Mean squared prediction error in the spatial linear model with estimated covariance parameters

Abstract

Abstract. The problem considered is that of predicting the value of a linear functional of a random field when the parameter vector 0 of the covariance function (or generalized covariance function) is unknown. The customary predictor when 0 is unknown, which we call the EBLUP, is obtained by substituting an estimator 0 for 0 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 0 for 0 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.

Height Model Integration Using Alos Palsar, X Sar, SRTM C, and Icesat/Glas

“…The next process is to generate HEM and display the height error of the height model. Height error was made of a standard deviation or vertical error in the height model data (Zimmerman and Cressie, 1992). Height error can be made from its own data.…”

Corrected prediction intervals for change detection in paired watershed studies

“…A small sample size is not an issue in our example, in which n = 414 for the calibration time period. For spatially autocorrelated data, in addition to the assumptions necessary for equation (A3), Zimmerman and Cressie (1992) suggest using the Prasad-Rao MSE estimator when the spatial correlation is known or estimated to be weak, and when the estimated variance-covariance matrix of the observed data is known to be negatively biased. The estimated value of the first autoregressive parameter for our example is 0.996 (95% CI: 0.90, 1.09), which is consistent with values of strong autocorrelation as seen in Vijapurkar and Gotway (2001).…”

“…Given this unbiased estimator for y|x j and again that the variance and covariance parameters are known, the prediction mean-squared error (MSE) can be computed as Jeske 1992, Zimmerman andCressie 1992):…”

“…A common method to obtain prediction variances, here after referred to as the plug-in estimator, has been to estimate the variance-covariance parameters, and simply plug-in these values to equation (8) as fixed and known (Das et al 2004, Schabenberger andGotway 2005), however, this estimator can be negatively biased (Kackar and Haarville 1984, Harville and Jeske 1992, Zimmerman and Cressie 1992, Schabenberger and Gotway 2005. The origin of this bias and the derivation leading to the .…”

Smoothing Regional Maps Using Empirical Bayes Predictors

“…However, whgn 8 is also unknown and estimated by 8, the empirical Bayes predictor p(Y; 8) is typically no longer necessarily linear nor Bayes. (Sometimes, the empirical Bayes Using analogous arguments to Zimmerman and Cressie (1991), it is straightforward to show that E@(Y; 6)) = E ( Z ) = Xp , (30) rovided 6 i : even (that is, 6(-Y) = 6(Y)) and translation invariant (that is, &Y +Xw) = 8(Y) for every p x 1 vector w). Notice that the m.1.…”