Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It is realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, 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 paper, the construction of confidence sets for the level set of linear regression is considered. In particular, exact $1-\alpha$ level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets are closely connected with the corresponding $1-\alpha$ 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 paper is widely applicable. Two examples are given to illustrate the method.
翻译:回归模型是统计的成份,关于回归函数的估计有大量文献。近年来,在回归分析中,最终目的可能是估算回归函数的一组水平,而不是估算回归函数本身。迄今为止,关于水平集估算的出版工作主要侧重于非对称回归,特别是点估算。本文考虑了为水平线性回归集构建信心套数的问题。特别是,为正常-eror线性回归构建了准确的1美元水平上方、下方和双面信任套数。这表明这些信任套数与相应的1美元/alpha$水平的同步信任带密切相关。还据指出,当平均反应取决于线性预测,通过单线性连接函数,包括普通线性模型、线性混合模型和一般线性混合模型时,构建方法很容易适用于其他参数回归模型,因此,本文中建议的方法是广泛适用的。有两种例子说明该方法。