It is well known that the maximum likelihood estimator (MLE) is inadmissible when estimating the multidimensional Gaussian location parameter. We show that the verdict is much more subtle for the binary location parameter. We consider this problem in a regression framework by considering a ridge logistic regression (RR) with three alternative ways of shrinking the estimates of the event probabilities. While it is shown that all three variants reduce the mean squared error (MSE) of the MLE, there is at the same time, for every amount of shrinkage, a true value of the location parameter for which we are overshrinking, thus implying the minimaxity of the MLE in this family of estimators. Little shrinkage also always reduces the MSE of individual predictions for all three RR estimators; however, only the naive estimator that shrinks toward 1/2 retains this property for any generalized MSE (GMSE). In contrast, for the two RR estimators that shrink toward the common mean probability, there is always a GMSE for which even a minute amount of shrinkage increases the error. These theoretical results are illustrated on a numerical example. The estimators are also applied to a real data set, and practical implications of our results are discussed.

中文翻译：

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Logistic回归中岭估计量的均方误差

众所周知，在估计多维高斯位置参数时，最大似然估计器（MLE）是不可接受的。我们表明，对于二进制位置参数而言，结论更加微妙。我们通过考虑岭逻辑回归（RR）和缩小事件概率估计值的三种替代方法，在回归框架中考虑此问题。虽然显示了所有三个变体都降低了MLE的均方误差（MSE），但对于每种收缩量，同时存在我们过度收缩的位置参数的真实值，从而暗示了极小值在这个估计量系列中的MLE。很少的收缩也总是会降低所有三个RR估计量的单个预测的MSE；然而，对于任何广义MSE（GMSE），只有缩小到1/2的天真估计量才保留此属性。相反，对于两个朝着共同平均概率收缩的RR估计量，总会有一个GMSE，即使是很小的收缩量也会增加误差。数值示例说明了这些理论结果。估计量也适用于真实数据集，并讨论了我们结果的实际含义。