## Local polynomial kernel regression for generalized linear models and quasi-likelihood functions (1995)

Venue: | Journal of the American Statistical Association,90 |

Citations: | 57 - 7 self |

### BibTeX

@ARTICLE{Fan95localpolynomial,

author = {Jianqing Fan and Nancy E. Heckman and M. P. Wand},

title = {Local polynomial kernel regression for generalized linear models and quasi-likelihood functions},

journal = {Journal of the American Statistical Association,90},

year = {1995},

pages = {141--150}

}

### Years of Citing Articles

### OpenURL

### Abstract

were introduced as a means of extending the techniques of ordinary parametric regression to several commonly-used regression models arising from non-normal likelihoods. Typically these models have a variance that depends on the mean function. However, in many cases the likelihood is unknown, but the relationship between mean and variance can be specified. This has led to the consideration of quasi-likelihood methods, where the conditionallog-likelihood is replaced by a quasi-likelihood function. In this article we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to these more general contexts. In the ordinary regression case local polynomial fitting has been seen to possess several appealing features in terms of intuitive and mathematical simplicity. One noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators. These properties are shown to carryover to the generalized linear model and quasi-likelihood model. The end result is a class of kernel type estimators for smoothing in quasi-likelihood models. These estimators can be viewed as a straightforward generalization of the usual parametric estimators. In addition, their simple asymptotic distributions allow for simple interpretation

### Citations

1329 | Generalized Additive Models - Hastie, Tibshirani - 1990 |

235 |
Design-adaptive Nonparametric Regression
- Fan
- 1992
(Show Context)
Citation Context ...the case P = 0, the estimator for f-L( x) is simply a weighted average: (3.3). Although this estimator is intuitive, it nevertheless suffers serious drawbacks such as large biases and low efficiency (=-=Fan 1992-=-b). Figure 1 and its accompanying discussion in Section 1 show how simple r,(' i P, h) and jJ,('jP, h) are to understand intuitively. Notice that the bandwidth h governs the amount of smoothing being ... |

226 | Spline smoothing and nonparametric regression - EUBANK - 1988 |

213 |
Smooth Regression Analysis
- WATSON
(Show Context)
Citation Context ...ibshirani and Hastie (1987) baSed their generalization on the "running lines" smoother. Staniswalis (1989) carried out a similar generalization of the Nadaraya-Watson kernel estimator (Nadaraya 1964, =-=Watson 1964-=-) which is equivalent to local constant fitting with a kernel weight. In an important further extension of the generalized linear model one only needs to model the conditional variance of the response... |

206 |
Generalized Linear Models, Second Edition
- McCullagh, Nelder
- 1989
(Show Context)
Citation Context ...ernel Regression for Generalized Linear Models and Quasi-Likelihood Functions JIANQING FAN, NANCY E. HECKMAN and M. P. WAND* 20th November, 1992 Generalized linear models (Wedderburn and NeIder 1972, =-=McCullagh and NeIder 1988-=-) were introduced as a means of extending the techniques of ordinary parametric regression to several commonly-used regression models arising from non-normal likelihoods. Typically these models have a... |

174 |
A reliable data-based bandwidth selection method for kernel density estimation
- Sheather, Jones
- 1991
(Show Context)
Citation Context ...imates of unknown quantities to a formula for the asymptotically optimal bandwidth. This has been shown to be more stable in both theoretical and practical performance (see e.g. Park and Marron 1990, =-=Sheather and Jones 1991-=-). Indeed, Fan and Marron (1992) show that, in the density estimation case, the plug-in selector is an asymptotically efficient method from a semiparametric point of view. For odd degree polynomial fi... |

141 |
Multivariate locally weighted least squares regression
- Ruppert, Wand
- 1994
(Show Context)
Citation Context ...ry(x; 0, h) = g(jL(x; 0, h)) where jL(x; 0, h) has the explicit weighted average expression given by (3.3). Therefore, standard results for the conditional mean squared error of jL(x; 0, h) (see e.g. =-=Ruppert and Wand 1992-=-) lead to: Theorem le. Let p = °and suppose that Conditions (1)-(5) stated in the Appendix are satisfied. Assume that h = hn the interior of supp(f) then -+ °and nh -+ 00 as n -+ 00. If x is a fixed p... |

131 |
Local Linear Regression Smoothers and Their Minimax Efficiencies.” The Annals of Statistics 21(1
- Fan
- 1993
(Show Context)
Citation Context ...the case P = 0, the estimator for f-L( x) is simply a weighted average: (3.3). Although this estimator is intuitive, it nevertheless suffers serious drawbacks such as large biases and low efficiency (=-=Fan 1992-=-b). Figure 1 and its accompanying discussion in Section 1 show how simple r,(' i P, h) and jJ,('jP, h) are to understand intuitively. Notice that the bandwidth h governs the amount of smoothing being ... |

71 | Asymptotic analysis of penalized likelihood and related estimators,” Ann - Cox, O’Sullivan - 1990 |

65 | Automatic Smoothing of Regression Functions in Generalized Linear Models - O’Sullivan, Yandell, et al. - 1986 |

63 |
On estimating regression. Theory of Probability and its
- Nadaraya
- 1964
(Show Context)
Citation Context ...livan (1990). Tibshirani and Hastie (1987) baSed their generalization on the "running lines" smoother. Staniswalis (1989) carried out a similar generalization of the Nadaraya-Watson kernel estimator (=-=Nadaraya 1964-=-, Watson 1964) which is equivalent to local constant fitting with a kernel weight. In an important further extension of the generalized linear model one only needs to model the conditional variance of... |

63 | Local likelihood estimation - Hastie, Tibshirani - 1987 |

59 | Quasilikelihood estimation in semiparametric models - Severini, Staniswalis - 1994 |

49 | Kernels for nonparametric curve estimation - Gasser, Mller, et al. - 1985 |

45 | Comparison of data-driven bandwidth selectors
- Park, Marron
- 1990
(Show Context)
Citation Context ...d on "plugging-in" estimates of unknown quantities to a formula for the asymptotically optimal bandwidth. This has been shown to be more stable in both theoretical and practical performance (see e.g. =-=Park and Marron 1990-=-, Sheather and Jones 1991). Indeed, Fan and Marron (1992) show that, in the density estimation case, the plug-in selector is an asymptotically efficient method from a semiparametric point of view. For... |

42 | Quasi likelihood and optimal estimation - Godambe, Heyde - 1987 |

32 | On the kernel estimate of a regression function in likelihood based models - Staniswalis - 1989 |

30 | Semi-parametric generalized linear models - Green, Yandell - 1985 |

26 |
Smoothing by weighted averaging of rounded points
- Härdle, Scott
- 1992
(Show Context)
Citation Context ... could involve a version of the method of scoring for the minimization of (3.2) and application of discretization ideas for computation of the kernel-type estimators required for each iteration (e.g. =-=HardIe and Scott, 1992-=-). Bandwidth selection rules of the type considered by Park and Marron (1990) and Sheather and Jones (1991) could also be developed. The theory for estimating higherorder derivatives of TJ derived in ... |

24 | Some Remarks on Overdispersion - Cox - 1983 |

15 | Applied Nonparametric Regression - HardIe - 1990 |

8 | Best possible constant for bandwidth selection - Fan, Marron - 1992 |

5 | Generalized linear models - Neider, Wedderburn - 1972 |

2 | Spatial and design adaptation: Variable order approximation in function estimation - Fan, Gijbels - 1992 |

2 | Graphical understanding of higher order kernels. Mimeo Series #2082 - Marron - 1992 |

1 | Asymptotics for Least Absolute Deviation Regression Estimators," Econometric Theory - unknown authors - 1991 |