  The James-Stein estimator is a biased estimator -- for a finite number of
samples its expected value is not the true mean. The maximum-likelihood
estimator (MLE), is unbiased and asymptotically optimal. Yet, when estimating
the mean of $3$ or more normally-distributed random variables, the James-Stein
estimator has a smaller total (expected) error than the MLE. We introduce the
James-Stein estimator to the field of quantum metrology, from both the
frequentist and Bayesian perspectives. We characterise the effect of quantum
phenomena on the James-Stein estimator through the lens of quantum Gaussian
sensing, the task of estimating the mean of an unknown multivariate quantum
Gaussian state. We find that noiseless entanglement or coherence improves
performance of the James-Stein estimator, but diminishes its advantage over the
MLE. In the presence of noise, the James-Stein advantage is restored. Quantum
effects can also boost the James-Stein advantage. We demonstrate this by
investigating multivariate postselective metrology (generalised weak-value
amplification), a strategy that uses quantum effects to measure parameters with
imperfect detectors. Simply by post-processing measured data differently, our
techniques reduce errors in quantum experiments.
