Newton’s Method and Scoring

  • Kenneth Lange
Part of the Statistics for Biology and Health book series (SBH)

Abstract

This chapter explores some alternatives to maximum likelihood estimation by the EM algorithm. Newton’s method and scoring usually converge faster than the EM algorithm. However, the trade-offs of programming ease, numerical stability, and speed of convergence are complex, and statistical geneticists should be fluent in a variety of numerical optimization techniques for finding maximum likelihood estimates. Outside the realm of maximum likelihood, Bayesian procedures have much to offer in small to moderate-sized problems. For those uncomfortable with pulling prior distributions out of thin air, empirical Bayes procedures can be an appealing compromise between classical and Bayesian methods. This chapter illustrates some of these well-known themes in the context of allele frequency estimation and linkage analysis.

Keywords

Exponential Family Multinomial Distribution Dirichlet Distribution Observe Information Matrix Tandem Repeat Locus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bernardo JM (1976) Algorithm AS 103: psi (digamma) function. ApplStatist 25:315–317Google Scholar
  2. [2]
    Bradley EL (1973) The equivalence of maximum likelihood and weighted least squares estimates in the exponential family. J Amer Stat Assoc 68: 199–200MATHGoogle Scholar
  3. [3]
    Charnes A, Frome EL, Yu PL (1976) The equivalence of generalized least squares and maximum likelihood in the exponential family. J Amer Stat Assoc 71:169–171MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Conn AR, Gould NIM, Toint PL (1991) Convergence of quasi-Newton matrices generated by the symmetric rank one update. Math Prog 50:177–195MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Davidon WC (1959) Variable metric methods for minimization. AEC Research and Development Report ANL-5990, Argonne National LaboratoryCrossRefGoogle Scholar
  6. [6]
    Edwards A, Hammond HA, Jin L, Caskey CT, Chakraborty R (1992) Genetic variation at five trimeric and tetrameric tandem repeat loci in four human population groups. Genomics 12:241–253CrossRefGoogle Scholar
  7. [7]
    Efron B, Hinkley DV (1978) Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika 65:457–487MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Hille E (1959) Analytic Function Theory Vol1. Blaisdell Ginn, New YorkMATHGoogle Scholar
  9. [9]
    Jennrich RI, Moore RH (1975) Maximum likelihood estimation by means of nonlinear least squares. Proceedings of the Statistical Computing Section: American Statistical Association 57–65Google Scholar
  10. [10]
    Khalfan HF, Byrd RH, Schnabel RB (1993) A theoretical and experimental study of the symmetric rank-one update. SIAM J Optimization 3:1–24MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Kingman JFC (1993) Poisson Processes. Oxford University Press, OxfordMATHGoogle Scholar
  12. [12]
    Lange K (1995) Applications of the Dirichlet distribution to forensic match probabilities. Genetica 96:107–117CrossRefGoogle Scholar
  13. [13]
    Lee PM (1989) Bayesian Statistics: An Introduction. Edward Arnold, London.MATHGoogle Scholar
  14. [14]
    Miller KS (1987) Some Eclectic Matrix Theory. Robert E Krieger Publishing, Malabar, FLMATHGoogle Scholar
  15. [15]
    Mosimann JE (1962) On the compound multinomial distribution, the multivariate β-distribution, and correlations among proportions. Biometrika 49:65–82MATHMathSciNetGoogle Scholar
  16. [16]
    Ott J (1985) Analysis of Human Genetic Linkage. Johns Hopkins University Press, BaltimoreGoogle Scholar
  17. [17]
    Rao CR (1973) Linear Statistical Inference and its Applications, 2nd ed. Wiley, New YorkMATHCrossRefGoogle Scholar
  18. [18]
    Schneider BE (1978) Algorithm AS 121: trigamma function. Appl Statist 27:97–99CrossRefGoogle Scholar
  19. [19]
    Yasuda N (1968) Estimation of the inbreeding coefficient from phenotype frequencies by a method of maximum likelihood scoring. Biometrics 24:915–934CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Kenneth Lange
    • 1
  1. 1.Department of Biostatistics and MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations