Environmental and Ecological Statistics

, Volume 5, Issue 4, pp 391–402 | Cite as

Non-parametric MLE for Poisson species abundance models allowing for heterogeneity between species

  • James L. Norris
  • Kenneth H Pollock


The proper management of an ecological population is greatly aided by solid information about its species' abundances. For the general heterogeneous Poisson species abundance setting, we develop the non-parametric mle for the entire probability model, namely for the total number N of species and the generating distribution F for the expected values of the species' abundances. Solid estimation of the entire probability model allows us to develop generator-based measures of ecological diversity and evenness which have inferences over similar regions. Also, our methods produce a solid goodness-of-fit test for our model as well as a likelihood ratio test to examine if there is heterogeneity in the expected values of the species' abundances. These estimates and tests are examined, in detail, in the paper. In particular, we apply our methods to important data from the National Breeding Bird Survey and discuss how our methods can also be easily applied to sweep net sampling data. To further examine our methods, we provide simulations for several illustrative situations.

bootstrap ecological diversity and evenness goodness-of-fit test likelihood ratio test mixture model 


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  1. Boulinier, T., Nichols, J.D., Pollock, K.H., Sauer, J.R. and Hines, J.E. (1996) On the use of closed models of capture-recapture to estimate bird species richness: a test with large scale data. Unpublished manuscript.Google Scholar
  2. Bulmer, M. (1974) On fitting the Poisson lognormal distribution of species-abundance data. Biometrics, 30, 101–10.Google Scholar
  3. Burnham, K.P. and Overton, W.S. (1979) Robust estimation of population size when capture probabilities vary among animals. Ecology, 60(5), 927–36.Google Scholar
  4. Bystrak, D. (1981) The North American Breeding Bird Survey. Studies in Avian Biology, 6, 34–41.Google Scholar
  5. Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum likelihood estimation from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.Google Scholar
  6. Efron, B. (1981) Non-parametric estimates of standard error: The jackknife, the bootstrap and other methods. Biometrika, 68, 589–99.Google Scholar
  7. Efron, B. and Thisted, R. (1976) Estimating the number of unseen species: how many words did Shakespeare know? Biometrika, 63(3), 435–47.Google Scholar
  8. Engen, S. (1978) Stochastic Abundance Models, Chapman and Hall, London.Google Scholar
  9. Fisher, R.A., Corbet, A.S. and Williams, C.B. (1943) The relation between the number of species and the number of individuals in a random sample from an animal population. Journal of Animal Ecology, 12, 42–58.Google Scholar
  10. Kempton, R.A. and Taylor, L.R. (1974) Log-series and log-normal parameters as diversity discriminants for the Lepidoptera. Journal of Animal Ecology, 43, 381–99.Google Scholar
  11. Lee, S. and Chao, A. (1994) Estimating population size via sample coverage for closed capture-recapture models. Biometrics, 50, 88–97.Google Scholar
  12. Lindsay, B.G. and Roeder, K. (1992) Residual diagnostics for mixture models. Journal of the American Statistical Association, 87, 785–94.Google Scholar
  13. Ludwig, J.A. and Reynolds, J.F. (1988) Statistical Ecology: A Primer on Methods and Computing, John Wiley and Sons, New York.Google Scholar
  14. Mehninick, E.F. (1964) A comparison of some species individuals diversity indices applied to samples of field insects. Ecology, 45, 859–61.Google Scholar
  15. Norris, J.L. and Pollock, K.P. (1996) Non-parametric MLE under two closed capture-recapture models with heterogeneity. Biometrics, 52, 639–49.Google Scholar
  16. Preston, F.W. (1948) The commonness and rarity of species, Ecology, 29, 254–83.Google Scholar
  17. Robbins, C.S., Bystrak, D. and Geissler, P.H. (1986) The breeding bird survey: its first fifteen years, 1965–1979. U.S. Fish and Wildlife Service, Resource Publication 157, Washington, D.C., USA.Google Scholar
  18. Romano, J.P. (1988) A bootstrap revival of some non-parametric distance tests. Journal of the American Statistical Association, 83, 698–708.Google Scholar
  19. Shannon, C E. and Weaver, W. (1949) The Mathematical Theory of Communication, University Illinois Press, Urbana, Il.Google Scholar
  20. Sichel, H. (1991) Modelling species-abundance frequencies and species-individual functions with the generalized inverse Gaussian-Poisson distribution law. Unpublished manuscript, University of the Eitwatersrand, South Africa, Department of Statistics and Actuarial Science.Google Scholar
  21. Simpson, E.H. (1949) Measurement of diversity. Nature, 163, 688.Google Scholar
  22. Taylor, L.R., Kempton, R.A. and Woiwod, I.P. (1976) Diversity statistics and the log-series model. Journal of Animal Ecology, 45, 255–72.Google Scholar
  23. Titterington, D. M. (1991) Some recent research in the analysis of mixture distributions. Statistics, 21, 619–41.Google Scholar
  24. Titterington, D.M., Smith, A.F.M. and Makov, U.E. (1985) Statistical Analysis of Finite Mixture Distributions, John Wiley and Sons, Chichester, England.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • James L. Norris
    • 1
  • Kenneth H Pollock
    • 2
  1. 1.Department of Mathematics and Computer ScienceWake Forest UniversityWinston-SalemUSA
  2. 2.Biostatistics ProgramNorth Carolina State UniversityRaleighUSA

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