## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 6 (2017), 2736-2763.

### Optimal sequential detection in multi-stream data

#### Abstract

Consider a large number of detectors each generating a data stream. The task is to detect online, distribution changes in a small fraction of the data streams. Previous approaches to this problem include the use of mixture likelihood ratios and sum of CUSUMs. We provide here extensions and modifications of these approaches that are optimal in detecting normal mean shifts. We show how the (optimal) detection delay depends on the fraction of data streams undergoing distribution changes as the number of detectors goes to infinity. There are three detection domains. In the first domain for moderately large fractions, immediate detection is possible. In the second domain for smaller fractions, the detection delay grows logarithmically with the number of detectors, with an asymptotic constant extending those in sparse normal mixture detection. In the third domain for even smaller fractions, the detection delay lies in the framework of the classical detection delay formula of Lorden. We show that the optimal detection delay is achieved by the sum of detectability score transformations of either the partial scores or CUSUM scores of the data streams.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 6 (2017), 2736-2763.

**Dates**

Received: June 2015

Revised: July 2016

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1513328589

**Digital Object Identifier**

doi:10.1214/17-AOS1546

**Mathematical Reviews number (MathSciNet)**

MR3737908

**Zentralblatt MATH identifier**

06838149

**Subjects**

Primary: 62G10: Hypothesis testing 62L10: Sequential analysis

**Keywords**

Average run length CUSUM detectability score detection delay mixture likelihood ratio sparse detection stopping rule

#### Citation

Chan, Hock Peng. Optimal sequential detection in multi-stream data. Ann. Statist. 45 (2017), no. 6, 2736--2763. doi:10.1214/17-AOS1546. https://projecteuclid.org/euclid.aos/1513328589