Estimation of smooth functionals of location parameter in Gaussian and Poincaré random shift models. (Q2051009)

The paper under review deals with the problem of estimating smooth functionals of location parameters in random noise models in Banach spaces. The authors study how the mean squared error rates in this problem depend on the smoothness of the functionals and determine how much smoothness is needed for efficient estimation. More specifically, they extend the results of [\textit{V. Koltchinskii} and \textit{M. Zhilova}, Ann. Inst. Henri Poincaré, Probab. Stat. 57, No. 1, 351--386 (2021; Zbl 1469.62430)] from Gaussian shift models to more general random shift models with noise having a distribution satisfying a Poincaré inequality (in particular, log-concave distribution). This is done both in the case of known noise distribution and also in the case when the distribution of the noise is Gaussian and the covariance is an unknown nuisance parameter.