1. Introduction

Obtaining information about a stigmatized (induced abortion, use of drugs, tax evasion, etc.) attribute rampant in a human population is a complicated issue. Direct interrogating approach generally lead to doctoring of the true answers. The reason may be fear of social disgrace or counter attacks. But due to socioeconomic reasons, information about prevalence of such attributes in the population becomes essential. Warner [22] introduced an ingenious method of survey to obtain information about stigmatized attributes by guaranteeing confidentiality to the respondents. Numerous developments and improvements on Warner’s Randomized Response Technique have been suggested by many researchers. Greenberg et al. [9], Folsom et al. [8], Christofides [7], Mangat [16], Kim and Warde [15], Adebola and Adepetun [1], Adebola and Adepetun [2], Adebola and Adepetun [3], Adepetun and Adebola [4] are some of the many to be mentioned.

At times, prior information about the unknown parameter may be available and can be used along with the sample information for the determination of that unknown parameter. This is called the Bayesian approach of estimation. The work on Bayesian analysis of randomized response techniques is not very enormous. Nonetheless, attempts have been made on the Bayesian analysis of randomized response techniques. Winkler and Franklin [23], Spurrier and Padgett [20], O’Hagan [18], Oh [19], Migon and Tachibana [17], and Unnikrishnan and Kunte [21], Barabesi and Marcheselli [5, 6], Hussain and Shabbir [10, 11], Hussain et al [13], Hussain and Shabbir [12] and Kim et al. [14] are the major references on the Bayesian analysis of the Randomized Response Techniques.

2. Presentation of the Existing Technique

Hussain and Shabbir [12] in their paper presented a Bayesian analysis to the Randomized Response Technique (RRT) proposed by Kim and Warde [15] using a simple beta prior distribution to estimate the population proportion of respondents possessing stigmatized attribute.

Let the simple beta prior be defined as follows

(2.1)

where are the shape parameters of the distribution and is the population proportion of respondents possessing the stigmatized attribute.

By letting denotes the total number of the yes response in a sample of size n drawn from the population with simple random sampling with replacement (srswr). The conditional distribution of given is

(2.2)

where is the probability of “yes response” in a sample of size n and and are the pre-assigned probabilities respectively.

Then

Let

Using binomial series expansion

(2.3)

The joint density function of and X was derived as follows

(2.4)

The marginal probability density function was found using

(2.5)

Thus, the posterior distribution of given X was

(2.6)

Under the Square error loss, the Bayes estimator i.e the posterior mean was found using

(2.7)

The bias as well as the Mean Square Error (MSE) of corresponding to the sample of size n was given by

(2.8)

(2.9)

3. Presentation of the Proposed Techniques

In this section, we present a Bayesian analysis to Kim and Warde [15] Randomized Response Technique using both Kumaraswamy and Generalised Beta prior distributions as our alternative prior distributions in addition to the simple Beta prior distribution used by Hussain and Shabbir [12].

3.1. Estimation of Using Kumaraswamy Prior

The Kumaraswamy prior distribution of is given as

(3.1)

The joint density function of and X with Kumaraswamy Prior is as follows

(3.2)

The marginal probability density function is found using

Recall that

Then

(3.3)

The posterior distribution is

Thus the posterior mean is

(3.4)

The bias as well as Mean Square Error (MSE) of is computed as

(3.5)

(3.6)

3.2. Estimation of Using Generalised Beta Prior

The Generalised Beta prior is defined as

(3.7)

where are the shape parameters of the prior distribution as given in formula (3.7)

We recall from Binomial series expansion that

So that

The joint density function of and X with generalized beta prior is

which simplify to

(3.8)

where

The marginal probability density function is

(3.9)

Thus, the posterior distribution of given X is

The posterior mean which is the Bayes estimator is found by using

(3.10)

The bias of is

(3.11)

The Mean Square Error (MSE) of is

(3.12)

4. Presentation and Comparison of Results

In this section, we present as well as compare our results with the existing Hussain and Shabbir [12] under the same values of parameters in the estimators at different sample sizes. In order to overcome the computational difficulties and generate these results, we have written computer programs using R-statistical software. To save spaces, we present few results in tables and figures as follows:

Table 4.1a. Table showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Table 4.1b. Table showing the Absolute Bias for Kim and Warde [15] RRT at

Figure 4.1a. Graph showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Figure 4.1b. Graph showing the Absolute Bias for Kim and Warde [15] RRT at

Figure 4.2a. Graph showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Figure 4.2b. Graph showing the Absolute Bias for Kim and Warde [15] RRT at

Table 4.2a. Table showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Table 4.2b. Table showing the Absolute Bias for Kim and Warde [15] RRT at

Table 4.3a. Table showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Table 4.3b. Table showing the Absolute Bias for Kim and Warde [15] RRT at

Figure 4.3a. Graph showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Figure 4.3b. Graph showing the Absolute Bias for Kim and Warde [15] RRT at

From the results presented in the above tables and figures 4.1a to 4.3b, we can deduce that the proposed Bayesian estimators obtained using alternative priors are more efficient than the usual Bayesian estimator when a simple beta prior is used for obtaining high response from respondents with respect to the stigmatized attribute for small as well as moderate sample sizes respectively.

Table 4.4a. Table showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Table 4.4b. Table showing the Absolute Bias for Kim and Warde [15] RRT at

Figure 4.4a. Graph showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Figure 4.4b. Graph showing the Absolute Bias for Kim and Warde [15] RRT at

Table 4.5a. Table showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Table 4.5b. Table showing the Absolute Bias for Kim and Warde [15] RRT at

Figure 4.5a. Graph showing the Mean Square Errors (MSEs) for Kim and Warde [15] RRT at

Figure 4.5b. Graph showing the Absolute Bias for Kim and Warde [15] RRT at

From the results of tables and figures 4.4a to 4.5b, we observed the reverse case for the performances of the proposed Bayesian estimators in obtaining response from respondents possessing stigmatized attribute for large sample sizes. Hence, the proposed Bayesian estimators were not suitable in this case.

5. Conclusions

We have presented the Bayesian estimation of the population proportion when the data were gathered through the Kim and Warde [15] Randomized Response Technique using both Kumaraswamy (KUMA) and Generalised (GLS) Beta priors as our alternative prior distributions in addition to simple Beta prior distribution used by Hussain and Shabbir [12]. We presented our results in tables and figures for some selected values of the design parameters and population proportion. We observed that for relatively small sample as well as moderate sample size, the proposed Bayesian estimators performed significantly better than that of Hussain and Shabbir [12]. However, the reverse was the case for very large sample sizes.