## Annals of Statistics

### Permutational Extreme Values of Autocorrelation Coefficients and a Pitman Test Against Serial Dependence

#### Abstract

Normal approximations, as provided by permutational central limit theorems, conditionally can be arbitrarily bad. Such approximations therefore are poorly suited to the construction of critical values for Pitman (permutation) tests. A classical remedy consists in substituting a beta approximation (over the appropriate conditional interval range) for the normal one. Whereas deriving permutational extreme values for usual, nonserial statistics is generally straightforward, the corresponding problem for serial statistics (e.g., autocorrelation coefficients), however, appears somewhat more difficult. This problem, which is shown to reduce to a particular case of the well-known travelling salesman problem, is explicitly solved here for the autocorrelation coefficient of order one, allowing for a simple computation of permutational critical values for Pitman tests against serial dependence. The case of higher order autocorrelations is, however, of a different nature and requires another approach.

#### Article information

Source
Ann. Statist., Volume 20, Number 1 (1992), 523-534.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348536

Digital Object Identifier
doi:10.1214/aos/1176348536

Mathematical Reviews number (MathSciNet)
MR1150358

Zentralblatt MATH identifier
0746.62050

JSTOR