Multiple frame agricultural surveys. v. 1: Current surveys based on area and list sampling methods. (FAO Statistical Development Series - 7) (1996)
 PART III: THE MULTIPLE FRAME SURVEY DESIGN
 CHAPTER 10 - MULTIPLE FRAME ESTIMATORS AND THE LIST OF SPECIAL HOLDINGS
 10.1 Multiple Frame Estimators 10.2 Construction of the Complementary List of Special Holdings 10.3 Data Collection for the List of Special Holdings 10.4 Examples of Multiple Frame Estimators and their Variances

### 10.1 Multiple Frame Estimators

A multiple frame estimator is the sum of an area sample estimator (using data collected in the sample segments) plus the total data collected from the special list of holdings. If the list is sampled, the multiple frame estimator is the sum of the estimators from both frames, the area sample frame and the list of special holdings. As already mentioned, for variables with a distribution such that a large percentage of the total is concentrated in a relatively small number of holdings, it is often convenient to use multiple frame estimators in order to improve the precision of the direct area sample estimators. Standard techniques are used to make adjustments for missing data.

The area sample estimator should not consider the tracts corresponding to holdings of the list of special holdings, in order to avoid duplications. Therefore, all holdings with tracts included in the sample segments should be compared with those of the special list of holdings, in order to eliminate duplications from the area sample.

For a survey variable, if the list of special holdings is completely enumerated, then the variance of the multiple frame estimator is equal to the variance of the area sample estimator considered. If the list is sampled, the variance of the list sample is added to the area sample variance to arrive at the total multiple frame variance.

If the list of special holdings is completely enumerated, the variance of the multiple frame estimator is equal to the variance of the area sample estimator, but the CV of the multiple frame estimator will always be less than or equal to the CV of the area sample estimator. Multiple frame surveys obtain therefore more precise estimates than area sample surveys for those variables partially accounted with the list of special holdings.