Author/Editor		Blagus, Rok; Goeman, Jelle J.
Title		Mean squared error of ridge estimators in logistic regression
Type		članek
Publication year		2019
Volume		str. str.
ISSN		1467-9574
Language		eng
Abstract		It is well known that maximum likelihood estimator (MLE) is inadmissible when estimating the multi-dimensional 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 regression 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, at the same time there is, for every amount of shrinkage, a true value of the location parameter for which we are overshrinking, thus implying the minimaxity of 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 towards 1/2 retains this property for any generalized MSE (GMSE). In contrast, for the two RR estimators that shrink towards 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 dataset and practical implications of our results are discussed.
Keywords		logistic regression multi-dimensional Gaussian location parameter likelihood logistična regresija večdimenzionalni Gaussov lokacijski parameter verjetnost