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In this paper, we extend the optional randomized response technique (ORRT) developed by Chaudhuri and Mukerjee [Optionally randomized response techniques. Bull. Calcutta Statist. Assoc. 1985;34:225–230; Randomized response: theory and techniques. New York: Marcel Dekker, Inc.; 1988] to the situation of estimating the proportion of two sensitive characteristics and their overlap. Lee, Sedory and Singh [Estimating at least seven measures of qualitative variables from a single sample using randomized response technique. Stat Prob Lett. 2013;83(1):399–409; Estimation of odds ratio, attributable risk, relative risk, correlation coefficient and other parameters using randomized response techniques. Behaviormetrika. 2021;48:371–392.] have shown that their crossed model performs better than their simple model from an efficiency point of views. Here we investigated a further improvement in the crossed model along the lines of Chaudhuri and Mukerjee [Optionally randomized response techniques. Bull. Calcutta Statist. Assoc. 1985;34:225–230; Randomized response: theory and techniques. New York: Marcel Dekker, Inc.; 1988]. New unbiased estimators are proposed, their variance expressions are derived and estimators of variances are suggested. Lastly, we carry out a simulation study to investigate the behaviour of the proposed estimators with respect to their competitors. [ABSTRACT FROM AUTHOR]