Estimates based on preliminary data from a specific subsample and from respondents not included in the subsample

Various approaches to obtaining estimates based on preliminary data are outlined. A case is then considered which frequently arises when selecting a subsample of units, the information for which is collected within a deadline that allows preliminary estimates to be produced. At the moment when these estimates have to be produced it often occurs that, although the collection of data on subsample units is still not complete, information is available on a set of units which does not belong to the sample selected for the production of the preliminary estimates. An estimation method is proposed which allows all the data available on a given date to be used to the full and the expression of the expectation and variance are derived. The proposal is based on two-phase sampling theory and on the hypothesis that the response mechanism is the result of random processes whose parameters can be suitably estimated. An empirical analysis of the performance of the estimator on the Italian Survey on building permits concludes the work.