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首页> 外文期刊>Annals of the Institute of Statistical Mathematics >Characterizing the optimal solutions to the isotonic regression problem for identifiable functionals
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Characterizing the optimal solutions to the isotonic regression problem for identifiable functionals

机译：表征可识别泛函的等渗回归问题的最佳解

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摘要

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.
机译：通常，回归问题的解是给定损失准则的最小化器，并且取决于指定的损失函数。非参数等渗回归问题很特殊，因为只需指定一个泛函即可找到最优解。然后，只要损失函数具有贝叶斯作用所请求的功能，这些解决方案就会同时成为所有损失函数下的最小化器。对于泛函，唯一的要求是可以通过识别函数来定义它，示例包括期望泛函、分位数泛函和期望泛函。推广经典结果，我们通过严格地将可识别泛函视为集合值来表征可识别泛函的等渗回归问题的最优解。在完全或部分有序解释变量的情况下，结果成立。对于总订单，我们表明，由pool-adjacent-vioroators算法产生的任何解决方案都是最佳的。

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