Abstract
In this paper some properties and analytic expressions regarding the Poisson lognormal distribution such as moments, maximum likelihood function and related derivatives are discussed. The author provides a sharp approximation of the integrals related to the Poisson lognormal probabilities and analyzes the choice of the initial values in the fitting procedure. Based on these he describes a new procedure for carrying out the maximum likelihood fitting of the truncated Poisson lognormal distribution. The method and results are illustrated on real data. The computer program for calculations is freely available.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bulmer MG (1974). On fitting the Poisson lognormal distribution to species-abundance data. Biometrics 30: 101–110
Corbet AS (1942). The distribution of butterflies in the Malay Peninsula. Proc Roy Entomol Soc Lond (A) 16: 101–116
Dennis B and Patil GP (1979). Species abundance, diversity and environmental predictability. In: Grassle, JF, Patil, GP, Smith, W, and Taillie, C (eds) Ecological diversity in theory and practice, pp. International Co-operative Publishing House, Fairland
Engen S (2001). A dynamic and spatial model with migration generating the log-Gaussian field of population densities. Math Biosci 173: 85–102
Engen S and Lande R (1995). Population dynamic models generating the lognormal species abundance distribution. Math Biosci 132: 169–183
Fisher RA, Corbet AS and Williams CB (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. J Anim Ecol 12: 42–58
Grundy RM (1951). The expected frequencies in a sample of an animal population in which the abundances are lognormally distributed. Biometrika 38: 427–434
Kempton RA and Taylor LR (1974). Log-series and log-normal parameters as diversity discriminants for the lepidoptera. J Anim Ecol 43: 381–399
MacArthur RH (1957). On the relative abundance of bird species. Proc Nat Acad Sci USA 43: 293–295
Marquardt DW (1963). An algorithm for least squares estimation of nonlinear parameters. J Soc Ind Appl Math 11: 31–441
Press WH, Flannery BP, Teukolsky SA and Vetterling WT (2002). Numerical recipes in C: the art of scientific computing (2nd edn.). Cambridge University Press, Cambridge
Preston FW (1948). The commonness and rarity, of species. Ecology 29: 54–283
Shaban SA (1988). Poisson-lognormal distributions. In: Crow, EL and Shimizu, K (eds) Lognormal distributions: theory and applications, pp 195–210. Marcel Dekker, New York and Basel
Williams CB (1964). Patterns in the balance of nature. Academic Press, London
Williamson M and Gaston KJ (2005). The lognormal distribution is not an appropriate null hypothesis for the species-abundance distribution. J Anim Ecol 74: 409–422
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Izsák, R. Maximum likelihood fitting of the Poisson lognormal distribution. Environ Ecol Stat 15, 143–156 (2008). https://doi.org/10.1007/s10651-007-0044-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10651-007-0044-x