The Annals of Statistics

Local quasi-likelihood with a parametric guide

Jianqing Fan, Yichao Wu, and Yang Feng

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Generalized linear models and the quasi-likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response. Nonparametric methods need no explicit parametric specification, and the resulting model is completely determined by the data themselves. However, nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in Fan, Heckman and Wand [J. Amer. Statist. Assoc. 90 (1995) 141–150]. In this work, we propose a unified family of parametrically-guided nonparametric estimation schemes. This combines the merits of both parametric and nonparametric approaches and enables us to incorporate prior knowledge. Asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart.

Article information

Ann. Statist., Volume 37, Number 6B (2009), 4153-4183.

First available in Project Euclid: 23 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Generalized linear model local polynomial smoothing parametric guide quasi-likelihood method


Fan, Jianqing; Wu, Yichao; Feng, Yang. Local quasi-likelihood with a parametric guide. Ann. Statist. 37 (2009), no. 6B, 4153--4183. doi:10.1214/09-AOS713.

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  • Cheng, M.-Y. and Hall, P. (2003). Reducing variance in nonparametric surface estimation. J. Multivariate Anal. 86 375–397.
  • Cheng, M.-Y., Peng, L. and Wu, J.-S. (2007). Reducing variance in univariate smoothing. Ann. Statist. 35 522–542.
  • Cox, D. D. and O’Sullivan, F. (1990). Asymptotic analysis of penalized likelihood and related estimators. Ann. Statist. 18 1676–1695.
  • Fan, J. (1993). Local linear regression smoothers and their minimax efficiency. Ann. Statist. 21 196–216.
  • Fan, J., Farmen, M. and Gijbels, I. (1998). Local maximum likelihood estimation and inference. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 591–608.
  • Fan, J. and Gijbels, I. (1995). Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation. J. R. Stat. Soc. Ser. B Stat. Methodol. 57 371–394.
  • Fan, J., Heckman, N. E. and Wand, M. P. (1995). Local polynomial kernel regression for generalized linear models and quasi-likelihood functions. J. Amer. Statist. Assoc. 90 141–150.
  • Gasser, T., Müller, H.-G. and Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 47 238–252.
  • Glad, I. K. (1998). Parametrically guided non-parametric regression. Scand. J. Statist. 25 649–668.
  • Green, P. J. and Yandell, B. S. (1985). Semiparametric generalized linear models. In Lecture Notes in Statistics 32 44–55. Springer, Berlin.
  • Hjort, N. L. and Glad, I. K. (1995). Nonparametric density estimation with a parametric start. Ann. Statist. 23 882–904.
  • Huang, L.-S. and Fan, J. (1999). Nonparametric estimation of quadratic regression functionals. Bernoulli 5 927–949.
  • Hurvich, C. M. and Tsai, C.-L. (1995). Model selection for extended quasi-likelihood models in small samples. Biometrics 51 1077–1084.
  • Martins-Filho, C., Mishra, S. and Ullah, A. (2008). A class of improved parametrically guided nonparametric regression estimators. Econometric Rev. 27 542–573.
  • McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman & Hall, London.
  • Naito, K. (2004). Semiparametric density estimation by local L2-fitting. Ann. Statist. 32 1162–1191.
  • Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized linear models. J. Roy. Statist. Soc. Ser. A 135 370–384.
  • O’Sullivan, F., Yandell, B. and Raynor, W. (1986). Automatic smoothing of regression functions in generalized linear models. J. Amer. Statist. Assoc. 81 96–103.
  • Staniswalis, J. G. (1989). The kernel estimate of a regression function in likelihood-based models. J. Amer. Statist. Assoc. 84 276–283.
  • Stone, C. J. (1975). Nearest neighbor estimators of a nonlinear regression function. In Proc. Computer Science and Statistics, 8th Annual Symposium on the Interface 413–418. Health Sciences Computer Facility, UCLA.
  • Stone, C. J. (1977). Consistent nonparametric regression, with discussion. Ann. Statist. 5 549–645.
  • Tibshirani, R. and Hastie, T. (1987). Local likelihood estimation. J. Amer. Statist. Assoc. 82 559–568.
  • Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models, and the gauss-Newton method. Biometrika 61 439–447.
  • White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica 50 1–25.