The Use of Calibration Weighting for Variance Estimation Under Systematic Sampling: Applications to Forest Cover Assessment

  • Lorenzo FattoriniEmail author
  • Timothy G. Gregoire
  • Sara Trentini


The purpose of this note is to propose a variance estimator under non-measurable designs that exploits the existence of an auxiliary variable well correlated with the survey variable of interest. Under non-measurable designs, the Sen–Yates–Grundy variance estimator generates a downward bias that can be reduced using a calibration weighting based on the auxiliary variable. Conditions of approximate unbiasedness for the resulting calibration estimator are given. The application to systematic sampling is considered. The proposal proves to be effective for estimating the variance of the forest cover estimator in remote sensing-based surveys, owing to the strong correlation between the reference data, available from a systematic sample, and the satellite map data, available for the whole population and hence exploited as an auxiliary variable. Supplementary materials accompanying this paper appear online.


Calibration estimation Forest cover estimation Non-measurable designs Pseudo-designs 

Supplementary material

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  1. Bartolucci F, Montanari GE (2006) A new class of unbiased estimator for the variance of the systematic sample mean. Journal of Statistical Planning and Inference 136:1512–1525.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Corona P, Fattorini L, Pagliarella MC (2015) Sampling strategies for estimating forest cover from remote sensing-based two-stage inventories. Forest Ecosystems 2:18CrossRefGoogle Scholar
  3. Fattorini L (2006) Applying the Horvitz-Thompson criterion in complex designs: a computer-intensive perspective for estimating inclusion probabilities. Biometrika 93:269–278MathSciNetCrossRefzbMATHGoogle Scholar
  4. Grafström A (2012) Spatial correlated Poisson sampling. Journal of Statistical Planning and Inference 142:139–147MathSciNetCrossRefzbMATHGoogle Scholar
  5. Grafström A, Tillé Y (2013) Doubly balanced spatial sampling with spreading and restitution of auxiliary totals. Environmetrics 24:120–131MathSciNetCrossRefGoogle Scholar
  6. Grafström A, Lundström NLP, Schelin L (2012) Spatially Balanced Sampling through the Pivotal Method. Biometrics 68:514–520MathSciNetCrossRefzbMATHGoogle Scholar
  7. Gregoire TG, Næsset E, McRoberts RE, Ståhl G, Andersen HE, Ene L, Nelson R (2016) Statistical rigor in lidar-assisted estimation of aboveground forest biomass. Remote Sensing of Environment 173: 98–106.CrossRefGoogle Scholar
  8. Gregoire TG, Valentine HT (2008) Sampling Strategies for Natural Resources and the Environment. Chapman & Hall/CRC.Google Scholar
  9. Hansen MC, Potapov PV, Moore R, Hancher M, Turubanova SA, Tyukavina A, Thau D, Stehman SV, Goetz SJ, Loveland TR, Kommareddy A, Egorov A, Chini L, Justice CO, Townshend JRG (2013) High-Resolution Global Maps of 21st-Century Forest Cover Change. Science 342:850–853CrossRefGoogle Scholar
  10. Hasel AA (1938) Sampling error in timber surveys. Journal of Agricultural Research 57: 713–736.Google Scholar
  11. Hedayat AS, Sinha BK (1991) Design and Inference in Finite Population Sampling. Wiley, New YorkzbMATHGoogle Scholar
  12. Langsæter A (1926) Om beregning av middelfeilen ved regelmessige linjetakseringer (About calculation of standard error for systematic strip survey). Meddelelser fra Det Norske Skog-forsøksvesen 2: 5–43.Google Scholar
  13. Matérn B (1960) Spatial variation: stochastic models and their application to some problems in forest surveys and other sampling investigations. Technical Report 5, Meddelanden från Statens Skogsforskninstitut, Stockholm, Sweden.Google Scholar
  14. Montanari GE, Bartolucci F (1998) On estimating the variance of the systematic sample mean. Journal of the Italian Statistical Society 7:185–196.CrossRefGoogle Scholar
  15. Neyman J (1934) On two different aspects of the representative method: the method of stratified sampling and the method of purposive selection. Journal of the Royal Statistical Society 97: 558–606.CrossRefzbMATHGoogle Scholar
  16. Opsomer JD, Francisco-Fernandez M, Li X (2012) Model-based non-parametric variance estimation for systematic sampling. Scandinavian Journal of Statistics 39: 528–542.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Osborne JG (1942) Sampling errors of systematic and random surveys of cover type areas. Journal of the American Statistical Association 37: 256–264.CrossRefGoogle Scholar
  18. Stevens DJ, Olsen AR (2004) Spatially balanced sampling of natural resources. Journal of the American Statistical Association 99:262–278MathSciNetCrossRefzbMATHGoogle Scholar
  19. Sannier C, Mc Roberts RE, Fichet LV, Makaga EMK (2014) Using regression estimator with Landsat data to estimate proportion forest cover and net proportion deforestation in Gabon. Remote Sensing of Environment 151:138–148CrossRefGoogle Scholar
  20. Särndal CE, Lundström S (2005) Estimation in Surveys with Nonresponse. Wiley, New York.CrossRefzbMATHGoogle Scholar
  21. Särndal CE, Swensson B, Wretman J (1992) Model Assisted Survey Sampling. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
  22. Tomppo LM, Gschwantner RE, McRoberts RE Eds (2010) National forest inventories: pathways for common reporting. Springer, HeidelbergGoogle Scholar
  23. UN-REDD (2013) National Forest Monitoring Systems: Monitoring and Measurement, Reporting and Verification (M & MRV) in the context of REDD+ Activities. FAO, RomeGoogle Scholar
  24. Wolter KM (1984) An investigation of some estimators of variance for systematic sampling. Journal of the American Statistical Association 79:781–790.MathSciNetCrossRefGoogle Scholar
  25. Wolter KM (2007) Introduction to Variance Estimation (\(2^{{\rm nd}}\) edn). Springer-Verlag, New York.Google Scholar

Copyright information

© International Biometric Society 2018

Authors and Affiliations

  • Lorenzo Fattorini
    • 1
    Email author
  • Timothy G. Gregoire
    • 2
  • Sara Trentini
    • 1
  1. 1.Department of Economics and StatisticsUniversity of SienaSienaItaly
  2. 2.School of Forestry and Environmental StudiesYale UniversityNew HavenUSA

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