## The Annals of Statistics

### A Nonparametric Control Chart for Detecting Small Disorders

#### Abstract

We consider sequential observation of independent random variables $X_1,\cdots, X_N$ whose distribution changes from $F$ to $G$ after the first $\lbrack N\theta \rbrack$ variables. The object is to detect the unknown change-point quickly without too many false alarms. A nonparametric control chart based on partial weighted sums of sequential ranks is proposed. It is shown that if the change from $F$ to $G$ is small, then as $N \rightarrow \infty$, the appropriately scaled and linearly interpolated graph of partial rank sums converges to a Brownian motion on which a drift sets in at time $\theta$. Using this, the asymptotic performance of the one-sided control chart is compared with one based on partial sums of the $X$'s. Location change, scale change and contamination are considered. It is found that for distributions with heavy tails, the control chart based on ranks stops more frequently and faster than the one based on the $X$'s. Performance of the two procedures are also tested on simulated data and the outcomes are compatible with the theoretical results.

#### Article information

Source
Ann. Statist., Volume 9, Number 3 (1981), 544-554.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345458

Digital Object Identifier
doi:10.1214/aos/1176345458

Mathematical Reviews number (MathSciNet)
MR615430

Zentralblatt MATH identifier
0503.62077

JSTOR