Admissible and minimax estimation of the parameters of the selected normal population in two-stage adaptive designs under reflected normal loss function

In clinical research, one of the key problems is to estimate the effect of the best treatment among the given k treatments in two-stage adaptive design. Suppose the effects of two treatments have normal distributions with means θ1 and θ2, respectively, and common known variance σ2. In the first stage, random samples of size n1 with means X1 and X2 are chosen from the two populations. Then the population with the larger or smaller sample mean XM is selected, and a random sample of size n2 with mean YM is chosen from this population in the second stage of design. Our aim is to estimate the mean θM or θJ of the selected population based on XM and YM in two-stage adaptive design under the reflected normal loss function. We obtain minimax estimators of θM and θJ, and then provide some sufficient conditions for the inadmissibility of estimators of θM and θJ. Theoretical results are augmented with a simulation study as well as a real data application.

Minimax quantum state estimation under Bregman divergence

We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.