On efficiency of using distinct respondents in a randomized response survey

It is well known that the sample mean based on the distinct sample units in simple random sampling with replacement is more efficient than the sample mean based on all units selected including repetitions (Murthy 1967, pp. 65-66). Seth and Rao (1964) showed that the mean of the distinct units is less efficient than the sample mean in sampling without replacement under the same average sampling cost. Under Warner’s (1965) method of randomized response we compare simple random sampling without replacement and sampling with replacement when only the distinct number of units in the sample are considered.