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.

本文关注于二进制纵向数据的部分线性单指标模型（PLSIM）的最佳估计策略。响应和协变量之间的拟合模型可能会导致复杂性，并且线性项可能不足以表示这种关系。在这种情况下，最好同时包含线性和非线性项的PLSIM。本文的目的是开发优化的估计策略，例如预测试和收缩方法，以分析PLSIM下的二进制纵向数据，其中某些回归参数受到限制。我们使用核估计方程来估计非参数分量，然后使用轮廓估计方程来估计无限制和受限制的估计量。要应用预测试和收缩方法，我们拟合两个模型：一个包含所有协变量，另一个包含基于辅助信息的回归参数。然后将无限制和受限制的估算器进行最佳组合，以得到预测试和收缩率估算器。我们还根据偏倚和风险推导了估计量的渐近性质。还进行了蒙特卡洛模拟，以检验拟议估计量与无限制估计量的相对性能。一个经验应用也可以用来说明我们方法论的有效性。还进行了蒙特卡洛模拟，以检验拟议估计量与无限制估计量的相对性能。一个经验应用也可以用来说明我们方法论的有效性。还进行了蒙特卡洛模拟，以检验拟议估计量与无限制估计量的相对性能。一个经验应用也可以用来说明我们方法论的有效性。