Abstract
We prove minimax bounds for estimating Gaussian location mixtures on under the squared and the squared Hellinger loss functions. Under the squared loss, we prove that the minimax optimal rate is upper and lower bounded by a constant multiple of . Under the squared Hellinger loss, we consider two subclasses based on the behavior of the tails of the mixing measure. When the mixing measure has a sub-Gaussian tail, the minimax rate under the squared Hellinger loss is bounded from below by , which implies that the optimal minimax rate is between and the upper bound obtained by [11]. On the other hand, when the mixing measure is only assumed to have a bounded moment for a fixed , the minimax rate under the squared Hellinger loss is bounded from below by . This rate is minimax optimal up to logarithmic factors.
Funding Statement
The first author was supported by NRF-2020R1F1A1A01069632. The second author was supported by NSF CAREER Grant DMS-16-54589.
Citation
Arlene K. H. Kim. Adityanand Guntuboyina. "Minimax bounds for estimating multivariate Gaussian location mixtures." Electron. J. Statist. 16 (1) 1461 - 1484, 2022. https://doi.org/10.1214/21-EJS1975
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