## The Annals of Statistics

### Predicting integrals of random fields using observations on a lattice

Michael L. Stein

#### Abstract

For a stationary random field Z on $\mathbb{R}^d$, this work studies the asymptotic behavior of predictors of $\int v(x)Z(x)dx$ based on observations on a lattice as the distance between neighbors in the lattice tends to 0. Under a mild condition on the spectral density of Z, an asymptotic expression for the mean-squared error of a predictor of $\int v(x)Z(x)dx$ based on observations on an infinite lattice is derived. For predicting integrals over the unit cube, a simple predictor based just on observations in the unit cube is shown to be asymptotically optimal if v is sufficiently smooth and Z is not too smooth. Modified predictors extend this result to smoother processes.

#### Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 1975-1990.

Dates
First available in Project Euclid: 15 October 2002

https://projecteuclid.org/euclid.aos/1034713643

Digital Object Identifier
doi:10.1214/aos/1034713643

Mathematical Reviews number (MathSciNet)
MR1389861

Zentralblatt MATH identifier
0856.62083

#### Citation

Stein, Michael L. Predicting integrals of random fields using observations on a lattice. Ann. Statist. 23 (1995), no. 6, 1975--1990. doi:10.1214/aos/1034713643. https://projecteuclid.org/euclid.aos/1034713643

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