Generalized Partially Linear Single-Index Models

R. J. Carroll; Jianqing Fan; Irène Gijbels; M. P. Wand

doi:10.1080/01621459.1997.10474001

Generalized Partially Linear Single-Index Models

R. J. Carroll, Jianqing Fan, Irène Gijbels, M. P. Wand

Research output: Contribution to journal › Article › peer-review

672 Scopus citations

Abstract

The typical generalized linear model for a regression of a response Y on predictors (X, Z) has conditional mean function based on a linear combination of (X, Z). We generalize these models to have a nonparametric component, replacing the linear combination α^T₀X + β^T₀Z by η₀(α^T₀X) + β^T₀Z, where η₀(·) is an unknown function. We call these generalized partially linear single-index models (GPLSIM). The models include the “single-index” models, which have β₀ = 0. Using local linear methods, we propose estimates of the unknown parameters (α₀, β₀) and the unknown function η₀(·) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.

Original language	English (US)
Pages (from-to)	477-489
Number of pages	13
Journal	Journal of the American Statistical Association
Volume	92
Issue number	438
DOIs	https://doi.org/10.1080/01621459.1997.10474001
State	Published - Jun 1 1997
Externally published	Yes

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

Keywords

Asymptotic theory
Generalized linear models
Kernel regression
Local estimation
Local polynomial regression
Nonparametric regression
Quasi-likelihood

Access to Document

10.1080/01621459.1997.10474001

Cite this

@article{e3bedbcbd8844fe883d4dbf258f5ff75,

title = "Generalized Partially Linear Single-Index Models",

abstract = "The typical generalized linear model for a regression of a response Y on predictors (X, Z) has conditional mean function based on a linear combination of (X, Z). We generalize these models to have a nonparametric component, replacing the linear combination αT0X + βT0Z by η0(αT0X) + βT0Z, where η0(·) is an unknown function. We call these generalized partially linear single-index models (GPLSIM). The models include the “single-index” models, which have β0 = 0. Using local linear methods, we propose estimates of the unknown parameters (α0, β0) and the unknown function η0(·) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.",

keywords = "Asymptotic theory, Generalized linear models, Kernel regression, Local estimation, Local polynomial regression, Nonparametric regression, Quasi-likelihood",

author = "Carroll, {R. J.} and Jianqing Fan and Ir{\`e}ne Gijbels and Wand, {M. P.}",

year = "1997",

month = jun,

day = "1",

doi = "10.1080/01621459.1997.10474001",

language = "English (US)",

volume = "92",

pages = "477--489",

journal = "Journal of the American Statistical Association",

issn = "0162-1459",

publisher = "Taylor and Francis Ltd.",

number = "438",

}

TY - JOUR

T1 - Generalized Partially Linear Single-Index Models

AU - Carroll, R. J.

AU - Fan, Jianqing

AU - Gijbels, Irène

AU - Wand, M. P.

PY - 1997/6/1

Y1 - 1997/6/1

N2 - The typical generalized linear model for a regression of a response Y on predictors (X, Z) has conditional mean function based on a linear combination of (X, Z). We generalize these models to have a nonparametric component, replacing the linear combination αT0X + βT0Z by η0(αT0X) + βT0Z, where η0(·) is an unknown function. We call these generalized partially linear single-index models (GPLSIM). The models include the “single-index” models, which have β0 = 0. Using local linear methods, we propose estimates of the unknown parameters (α0, β0) and the unknown function η0(·) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.

AB - The typical generalized linear model for a regression of a response Y on predictors (X, Z) has conditional mean function based on a linear combination of (X, Z). We generalize these models to have a nonparametric component, replacing the linear combination αT0X + βT0Z by η0(αT0X) + βT0Z, where η0(·) is an unknown function. We call these generalized partially linear single-index models (GPLSIM). The models include the “single-index” models, which have β0 = 0. Using local linear methods, we propose estimates of the unknown parameters (α0, β0) and the unknown function η0(·) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.

KW - Asymptotic theory

KW - Generalized linear models

KW - Kernel regression

KW - Local estimation

KW - Local polynomial regression

KW - Nonparametric regression

KW - Quasi-likelihood

UR - http://www.scopus.com/inward/record.url?scp=0031536017&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031536017&partnerID=8YFLogxK

U2 - 10.1080/01621459.1997.10474001

DO - 10.1080/01621459.1997.10474001

M3 - Article

AN - SCOPUS:0031536017

SN - 0162-1459

VL - 92

SP - 477

EP - 489

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 438

ER -

Generalized Partially Linear Single-Index Models

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this