In this chapter, the authors investigate the performance of the Gjestvang and Singh randomized response model for estimating the mean of a sensitive variable using ranked set sampling along the lines of Bouza. The proposed estimator is found to be unbiased, and a variance expression is derived. Then a simulation study is carried out to judge the magnitude of relative efficiency in various situations. At the end, the proposed model is assessed based on real secondary data applications. A set of SAS codes is also included.
Top1. Introduction
Warner (1965) proposed an estimator for estimating the prevalence of a sensitive attribute in a population that utilized a device by implementing the idea of randomized response. The goal of his method was to reduce non-response and evasive-answer biases by maintaining the privacy of interviewees during an in person survey. While his procedure allowed for the estimation of the proportion of a population having or not having a stigmatizing characteristic, it does not address the issue of estimating the mean of a sensitive quantitative variable, such as the number of abortions, income, drug usage, etc. To overcome this limitation Horvitz et al. (1967) and Greenberg et al. (1971) extended the work of Warner and by considering quantitative variables rather than qualitative ones. Himmelfarb and Edgel (1980) introduced the concept of an additive scrambling randomization response model, which utilized a scrambling variable having a known distribution in order to estimate the mean of a quantitative sensitive variable. Later Eichhorn and Hayre (1983) developed the concept of a multiplicative randomized response model for the same purpose. Chaudhuri and Stenger (1992) suggested a way to incorporate both ideas of additive and multiplicative models in a single model. Bouza (2009) investigated use of Chaudhuri and Stenger (1992) randomized response technique in ranked set sampling, while Bouza et al. (2017) extended the Saha‘s randomized response technique for ranked set sampling. These two investigations by Bouza motivated the authors to work on similar lines for Gjestvang and Singh (2009) model. This later model was originally used by Gjestvang and Singh (2006) for estimating the proportion of a sensitive characteristic, further detail can be found in Singh (2020).
In the next section, we propose a new estimator of the population mean of a sensitive variable, then show the unbiasedness of the proposed estimator and derive the variance expression.
Top2. Proposed Estimator
Consider a population Ω of N units with the values of the stigmatizing variable Y as Y1,Y2,…,YN. Then our aim is to estimate the population mean 
of the sensitive variable Y defined as:
(2.1)It may be worth pointing out that Ranked Set Sampling (RSS) was first introduced by McIntyre (1952). Bouza (2009) considered selecting a sample s of n people from a population Ω by using Ranked Set Sampling. Let r be the overall number of replicates and then let m×m be the cumulative proportion of individuals specified for ranking through every replicate so that n=mr. One replication of ranked set sampling cycle is explained as follows. During the first step, choose a simple random sample of m individuals, ranking them by judgement assessment as 
; over the next session, designate the next sample of individuals a random sample of m individuals, ranking these again based on a judgement assessment as 
; and finally in the mth cycle, then choose a simple random sample of m individuals, classify these again as 
. Table 1 provides a visual cycle analysis of such a RSS process.
Table 1. Cycle of ranked set sampling
| y(11) | y(12) | ⋯ | y(1m) |
| y(21) | y(22) | ⋯ | y(2m) |
| ⋮ | ⋮ | | ⋮ |
| y(m1) | y(m2) | ⋯ | y(mm) |