Parameter Estimation for Truncated Exponential Families

  • Bradford R. Crain
  • Loren Cobb
Part of the NATO Advanced study Institutes Series book series (ASIC, volume 79)


The multiparameter exponential families of probability density functions include many commonly used densities such as the normal, gamma, and beta, and many others besides. We consider the general parameter estimation problem for these families, given observations that are restricted (i.e. truncated) to the closed interval [a,b]. We show how to find maximum likelihood estimators using Newton-Raphson iteration, but observe that in general this technique requires numerical integration within each iteration. Then we give a new non-iterative method for obtaining the desired estimators for the parameter vector. An example using the generalized Inverse Gaussian is given.

Key Words

Exponential families maximum likelihood non- iterative estimation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York.zbMATHGoogle Scholar
  2. Crain, B.R. (1976). Exponential models, maximum likelihood estimation, and the Haar condition. Journal of the American Statistical Association, 71, 737–740.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Lehman, E.L. (1959). Testing Statistical Hypotheses. Wiley, New York.Google Scholar
  4. Wilks, S.S. (1962). Mathematical Statistics. Wiley, New York.zbMATHGoogle Scholar
  5. Yang, G.L., Chen, T.C. (1978). On statistical methods of neuronal spike-train analysis. Mathematical Biosciences, 38, 1–34.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Zangwill, W.I. (1969). Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • Bradford R. Crain
    • 1
  • Loren Cobb
    • 2
  1. 1.Department of MathematicsPortland State UniversityPortlandUSA
  2. 2.Department of BiometryMedical University of South CarolinaCharlestonUSA

Personalised recommendations