In carrying out surveys involving sensitive characteristics, randomized response models have been considered among the best techniques since they provide the maximum privacy protection to the respondents and procure honest responses. Over the years, researchers have carried out studies on the estimation of proportions of the population possessing sensitive characteristics. However, there is a paucity of research studies that have addressed higher order interactions between these sensitive characters. In this article, we develop a new theory based on three proposed randomized response models which we name as: simple model, semi-crossed model, and fully crossed model. Twenty-one new unbiased estimators of seven parameters are introduced, their variance expressions are derived, and unbiased estimators of variances are developed. The three models are compared under various values of the parameters by computing the percent relative efficiency of one model over another model. The most efficient model is then applied to study the population proportions of three varieties of smoking habits among students, and their first- and second-order interactions. The last four sections (Ninth to Twelfth) are verifications of theoretical results using the Cramer–Rao lower bounds of variances for the developed 21 new estimators in randomized response sampling.

在进行涉及敏感特征的调查时，随机响应模型被认为是最好的技术之一，因为它们可以为受访者提供最大程度的隐私保护并获得诚实的响应。多年来，研究人员对具有敏感特征的人口比例进行了研究。但是，很少有研究涉及这些敏感字符之间的高阶相互作用。在本文中，我们基于三个提议的随机响应模型（分别称为简单模型，半交叉模型和全交叉模型）开发了一种新的理论。引入了七个新的七个参数的无偏估计量，推导了它们的方差表达式，并开发了方差的无偏估计量。通过计算一个模型相对于另一个模型的相对效率百分比，可以在各种参数值下比较这三个模型。然后，应用最有效的模型来研究学生中三种吸烟习惯的人口比例及其一阶和二阶交互作用。最后四个部分（第九至第十二个）是使用Cramer-Rao方差下限对已开发的21个新估计量在随机响应抽样中进行的理论结果验证。