###
MEAN SQUARED PREDICTION ERROR IN THE SPATIAL LINEAR

MODEL WITH ESTIMATED COVARIANCE PARAMETERS

###
DALE L. ZIMMERMAN^{1} AND NOEL CRESSIE^{2}

^{1} *Department of Statistics and Actuarial Science, University of Iowa,*

Iowa City, IA 52242, U.S.A.

^{2} *Department of Statistics, Iowa State University, Ames, IA 50011, U.S.A.*
(Received April 5, 1989; revised July 2, 1990)

**Abstract.**
The problem considered is that of predicting the
value of a linear functional of a random field when the parameter vector
*theta* of the covariance function (or generalized covariance function)
is unknown. The customary predictor when *theta* is unknown, which we
call the EBLUP, is obtained by substituting an estimator ^{^}*theta* for
*theta* in the expression for the best linear unbiased predictor (BLUP).
Similarly, the customary estimator of the mean squared prediction error
(MSPE) of the EBLUP is obtained by substituting ^{^}*theta* for *theta* in
the expression for the BLUP's MSPE; we call this the EMSPE. In this
article, the appropriateness of the EMSPE as an estimator of the EBLUP's
MSPE is examined, and alternative estimators of the EBLUP's MSPE for use
when the EMSPE is inappropriate are suggested. Several illustrative
examples show that the performance of the EMSPE depends on the strength of
spatial correlation; the EMSPE is at its best when the spatial correlation
is strong.

*Key words and phrases*:
Best linear unbiased prediction,
generalized covariances, geostatistics, kriging, spatial models.

**Source**
( TeX ,
DVI ,
PS )