## The Annals of Statistics

### Locally lattice sampling designs for isotropic random fields

Michael L. Stein

#### Abstract

For predicting $\int_G v(x)Z(x)dx$, where v is a fixed known function and Z is a stationary random field, a good sampling fesign should have a greater density of observations where v is relatively large in absolute value. Designs using this idea when $G = [0, 1]$ have been studied for some time. For G a region in two dimensions, very little is known about the statistical properties of cubature rules based on designs with varying density. This work proposes a class of designs that are locally parallelogram lattices but whose densities can vary. The asymptotic variance of the cubature error for these designs is obtained for a class of isotropic random fields and an asymptotically optimal sequence of cubature rules within this class is found. I conjecture that this sequence of cubature rules is asymptotically optimal with respect to all cubature rules.

#### Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 1991-2012.

Dates
First available in Project Euclid: 15 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034713644

Digital Object Identifier
doi:10.1214/aos/1034713644

Mathematical Reviews number (MathSciNet)
MR1389862

Zentralblatt MATH identifier
0856.62084

Subjects
Primary: 62M40: Random fields; image analysis
Secondary: 65D32: Quadrature and cubature formulas

#### Citation

Stein, Michael L. Locally lattice sampling designs for isotropic random fields. Ann. Statist. 23 (1995), no. 6, 1991--2012. doi:10.1214/aos/1034713644. https://projecteuclid.org/euclid.aos/1034713644

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