Statistical Methods and Applications

, Volume 11, Issue 2, pp 153–160 | Cite as

Adjusting the bias of adaptive sampling estimators of spatial dispersion indexes by the δ-method

  • Tonio Di Battista
Statistical Methods


It is common practice to investigate the spatial dispersion in a community of discrete individuals (like animals or plants). Usually, the study area is partitioned into spatial units of equal size and then the relationship between the first two moments of the variable representing the number of individuals in each plot is investigated. When the points are spread over a very wide area so that the population density is low but many points are concentrated inside a few units, then a suitable sample method for estimating the first two moments is adaptive sampling. However, since the more common dispersion indexes are non linear function of the first two moments, the resulting estimators are biased for finite samples. Accordingly, a procedure to adjust bias is required for small samples. In this paper a δ-method evaluation of the bias is proposed and the asymptotic distribution of the bias-corrected estimators is provided. Finally, a simulation study is performed in order to investigate the performance of the proposed procedure.

Key words

Spatial dispersion indexes adaptive sampling bias reduction δ-method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bishop YMM, Fienberg SE, Holland PW (1975) Discrete Multivariate Analysis: Theory and Practice, The MIT Press, pp 486–502Google Scholar
  2. Christman MC (2000) A Review of Quadrat-Based Sampling of Rade, Geographically Clustered Populations.Journal of Agricultural, Biological, and Environmental Statistics, 5, 168–201MathSciNetCrossRefGoogle Scholar
  3. David FN, Moore PG (1954) Notes on contagious distributions in plant populations.Annals of Botany 18, 47–53Google Scholar
  4. Douglas JB (1975) Clustering and aggregation.Sankhya 37 B, 398–417Google Scholar
  5. Fisher RA, Thornton HG, Mackenzie WA (1922) The accuracy of the planting method of estimating the density of bacterial populations.Annals of Applied Biology 9, 325–359CrossRefGoogle Scholar
  6. Lloyd M (1967) Mean crowding.Journal of Animal Ecology 36, 1–30CrossRefGoogle Scholar
  7. Rao CR (1973) Linear Statistical Inference and its application, 2nd edn. Wiley, New YorkGoogle Scholar
  8. Ripley BD (1981)Spatial Statistics. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  9. Rogers A (1974)Statistical analysis of spatial dispersion: the quadrat method. Pion, LondonGoogle Scholar
  10. Seber GAF (1982) A review of estimating animal abundance.II International Statistical Review 60, 129–166Google Scholar
  11. Seber GAF, Thompson SK (1994) Environmental Adaptive Sampling. In: Patil GP, Rao CR (eds)Handbook of Statistics. Elsiever Science Amsterdam, 12, 201–220Google Scholar
  12. Shao J, Tu D (1995) The Jackknife and Bootstrap. Berlin, SpringerzbMATHGoogle Scholar
  13. Thompson SK (1990) Adaptive cluster sampling.Journal of the American Statistical Associations 85, 412, 1050–1058CrossRefGoogle Scholar
  14. Thompson SK, Ramsey FL, Seber GAF (1992) An adaptive procedure for sampling animal populations.Biometrics 48, 1195–1199CrossRefGoogle Scholar
  15. Thompson SK, Seber GAF (1996)Adaptive Sampling. Wiley, New YorkzbMATHGoogle Scholar
  16. Withers CS (1987) Bias Reduction by Taylor Series.Communications in Statistics-Theory and Methods 16, 2369–2383zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2002

Authors and Affiliations

  • Tonio Di Battista
    • 1
  1. 1.Dipartimento di Metodi Quantitativi e Teoria EconomicaUniversità degli Studi “G. D'Annunzio”PescaraItaly

Personalised recommendations