The Quadratic Loss of Isotonic Regression Under Normality

The maximum likelihood estimator $\hat{\mu}$ of a nondecreasing regression function has been studied in detail in the literature. However, little is known about its quadratic loss pointwise. This paper shows that the mean square error of $\hat{\mu}_i$ is less than that of the usual estimator $\bar{X}_i$ for each $i$ when $\bar{X}_1,\cdots, \bar{X}_k$ are independent normal variates.