回归建模是统计学的主力,关于回归函数的估计有大量文献。近年来已经认识到,在回归分析中,最终目标可能是估计回归函数的水平集,ie,回归函数超过预定义水平的协变量值集合,而不是回归函数本身的估计。迄今为止,已发表的关于水平集估计的工作主要集中在非参数回归上,特别是在点估计上。在这篇文章中,考虑了线性回归水平集的置信度集的构造。特别是,1-α$$1-\\alpha$$上层,构建正态误差线性回归的下限和双侧置信度集。结果表明,这些置信集可以很容易地从相应的1-α$$1-\\α$$水平同时置信带构造。还指出,该构造方法很容易适用于其他参数回归模型,其中平均响应通过单调链接函数取决于线性预测因子,其中包括广义线性模型,线性混合模型和广义线性混合模型。因此,本文提出的方法具有广泛的适用性。使用线性和广义线性模型进行模拟研究以评估该方法,并使用实际示例来说明该方法。
Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It has been realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, ie, the set of covariate values for which the regression function exceeds a predefined level, instead of the estimation of the regression function itself. The published work on estimation of the level set has thus far focused mainly on nonparametric regression, especially on point estimation. In this article, the construction of confidence sets for the level set of linear regression is considered. In particular, 1 - α $$ 1-\\alpha $$ level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets can be easily constructed from the corresponding 1 - α $$ 1-\\alpha $$ level simultaneous confidence bands. It is also pointed out that the construction method is readily applicable to other parametric regression models where the mean response depends on a linear predictor through a monotonic link function, which include generalized linear models, linear mixed models and generalized linear mixed models. Therefore, the method proposed in this article is widely applicable. Simulation studies with both linear and generalized linear models are conducted to assess the method and real examples are used to illustrate the method.