Probability estimation when some observations are grouped

This paper considers the use of additional questions for decreasing survey non-response rates and an approach for estimating a probability based on the results obtained. In a survey, the respondents are asked to answer an original question and follow-up questions, where the answers for the follow-up questions are grouped answers for the original question. For example, respondents are asked to provide an exact number of incidents, but in cases of 'Do not know' or 'Refuse' responses, they are subsequently asked to pick an answer from a less specific categorical scale. The new estimator obtains smaller variance asymptotically and does not depend on a distribution family. This method is applied to income questions in a survey regarding injury prevention and behaviours. Another application is survey data on intimate partner violence, where some amendments were applied for incorporating post-stratification weights and for using non-random grouping. For additional illustration, an example of parameter estimation on artificially generated data is presented.