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Specification tests for the response distribution in generalized linear models

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Abstract

Goodness-of-fit tests are proposed for the case of independent observations coming from the same family of distributions but with different parameters. The most popular related context is that of generalized linear models (GLMs) where the mean of the distribution varies with regressors. In the proposed procedures, and based on suitable estimators of the parameters involved, the data are transformed to normality. Then any test for normality for i.i.d. data may be applied. The method suggested is in full generality as it may be applied to arbitrary laws with continuous or discrete distribution functions, provided that an efficient method of estimation exists for the parameters. We investigate by Monte Carlo the relative performance of classical tests based on the empirical distribution function, in comparison to a corresponding test which instead of the empirical distribution function, utilizes the empirical characteristic function. Standard measures of goodness-of-fit often used in the context of GLM are also included in the comparison. The paper concludes with several real-data examples.

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References

  • Chen G, Balakrishnan N (1995) A general purpose approximate goodness-of-fit test. J Qual Technol 27: 154–161

    Google Scholar 

  • Cox DR, Snell EJ (1968) A general definition of residuals (with discussion). J R Stat Soc B 30: 248–275

    MathSciNet  Google Scholar 

  • D’Agostino M, Stephens R (1986) Goodness-of-fit techniques. Marcel Dekker. Inc., New York

    MATH  Google Scholar 

  • Davison AC (2003) Statistical models. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Graph Stat 5: 236–244

    Google Scholar 

  • Epps TW, Pulley LB (1983) A test for normality based on the empirical characteristic function procedures. Biometrika 70: 723–726

    Article  MathSciNet  MATH  Google Scholar 

  • Faraway JJ (2006) Extending the Linear Model with R: generalized linear. Mixed effects and nonparametric regression models. Chapman & Hall/CRC

  • Faraway JJ (2011) Faraway: functions and datasets for books by Julian Faraway. http://cran.r-project.org/web/packages/faraway/index.html

  • Hallin M, Ingenbleek J-F (1983) The Swedish automobile portfolio in 1977. A statistical study. Scand Actuarial J 49–64

  • Hardin JW, Hilbe JM (2007) Generalized linear models and extensions, 2nd edn. Stata Press, College Station

    MATH  Google Scholar 

  • Henze N (1990) An approximation to the limit distribution of the Epps-Pulley test statistic for normality. Metrika 37: 7–18

    Article  MathSciNet  MATH  Google Scholar 

  • Hu B, Shao J (2008) Generalized linear model selection using R 2. J Stat Plan Infer 138: 3705–3712

    Article  MathSciNet  MATH  Google Scholar 

  • Loynes RM (1980) The Empirical distribution function of residuals from generalised regression. Ann Stat 8: 285–298

    Article  MathSciNet  MATH  Google Scholar 

  • McCullagh P, Nelder JA (1989) Generalized linear models. 2nd edn. Chapman and Hall, London

    MATH  Google Scholar 

  • Meintanis SG (2009) Goodness-of-fit testing by transforming to normality: comparison between classical and characteristic function-based methods. J Stat Comput Simul 79: 205–212

    Article  MathSciNet  MATH  Google Scholar 

  • Mittlböck M, Heinzl H (2002) Measures of explained variation in gamma regression models. Commun Stat Simul Comput 31: 61–73

    MATH  Google Scholar 

  • Paul SR, Deng D (2002) Score tests for goodness of fit of generalized linear models to sparse data. Sankhya 64: 179–191

    MathSciNet  MATH  Google Scholar 

  • R Development Core Team (2011) R: a language and environment for statistical computing. R Foundation for Statistical Computing. http://www.R-project.org

  • Shayib MA, Young DH (2002) Modified goodness of fit of tests in gamma regression. J Stat Comput Simul 33: 125–133

    Article  MathSciNet  Google Scholar 

  • Smyth G, with contributions from Hu Y, Dunn P, Phipson B (2011) statmod: Statistical Modeling. http://CRAN.R-project.org/package=statmod

  • Spinelli JJ, Lockhart RA, Stephens MA (2002) Tests for the response distribution in a Poisson regression model. J Statist Plan Infer 108: 137–154

    Article  MathSciNet  MATH  Google Scholar 

  • Thode HC (2002) Testing for normality. Marcel Dekker. Inc., New York

    Book  MATH  Google Scholar 

  • Venables WN, Ripley BD (2002) Modern applied statistics with S, 4th edn. Springer, New York

    MATH  Google Scholar 

  • Wood GR (2002) Generalized linear accident models and goodness of fit testing. Accident Anal Prevent 34: 417–427

    Article  Google Scholar 

  • Zheng B (2000) Summarizing the goodness of fit of generalized linear models for longitudinal data. Stat Med 19: 1265–1275

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Karlsruhe Institute of Technology (KIT), Kaiserstraße 89, 76133, Karlsruhe, Germany

    Bernhard Klar

  2. Department of Economics, National and Kapodistrian University of Athens, 8 Pesmazoglou Street, 105 59, Athens, Greece

    Simos G. Meintanis

Authors
  1. Bernhard Klar
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  2. Simos G. Meintanis
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Correspondence to Simos G. Meintanis.

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Klar, B., Meintanis, S.G. Specification tests for the response distribution in generalized linear models. Comput Stat 27, 251–267 (2012). https://doi.org/10.1007/s00180-011-0253-5

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  • DOI: https://doi.org/10.1007/s00180-011-0253-5

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