## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 5 (2018), 2635-2663.

### Thresholds for detecting an anomalous path from noisy environments

Shirshendu Chatterjee and Ofer Zeitouni

#### Abstract

We consider the “searching for a trail in a maze” composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side $n$ based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero mean off the anomalous path and mean $\mu_{n}$ on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. [*Ann. Statist.* **36** (2008) 1726–1757] showed that if the unknown directed path under the signal hypothesis has known initial location, then detection is possible (in the minimax sense) if $\mu_{n}\gg1/\sqrt{\log n}$, while it is not possible if $\mu_{n}\ll1/\log n\sqrt{\log\log n}$. In this paper, we show that this result continues to hold even when the initial location of the unknown path is not known. As is the case with Arias-Castro et al. [*Ann. Statist.* **36** (2008) 1726–1757], the upper bound here also applies when the path is undirected. The improvement is achieved by replacing the linear detection statistic used in Arias-Castro et al. [*Ann. Statist.* **36** (2008) 1726–1757] with a polynomial statistic, which is obtained by employing a multiscale analysis on a quadratic statistic to bootstrap its performance. Our analysis is motivated by ideas developed in the context of the analysis of random polymers in Lacoin [*Comm. Math. Phys.* **294** (2010) 471–503].

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 5 (2018), 2635-2663.

**Dates**

Received: May 2017

Revised: September 2017

First available in Project Euclid: 28 August 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1535443230

**Digital Object Identifier**

doi:10.1214/17-AAP1356

**Mathematical Reviews number (MathSciNet)**

MR3847969

**Zentralblatt MATH identifier**

06974761

**Subjects**

Primary: 62C20: Minimax procedures 62G10: Hypothesis testing

Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K37: Processes in random environments

**Keywords**

Detecting a chain of nodes in a network minimax detection random scenery

#### Citation

Chatterjee, Shirshendu; Zeitouni, Ofer. Thresholds for detecting an anomalous path from noisy environments. Ann. Appl. Probab. 28 (2018), no. 5, 2635--2663. doi:10.1214/17-AAP1356. https://projecteuclid.org/euclid.aoap/1535443230