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Spline-based quasi-likelihood estimation of mixed Poisson regression with single-index models

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Abstract

We consider spline-based quasi-likelihood estimation for mixed Poisson regression with single-index models. The unknown smooth function is approximated by B-splines, and a modified Fisher scoring algorithm is employed to compute the estimates. The spline estimate of the nonparametric component is shown to achieve the optimal rate of convergence, and the asymptotic normality of the regression parameter estimates is still valid even if the variance function is misspecified. The semiparametric efficiency of the model can be established if the variance function is correctly specified. The variance of the regression parameter estimates can be consistently estimated by a simple procedure based on the least-squares estimation. The proposed method is evaluated via an extensive Monte Carlo study, and the methodology is illustrated on an air pollution study.

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References

  • Borja-Aburto V, Castillejos M, Gold D, Bierzwinski S, Loomis D (1998) Mortality and ambient fine particles in southwest Mexico City, 1993–1995. Environ Health Perspect 106:849–855

    Article  Google Scholar 

  • Breslow N (1984) Extra-Poisson variation in log-linear models. Appl Stat 33:38–44

    Article  Google Scholar 

  • Breslow N (1990) Tests of hypotheses in overdispersed Poisson regression and other quasi-likelihood models. J Am Stat Assoc 85:565–571

    Article  Google Scholar 

  • Carroll R, Fan J, Gijbels I, Wand M (1997) Generalized partially linear single-Index models. J Am Stat Assoc 92:477–489

    Article  MathSciNet  MATH  Google Scholar 

  • Clayton D (1994) Some approaches to the analysis of recurrent event data. Stat Methods Med Res 3:244–262

    Article  Google Scholar 

  • Cui X, Härdle W, Zhu L (2011) The EFM approach for single-index models. Ann Stat 39:1658–1688

    Article  MathSciNet  MATH  Google Scholar 

  • Dean C, Lawless J, Willmot G (1989) A mixed Poisson–inverse-Gaussian regression model. Can J Stat 17:171–181

    Article  MathSciNet  MATH  Google Scholar 

  • Härdle W, Stoker E (1989) Investigating smooth multiple regression by the method of average derivatives. J Am Stat Assoc 84:986–995

    MathSciNet  MATH  Google Scholar 

  • Härdle W, Hall P, Ichimura H (1993) Optimal smoothing in single-index models. Ann Stat 21:157–178

    Article  MathSciNet  MATH  Google Scholar 

  • Hristache M, Juditsky A, Spokoiny V (2001) Direct estimation of the index coefficient in a single-index model. Ann Stat 29:595–623

    Article  MathSciNet  MATH  Google Scholar 

  • Huang J, Liu L (2006) Polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form. Biometrics 62:793–802

    Article  MathSciNet  MATH  Google Scholar 

  • Ichimura H (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J Econ 58:71–120

    Article  MathSciNet  MATH  Google Scholar 

  • Lawless J (1987) Negative binomial and mixed Poisson regression. Can J Stat 15:209–225

    Article  MathSciNet  MATH  Google Scholar 

  • Liang H, Liu X, Li R, Tsai C (2010) Estimation and testing for partially linear single-index models. Ann stat 38:3811–3866

    Article  MathSciNet  MATH  Google Scholar 

  • Lu M (2015) Spline estimation of generalised monotonic regression. J Nonparametr Stat 27:19–39

    Article  MathSciNet  MATH  Google Scholar 

  • Lu M, Loomis D (2013) Spline-based semiparametric estimation of partially linear Poisson regression with single-index models. J Nonparametr Stat 25:905–922

    Article  MathSciNet  MATH  Google Scholar 

  • Rosenberg P (1995) Hazard function estimation using \(B\)-splines. Biometrics 51:874–887

    Article  MATH  Google Scholar 

  • Schumaker L (1981) Spline functions: basic theory. Wiley, New York

    MATH  Google Scholar 

  • Stone C (1985) Additive regression and other nonparametric models. Ann Stat 13:689–705

    Article  MathSciNet  MATH  Google Scholar 

  • Wellner J, Zhang Y (2007) Two likelihood-based semiparametric estimation methods for panel count data with covariates. Ann of Stat 35:2106–2142

  • Xia Y, Härdle W (2006) Semi-parametric estimation of partially linear single-index models. J Multivar Anal 97:1162–1184

    Article  MathSciNet  MATH  Google Scholar 

  • Xia Y, Tong H, Li W (1999) On extended partially linear single-index models. Biometrika 86:831–842

    Article  MathSciNet  MATH  Google Scholar 

  • Xue L, Liang H (2010) Polynomial spline estimation for a generalized additive coefficient model. Scand J Stat 37:26–46

    Article  MathSciNet  MATH  Google Scholar 

  • Yu Y, Ruppert D (2002) Penalized spline estimation for partially linear single-index models. J Am Stat Assoc 97:1042–1054

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to the Editor, Professor Norbert Henze, and an anonymous referee for their constructive comments and suggestions that largely improved this manuscript from an earlier version.

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  1. School of Community Health Sciences, University of Nevada, Reno, Reno, NV, 89557, USA

    Minggen Lu

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  1. Minggen Lu
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Correspondence to Minggen Lu.

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Lu, M. Spline-based quasi-likelihood estimation of mixed Poisson regression with single-index models. Metrika 81, 1–17 (2018). https://doi.org/10.1007/s00184-017-0631-2

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