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Benchmarking a triplet of official estimates

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Abstract

The publication of official statistics at different levels of aggregation requires a benchmarking step. Difficulties arise when a benchmarking method needs to be applied to a triplet of related estimates, at multiple stages of aggregation. For ratios of totals, external benchmarking constraints for the triplet (numerator, denominator, ratio) are that the weighted sum of denominator/numerator/ratio estimates equals to a constant. The benchmarking weight, applied to the ratio estimates, is a function of the denominator estimates. For example, the United States Department of Agriculture’s National Agricultural Statistics Service’s county-level, end-of-season acreage, production and yield estimates need to aggregate to the corresponding agricultural statistics district-level estimates, which also need to aggregate to the corresponding prepublished state-level values. Moreover, the definition of yield, as the ratio of production to harvested acreage, needs to hold at the county level, the agricultural statistics district level and the state level. We discuss different methods of applying benchmarking constraints to a triplet (numerator, denominator, ratio), at multiple stages of aggregation, where the denominator and the ratio are modeled and the numerator is derived. County-level and agricultural statistics district-level, end-of-season acreage, production and yield estimates are constructed and compared using the different methods. Results are illustrated for 2014 corn and soybean in Indiana, Iowa and Illinois.

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Acknowledgements

The findings and conclusions in this preliminary publication have not been formally disseminated by the U.S. Department of Agriculture and should not be construed to represent any agency determination or policy. This research was supported in part by the intramural research program of the U.S. Department of Agriculture, National Agriculture Statistics Service. Dr. Nandram’s work was supported by a Grant from the Simons Foundation (#353953, Balgobin Nandram). The authors thank the reviewers for their comments and suggestions. The authors are grateful to Wendy Barboza, Valbona Bejleri, Dave Johnson and Linda Young from the National Agricultural Statistics Service.

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Correspondence to Andreea L. Erciulescu.

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Handling Editor: Pierre Dutilleul.

Appendices

Appendix A

A1. Fixed benchmarking weights

For a ratio benchmarking adjustment, the R11 subarea-level numerator estimates satisfy the benchmarking constraint. The proof is as follows

$$\begin{aligned} \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T1,B}= & {} \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T2,B} {\hat{\theta }}_{j}^{R,B}, \nonumber \\= & {} \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T2,B} K^{-1} \sum \limits _{k=1}^{K} {\theta }_{jk}^{R}, \hbox { by } {(6)}\nonumber \\= & {} \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T2,B} K^{-1} \sum \limits _{k=1}^{K} \theta _{jk}^{R, no adj} a_{R} \left( \sum \limits _{j=1}^{n_{c}} w_{jk}^{R} \theta _{jk}^{R, no adj}\right) ^{-1}, \hbox { by } {(4)}\nonumber \\= & {} a_{R} K^{-1} \sum \limits _{k=1}^{K} \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T2,B} \theta _{jk}^{R, no adj} \left( \sum \limits _{j=1}^{n_{c}} w_{jk}^{R} \theta _{jk}^{R, no adj}\right) ^{-1}, \nonumber \\&\hbox { since } a_R \hbox { is constant} \nonumber \\= & {} a_{R} K^{-1} \sum \limits _{k=1}^{K} \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T2,B} \theta _{jk}^{R, no adj} \left( \sum \limits _{j=1}^{n_{c}} a_{T2}^{-1} {\hat{\theta }}_{j}^{T2,B} \theta _{jk}^{R, no adj}\right) ^{-1}, \nonumber \\&\hbox { by definition of } w_{jk}^{R} \nonumber \\= & {} a_{R} a_{T2} K^{-1} \sum \limits _{k=1}^{K} \left( \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T2,B} \theta _{jk}^{R, no adj} \right) \left( \sum \limits _{j=1}^{n_{c}} {\hat{\theta }}_{j}^{T2,B} \theta _{jk}^{R, no adj}\right) ^{-1} , \nonumber \\&\hbox { since } a_{T2} \hbox { is constant} \nonumber \\= & {} a_{R} a_{T2} = a_{T1}. \end{aligned}$$

Similarly, the R11 area-level numerator estimates satisfy the benchmarking constraints. The proof follows immediately,

$$\begin{aligned} \sum \limits _{i=1}^m {\hat{\theta }}_{i}^{T1,B}= & {} \sum \limits _{i=1}^m {\hat{\theta }}_{i}^{T2,B} {\hat{\theta }}_{i}^{R,B}, \nonumber \\= & {} \sum \limits _{i=1}^m K^{-1} \sum \limits _{k=1}^{K} {\hat{\theta }}_{i}^{T2,B} {\theta }_{ik}^{R}, \hbox { by } {(6)}\nonumber \\= & {} \sum \limits _{i=1}^m K^{-1} \sum \limits _{k=1}^{K} a_{T2} w_i^R {\theta }_{ik}^{R}, \hbox { by definition of } w_{ik}^{R}\nonumber \\= & {} a_{T2} a_{R}= a_{T1}. \end{aligned}$$

The ratio estimator for the set of all subareas is the weighted sum of the subarea-level ratio estimators,

$$\begin{aligned} \sum \limits _{j=1}^{n_{c}} w_j^R {\hat{\theta }}_{j}^{R,B}= & {} \sum \limits _{j=1}^{n_{c}} w_j^R K^{-1} \sum \limits _{k=1}^{K} {\theta }_{jk}^{R}, \hbox { by } {(6)}\nonumber \\= & {} K^{-1} \sum \limits _{k=1}^{K} \sum \limits _{j=1}^{n_{c}} w_j^R {\theta }_{jk}^{R}, \hbox {exchanging summations}\nonumber \\= & {} K^{-1} \sum \limits _{k=1}^{K} \sum \limits _{j=1}^{n_{c}} w_{j}^R {\theta }_{jk}^{R, no adj} a_{R} \left( \sum \limits _{j=1}^{n_{c}} w_{j}^R \theta _{jk}^{R, no adj}\right) ^{-1}, \hbox { by } {(4)}\nonumber \\= & {} K^{-1} \sum \limits _{k=1}^{K} a_{R} ,\hbox { since } a_R \hbox { is constant} \nonumber \\= & {} a_{R} . \end{aligned}$$

A2. Random benchmarking weights

Note that, from the derivation of production total estimates, the benchmarking weights for the two totals are equal. Hence, \(w_j^{T1} = w_j^{T2} = w_{jk}^{T2},\) for all \(k=1,\ldots ,K\).

For a ratio benchmarking adjustment, the R12 subarea-level numerator estimates satisfy the benchmarking constraint. The proof is as follows

$$\begin{aligned} \sum \limits _{j=1}^{n_{c}} w_j^{T1} {\hat{\theta }}_{j}^{T1,B}= & {} \sum \limits _{j=1}^{n_{c}} w_j^{T1} K^{-1} \sum \limits _{k=1}^{K} {\theta }_{jk}^{T1}\nonumber \\= & {} \sum \limits _{j=1}^{n_{c}} w_j^{T1} K^{-1} \sum \limits _{k=1}^{K} {\theta }_{jk}^{T2} {\theta }_{jk}^{R}, \hbox { by } {(13)}\nonumber \\= & {} \sum \limits _{j=1}^{n_{c}} w_j^{T1} K^{-1} \sum \limits _{k=1}^{K} \sum \limits _{j=1}^{n_{c}} {\theta }_{jk}^{T2} \theta _{jk}^{R, no adj} a_{R} \left( \sum \limits _{j=1}^{n_{c}} w_{jk}^{R} \theta _{jk}^{R, no adj}\right) ^{-1} , \nonumber \\&\hbox { by } {(4)}\nonumber \\= & {} a_{R} K^{-1} \sum \limits _{k=1}^{K} \sum \limits _{j=1}^{n_{c}} w_j^{T1} a_{T2} w_{jk}^{R} \left( w_{jk}^{T2}\right) ^{-1} \theta _{jk}^{R, no adj} \nonumber \\&\left( \sum \limits _{j=1}^{n_{c}} w_{jk}^{R} \theta _{jk}^{R, no adj}\right) ^{-1} , \hbox { since } a_R \hbox { is constant and } \nonumber \\&\hbox { by definition of } w_{jk}^{R} \nonumber \\= & {} a_{R} a_{T2} K^{-1} \sum \limits _{k=1}^{K} \left( \sum \limits _{j=1}^{n_{c}} w_j^{T1}\left( w_{jk}^{T2}\right) ^{-1} w_{jk}^{R} \theta _{jk}^{R, no adj} \right) \nonumber \\&\left( \sum \limits _{j=1}^{n_{c}} w_{jk}^{R} \theta _{jk}^{R, no adj}\right) ^{-1} , \hbox { since } a_{T2} \hbox { is constant} \nonumber \\= & {} a_{R} a_{T2} K^{-1} \sum \limits _{k=1}^{K} \left( \sum \limits _{j=1}^{n_{c}} w_{jk}^{R} \theta _{jk}^{R, no adj} \right) \left( \sum \limits _{j=1}^{n_{c}} w_{jk}^{R} \theta _{jk}^{R, no adj}\right) ^{-1} , \nonumber \\&\hbox { since } w_j^{T1} = w_{jk}^{T2}, \hbox { for all } k=1,\ldots ,K \nonumber \\= & {} a_{R} a_{T2} = a_{T1}. \end{aligned}$$

The ratio estimator for the set of all subareas is the weighted sum of the subarea-level ratio estimators,

$$\begin{aligned} \sum \limits _{j=1}^{n_{c}} w_j^R {\hat{\theta }}_{j}^{R,B}= & {} \sum \limits _{j=1}^{n_{c}} w_j^R K^{-1} \sum \limits _{k=1}^{K} {\theta }_{jk}^{R}, \hbox { by } {(8)}\nonumber \\= & {} \sum \limits _{j=1}^{n_{c}} \left( K^{-1} \sum \limits _{k^{\prime }} w_{jk^{\prime }}^R\right) \left( K^{-1} \sum \limits _{k=1}^{K} {\theta }_{jk}^{R}\right) , \nonumber \\&\hbox { since } K^{-1} \sum \limits _{k^{\prime }} w_{jk^{\prime }}^R = \left( a_{T2}\right) ^{-1}{\hat{\theta }}_{j}^{T2,B} = w_j^R\nonumber \\\ne & {} K^{-2} \sum \limits _{k=1}^{K} \sum \limits _{k=1}^{K} \sum \limits _{j=1}^{n_{c}} w_{jk}^R {\theta }_{jk}^{R}\nonumber \\= & {} K^{-2} \sum \limits _{k=1}^{K} \sum \limits _{k=1}^{K} a_{R} = a_{R} . \end{aligned}$$

The subarea-level numerator total estimator is

$$\begin{aligned} {\hat{\theta }}_{j}^{T1,B} = K^{-1} \sum \limits _{k=1}^{K} \theta _{jk}^{T1}= & {} K^{-1} \sum \limits _{k=1}^{K} \theta _{jk}^{T2} \theta _{jk}^{R}, \hbox { by } {(13)}\nonumber \\\ne & {} \left( K^{-1} \sum \limits _{k^{\prime }}^{K} \theta _{jk^{\prime }}^{T2} \right) \left( K^{-1} \sum \limits _{k=1}^{K} \theta _{jk}^{R} \right) . \end{aligned}$$

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Erciulescu, A.L., Cruze, N.B. & Nandram, B. Benchmarking a triplet of official estimates. Environ Ecol Stat 25, 523–547 (2018). https://doi.org/10.1007/s10651-018-0416-4

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