## Electronic Journal of Statistics

### Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

#### Abstract

We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset $\mathcal{C}$ of $\mathbb{R}^{d}$. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension $d$ and variance $\frac{1}{n}$ giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for $\mathcal{C}$.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 3713-3735.

Dates
First available in Project Euclid: 7 November 2018

https://projecteuclid.org/euclid.ejs/1541559861

Digital Object Identifier
doi:10.1214/18-EJS1472

#### Citation

Blanchard, Gilles; Carpentier, Alexandra; Gutzeit, Maurilio. Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$. Electron. J. Statist. 12 (2018), no. 2, 3713--3735. doi:10.1214/18-EJS1472. https://projecteuclid.org/euclid.ejs/1541559861

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