2 for the normal super population and Tables 5.3 and 5.4 for the lognormal super population. When $\rho \mathrm{<mo>></mo>}$ 0.50, relative efficiencies $\left({\text{RE}}_{\text{RC}}\right)$ are higher in Table 5.1 $\left(N\mathrm{<mo>=</mo>\; 150}\right)$ than Table 5.2 $\left(N\mathrm{<mo>=</mo>}\text{1,000}\right).$ In Table 5.1, finite population correction factor is smaller than the finite population correction factor in Table 5.2. Hence, the reduction in MSPE is smaller in RSS estimator. Similar effect is also observed in Tables 5.3 and 5.4.

The simulation study also investigated the properties of the MSPE estimator of RSS sample mean estimator. Theoretical value of the MSPE estimator is given under the heading ${\sigma }_{\text{RSS}}^2$ when $\rho \mathrm{<mo>=</mo>}$ 1.0. The simulated (unbiased) MSPE estimate is given in columns 5 (6) in Tables 5.1-5.4. It is very clear that simulated and unbiased MSPE estimates are almost identical when $\rho \ne 1$ as expected. Under perfect ranking $\left(\rho \mathrm{<mo>=</mo>\; 1}\right)$ theoretical MSPE values, and the simulated and unbiased MSPE estimates are all close to each other within the simulation variation.

The coverage probabilities of the confidence intervals are given under the heading $C\left({\overline{Y}}_{\text{RC}}\right)$ in column 7 in Tables 5.1-5.4. In Tables 5.1 and 5.2, the coverage probabilities of the confidence intervals based on $t-$ approximation are reasonably close to the nominal coverage probability 0.950. On the other hand, the coverage probabilities in Tables 5.3 and 5.4 are smaller than the nominal coverage probability 0.95 for lognormal super population. The coverage probabilities are getting closer to nominal values when the sample size increases. This indicates that for skewed populations, sample sizes should be large enough to have a reasonable coverage probability for the confidence intervals.

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