Three common classes of kernel regression estimators are considered: the Nadaraya–Watson (NW) estimator, the Priestley–Chao (PC) estimator, and the Gasser–Müller (GM) estimator. It is shown that (i) the GM estimator has a certain monotonicity preservation property for any kernel $K$, (ii) the NW estimator has this property if and only the kernel $K$ is log concave, and (iii) the PC estimator does not have this property for any kernel $K$. Other related properties of these regression estimators are discussed.

中文翻译：

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核回归估计量的单调保性

考虑了三种常见的核回归估计器：Nadaraya-Watson（NW）估计器，Priestley-Chao（PC）估计器和Gasser-Müller（GM）估计器。结果表明：（i）GM估计量对于任何核都有一定的单调性保持性质。$ķ$，（ii）当且仅当内核时，NW估计量才具有此属性 $ķ$ 是对数凹入的，并且（iii）PC估计器对于任何内核都不具有此属性 $ķ$。讨论了这些回归估计量的其他相关属性。