Simultaneous Confidence Bands and Hypothesis Testing in Varying‐coefficient Models
基于局部多项式技术，推导了变系数模型中估计系数函数偏差最大值的渐近分布，并构建了同时置信带；同时提出了检验系数函数是否为零或常数的形式化方法，通过模拟和实例验证。
Regression analysis is one of the most commonly used techniques in statistics. When the dimension of independent variables is high, it is difficult to conduct efficient non‐parametric analysis straightforwardly from the data. As an important alternative to the additive and other non‐parametric models, varying‐coefficient models can reduce the modelling bias and avoid the “curse of dimensionality” significantly. In addition, the coefficient functions can easily be estimated via a simple local regression. Based on local polynomial techniques, we provide the asymptotic distribution for the maximum of the normalized deviations of the estimated coefficient functions away from the true coefficient functions. Using this result and the pre‐asymptotic substitution idea for estimating biases and variances, simultaneous confidence bands for the underlying coefficient functions are constructed. An important question in the varying coefficient models is whether an estimated coefficient function is statistically significantly different from zero or a constant. Based on newly derived asymptotic theory, a formal procedure is proposed for testing whether a particular parametric form fits a given data set. Simulated and real‐data examples are used to illustrate our techniques.