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Empirical likelihood inference for generalized additive partially linear models Rong Liu1

· Yichuan Zhao2

Received: 27 November 2019 / Accepted: 25 August 2020 © Sociedad de Estadística e Investigación Operativa 2020

Abstract Generalized additive partially linear models enjoy the simplicity of GLMs and the flexibility of GAMs because they combine both parametric and nonparametric components. Based on spline-backfitted kernel estimator, we propose empirical likelihood (EL)-based pointwise confidence intervals and simultaneous confidence bands (SCBs) for the nonparametric component functions to make statistical inference. Simulation study strongly supports the asymptotic theory and shows that EL-based SCBs are much easier for implementation and have better performance than Wald-type SCBs. We apply the proposed method to a university retention study and provide SCBs for the effect of the students information. Keywords Empirical likelihood · Kernel · Spline · Confidence bands · GAM Mathematics Subject Classification Primary 62G08 · 62G15; Secondary 62G32

1 Introduction For high-dimensional data with a response Y and a predictor vector X =   X 1 , . . . , X d2 , additive model introduced by Hastie and Tibshirani (1990) is an

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11749020-00731-1) contains supplementary material, which is available to authorized users.

B

Rong Liu [email protected] Yichuan Zhao [email protected]

1

Department of Mathematics and Statistics, The University of Toledo, Toledo, USA

2

Department of Mathematics and Statistics, Georgia State University, Atlanta, USA

123

R. Liu, Y. Zhao

effective nonparametric regression tool, which is E (Y |X) = m (X) , m (X) = c +

d2 

m α (X α ) .

(1.1)

α=1

n  When a data set Yi , Xi i=1 of size n is observed which follows model (1.1), unknown 2 can be estimated via kernel, B-spline and smoothcomponent functions {m α (xα )}dα=1 ing spline with a univariate convergence rate. Although a very useful framework, there are some situations where model (1.1) is not appropriate, such as binary or Poisson responses. Denote b as a known link function which is the derivative of the function b, one can instead use generalized additive model (GAM)  E (Y | X) = b



c+

d2 

 m α (X α ) .

(1.2)

α=1

But if some component functions do have parametric forms or there are some categorical regressors, then one needs to improve the model by introducing parametric components. This will lead to simplification, more interpretability and higher precision in statistical calibration. To handle this situation, GAM can be extended to the generalized additive partially linear model (GAPLM), E (Y | T, X) = b {m (T, X)} ,

(1.3)

  2 with m (T, X) = β  T+ dα=1 m α (X α ), β = (β0 , . . . , βd1 ) , T = T0 , . . . , Td1 ,   and X = X 1 , . . . , X d2 , where T0 = 1 and Tk ∈ R for all k ∈ {1, . . . , d1 }. Model (1.3) originates from the special case, where the probability density function of Yi conditional on Ti a