Computational Statistics

, Volume 21, Issue 3–4, pp 603–620

Semiparametric estimation of mean and variance functions for non-Gaussian data

Original Paper

Abstract

Flexible modelling of the response variance in regression is interesting for understanding the causes of variability in the responses, and is crucial for efficient estimation and correct inference for mean parameters. In this paper we describe methods for mean and variance estimation where the responses are modelled using the double exponential family of distributions and mean and dispersion parameters are described as an additive function of predictors. The additive terms in the model are represented by penalized splines. A simple and unified computational methodology is presented for carrying out the calculations required for Bayesian inference in this class of models based on an adaptive Metropolis algorithm. Application of the adaptive Metropolis algorithm is fully automatic and does not require any kind of pretuning runs. The methodology presented provides flexible methods for modelling heterogeneous Gaussian data, as well as overdispersed and underdispersed count data. Performance is considered in a variety of examples involving real and simulated data sets.

Keywords

Overdispersion modelling Double exponential models Generalized linear models Variance estimation 

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References

  1. Aerts M, Claeskens G (1997) Local polynomial estimation in multiparameter models. J Am Statist Assoc 92:1536–1545MATHMathSciNetCrossRefGoogle Scholar
  2. Besag J, Green PJ (1993) Spatial statistics and Bayesian computation (with discussion). J Roy Statist Soc B 16:395–407Google Scholar
  3. Carroll RJ, Ruppert D (1988) Transformations and weighting in regression. Chapman and Hall, LondonGoogle Scholar
  4. Carroll RJ, Ruppert D, Welsh AH (1998) Local estimating equations. J Am Statist Assoc 93:214–227MATHMathSciNetCrossRefGoogle Scholar
  5. Chan D, Kohn R, Nott D, Kirby C (2006) Locally-adaptive semiparametric estimation of the mean and variance functions in regression models. J Comp Graph Statist (to appear)Google Scholar
  6. Davidian M, Carroll RJ (1987) Variance function estimation. J Am Statist Assoc 82:1079–1091MATHMathSciNetCrossRefGoogle Scholar
  7. Davidian M, Carroll RJ (1988) A note on extended quasilikelihood. J Roy Statist Soc Ser B 50:74–82MathSciNetGoogle Scholar
  8. Dey D, Ravishanker N (2000) Bayesian approaches for overdispersion in generalized linear models. In: Dey DK, Ghosh SK, Mallick BK (eds)Generalized linear models: a Bayesian perspective. pp 73–90Google Scholar
  9. Efron B (1986) Double exponential families and their use in generalized linear regression. J Am Statist Assoc 81:709–721MATHMathSciNetCrossRefGoogle Scholar
  10. Faddy MJ (1997) Extended Poisson process modelling and analysis of count data. Biom J 39:431–440MATHGoogle Scholar
  11. Gelfand AE, Dey D, Peng F (1997) Overdispersed generalized linear models. J Statist Plan Inf 64:93–107MATHCrossRefGoogle Scholar
  12. Gelfand AE, Dalal SR (1990) A note on overdispersed exponential families. Biometrika 77:55–64MATHMathSciNetCrossRefGoogle Scholar
  13. Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian data analysis, 2nd edn. Chapman and Hall, LondonMATHGoogle Scholar
  14. Haario H, Saksman E, Tamminen J (2005) Componentwise adaptation for high dimensional MCMC. Comp Statist 20:265–274MATHMathSciNetGoogle Scholar
  15. Jorgensen B (1987) Exponential dispersion models (with discussion). J Roy Statist Soc Ser B 49:127–162MathSciNetGoogle Scholar
  16. Jorgensen B (1997) The theory of dispersion models. Chapman and Hall, LondonGoogle Scholar
  17. McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman and Hall, LondonMATHGoogle Scholar
  18. Moore DF, Tsiatis A (1991) Robust estimation of the variance in moment methods for extra-binomial and extra-Poisson variation. Biometrics 47:383–401MathSciNetCrossRefGoogle Scholar
  19. Nelder JA, Pregibon D (1987) An extended quasi-likelihood function. Biometrika 74:221–232MATHMathSciNetCrossRefGoogle Scholar
  20. Rigby RA, Stasinopoulos DM (2005) Generalized additive models for location, scale and shape. Appl Statist 52:507–554MathSciNetGoogle Scholar
  21. Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, CambridgeMATHGoogle Scholar
  22. Smyth GK (1989) Generalized linear models with varying dispersion. J Roy Statist Soc Ser B 51:47–60MathSciNetGoogle Scholar
  23. Smyth GK, Verbyla AP (1999) Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10:695–709CrossRefGoogle Scholar
  24. Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit (with discussion). J Roy Statist Soc Ser B 64:583–639MATHCrossRefGoogle Scholar
  25. Tweedie MCK (1947) Functions of a statistical variate with given means, with special reference to Laplace distributions. Proc Cambridge Phil Soc 49:41–49MATHMathSciNetCrossRefGoogle Scholar
  26. Weisberg S (1985) Applied Linear Regression. Wiley, New YorkMATHGoogle Scholar
  27. Yau P, Kohn R (2003) Estimation and variable selection in nonparametric heteroscedastic regression. Statist Comp 13:191–208MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of New South WalesSydneyAustralia

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