## Bernoulli

• Bernoulli
• Volume 24, Number 2 (2018), 1101-1127.

### Efficient estimation for generalized partially linear single-index models

#### Abstract

In this paper, we study the estimation for generalized partially linear single-index models, where the systematic component in the model has a flexible semi-parametric form with a general link function. We propose an efficient and practical approach to estimate the single-index link function, single-index coefficients as well as the coefficients in the linear component of the model. The estimation procedure is developed by applying quasi-likelihood and polynomial spline smoothing. We derive large sample properties of the estimators and show the convergence rate of each component of the model. Asymptotic normality and semiparametric efficiency are established for the coefficients in both the single-index and linear components. By making use of spline basis approximation and Fisher score iteration, our approach has numerical advantages in terms of computing efficiency and stability in practice. Both simulated and real data examples are used to illustrate our proposed methodology.

#### Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 1101-1127.

Dates
Revised: February 2016
First available in Project Euclid: 21 September 2017

https://projecteuclid.org/euclid.bj/1505980891

Digital Object Identifier
doi:10.3150/16-BEJ873

Mathematical Reviews number (MathSciNet)
MR3706789

Zentralblatt MATH identifier
06778360

#### Citation

Wang, Li; Cao, Guanqun. Efficient estimation for generalized partially linear single-index models. Bernoulli 24 (2018), no. 2, 1101--1127. doi:10.3150/16-BEJ873. https://projecteuclid.org/euclid.bj/1505980891

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#### Supplemental materials

• Supplement to “Efficient estimation for generalized partially linear single-index models”. We provide detailed proof for Lemma A.1, the statements and/or proofs of other technical lemmas and additional simulation results.