Quadratic inference functions for partially linear single-index models with longitudinal data

AMS 2000 subject classifications: 62G05 62G10 62G20 Keywords: Bias correction Generalized likelihood ratio Longitudinal data Partially linear single-index models QIF a b s t r a c t In this paper, we consider the partially linear single-index models with longitudinal data. We propose the bias-corrected quadratic inference function (QIF) method to estimate the parameters in the model by accounting for the within-subject correlation. Asymptotic properties for the proposed estimation methods are demonstrated. A generalized likelihood ratio test is established to test the linearity of the nonparametric part. Under the null hypotheses, the test statistic follows asymptotically a χ 2 distribution. We also evaluate the finite sample performance of the proposed methods via Monte Carlo simulation studies and a real data analysis. Partially linear single-index (PLSI) models describe both the linear relationship between a scalar response variable Y and a q-dimensional vector Z and the nonlinear relationship between Y and a p-dimensional vector X in the form Y = Z ⊤ θ + g(X ⊤ β) + ε, where g(·) is an unknown link function and ∥β∥ = 1 (∥ · ∥ denotes the Euclidean norm here). Model (1.1) has gained much attention in recent years. For example, Carroll et al. [3] studied the generalized partially linear single-index models, Xia and Härdle [21] proposed the MAVE method to estimate the parameters of PLSI models, Zhu and Xue [28] studied the confidence interval of parameters based on the empirical likelihood method. PLSI models avoid the problem of ''curse of dimensionality'' and are flexible enough to capture the hidden linear and nonlinear relationships between covariates and the response variable. More recently, the literature on the applications of partially linear single-index models for repeated measurements is available, especially in econometrics, biomedical research, and epidemiology. For example, Tian et al. [19] used the generalized penalized spline least squares method and assumed working correlation matrices to estimate the parameters and the unknown link function. Li et al. [13] constructed confidence intervals or confidence regions for PLSI models with longitudinal data based on empirical likelihood.