Nonparametric estimation when income is reported in bands and at points

We show how to estimate kernel density functions of distributions in which some of the responses are provided in brackets, by inverse probability weighting. We consider two cases, one where the data are CAR and where the data are not CAR. We show how the selection probabilities can be estimated by means of the EM algorithm without specifying a parametric distribution function for the variable. A Monte Carlo experiment shows that this procedure estimates the selection parameters fairly precisely. We apply these techniques to earnings data from South Africa’s first post-apartheid nationally representative survey, the 1994 October Household Survey.