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Characterizing the optimal solutions to the isotonic regression problem for identifiable functionals
Annals of the Institute of Statistical Mathematics ( IF 1 ) Pub Date : 2021-09-03 , DOI: 10.1007/s10463-021-00808-0
Alexander I. Jordan 1 , Anja Mühlemann 2 , Johanna F. Ziegel 2
Affiliation  

In general, the solution to a regression problem is the minimizer of a given loss criterion and depends on the specified loss function. The nonparametric isotonic regression problem is special, in that optimal solutions can be found by solely specifying a functional. These solutions will then be minimizers under all loss functions simultaneously as long as the loss functions have the requested functional as the Bayes act. For the functional, the only requirement is that it can be defined via an identification function, with examples including the expectation, quantile, and expectile functionals. Generalizing classical results, we characterize the optimal solutions to the isotonic regression problem for identifiable functionals by rigorously treating these functionals as set-valued. The results hold in the case of totally or partially ordered explanatory variables. For total orders, we show that any solution resulting from the pool-adjacent-violators algorithm is optimal.



中文翻译:

表征可识别泛函的等渗回归问题的最优解

通常,回归问题的解决方案是给定损失标准的最小值,并取决于指定的损失函数。非参数等渗回归问题很特殊,因为可以通过单独指定函数来找到最佳解决方案。只要损失函数具有贝叶斯行为所要求的函数,这些解决方案就会同时成为所有损失函数下的最小值。对于泛函,唯一的要求是它可以通过识别函数来定义,示例包括期望、分位数和期望泛函。概括经典结果,我们通过将这些泛函严格视为集合值来表征可识别泛函的等渗回归问题的最佳解决方案。结果在完全或部分有序的解释变量的情况下成立。对于总订单,我们表明由池相邻违反者算法产生的任何解决方案都是最佳的。

更新日期:2021-09-04
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