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Two-parameter link functions, with applications to negative binomial, Weibull and quantile regression

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  • V. F. Miranda-Soberanis

    (Auckland University of Technology)

  • Thomas W. Yee

    (University of Auckland)

Abstract

One-parameter link functions play a fundamental role in regression via generalized linear modelling. This paper develops the general theory for two-parameter links in the very large class of vector generalized linear models by using total derivatives applied to a composite log-likelihood within the Fisher scoring/iteratively reweighted least squares algorithm. We solve a four-decade old problem with an interesting history as our first example: the canonical link for negative binomial regression. The remaining examples are fitting Weibull regression using both the mean and quantile directly compared to GAMLSS, and performing quantile regression based on the Gaussian distribution. Numerical examples based on real and simulated data are given. The methods described here are implemented by the VGAM and VGAMextra R packages, available on CRAN. Supplementary materials for this article are available online.

Suggested Citation

  • V. F. Miranda-Soberanis & Thomas W. Yee, 2023. "Two-parameter link functions, with applications to negative binomial, Weibull and quantile regression," Computational Statistics, Springer, vol. 38(3), pages 1463-1485, September.
    • Handle: RePEc:spr:compst:v:38:y:2023:i:3:d:10.1007_s00180-022-01279-4
    DOI: 10.1007/s00180-022-01279-4
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    References listed on IDEAS

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    1. Angela Noufaily & M. C. Jones, 2013. "Parametric quantile regression based on the generalized gamma distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 62(5), pages 723-740, November.
    2. R. Thompson & R. J. Baker, 1981. "Composite Link Functions in Generalized Linear Models," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 30(2), pages 125-131, June.
    3. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    4. V. F. Miranda-Soberanis & T. W. Yee, 2019. "New Link Functions for Distribution–Specific Quantile Regression Based on Vector Generalized Linear and Additive Models," Journal of Probability and Statistics, Hindawi, vol. 2019, pages 1-11, May.
    5. Yee, Thomas W., 2010. "The VGAM Package for Categorical Data Analysis," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 32(i10).
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