General Restriction Estimator in Multisurvey Sampling

Abstract

In practice it may occur that two or more estimates of the same population parameter are available but that the estimates come from different surveys. A similar situation occurs when the estimates correspond to different parameters and the estimates have to obey a set of restrictions implied by the definitions of the underlying parameters or by theoretical considerations. For instance, quarterly data have to sum up to the yearly total, hence the sum of the four quarterly estimates should be equal to the yearly estimate if the estimates come from different surveys. At the macroeconomic level the aggregated demand for a good such as, for instance, energy must be equal to the aggregated supply of energy. Another example is that costs plus profits of all enterprises must be equal to their turnovers, whereas the sample observations might come from two or three (in)dependent surveys. Consider, for instance, the following situation with four samples that we will further elaborate on in Example 12.3. Let the variable x stand for the cost, y for the profit, and z for the turnover. Obviously, the population totals obey the restriction X + Y = Z. Furthermore, it is assumed that variable x is observed in samples 1 and 3, variable y in samples 2 and 3, while the variable z is only observed in sample 4.