Optimal shrinkage estimations in partially linear single-index models for binary longitudinal data

This paper focuses on the optimal estimation strategies of partially linear single-index models (PLSIM) for binary longitudinal data. Fitting model between the response and covariates may cause complexity and the linear terms may not be adequate to represent the relationship. In this situation, the PLSIM containing both linear and nonlinear terms is preferable. The objective of this paper is to develop optimal estimation strategies such as, pretest and shrinkage methods, for the analysis of binary longitudinal data under the PLSIM where some regression parameters are subject to restrictions. We estimate the nonparametric component using kernel estimating equations, and then use profile estimating equations to estimate the unrestricted and restricted estimators. To apply the pretest and shrinkage methods, we fit two models: one includes all covariates and the other restricts the regression parameters based on the auxiliary information. The unrestricted and restricted estimators are then combined optimally to get the pretest and shrinkage estimators. We also derive the asymptotic properties of the estimators in terms of biases and risks. Monte Carlo simulations are also conducted to examine the relative performance of the proposed estimators to the unrestricted estimator. An empirical application is also be used to illustrate the usefulness of our methodology.