Abstract
The problem considered is that of predicting the value of a linear functional of a random field when the parameter vector θ of the covariance function (or generalized covariance function) is unknown. The customary predictor when θ is unknown, which we call the EBLUP, is obtained by substituting an estimator Ĝj for θ 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 Ĝj for θ in the expression f 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.
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References
Bement, T. R. and Williams, J. S. (1969). Variance of weighted regression estimators when sampling errors are independent and heteroscedastic, J. Amer. Statist. Assoc., 64, 1369–1382.
Carroll, R. J., Wu, C. F. J. and Ruppert, D. (1988). The effect of estimating weights in weighted least squares, J. Amer. Statist. Assoc., 83, 1045–1054.
Cressie, N. (1985). Fitting variogram models by weighted least squares, Math. Geol., 17, 563–586.
Cressie, N. (1986). Kriging nonstationary data, J. Amer. Statist. Assoc., 81, 625–634.
Cressie, N. (1987). A nonparametric view of generalized covariances for kriging, Math. Geol., 19, 425–449.
Delfiner, P. (1976). Linear estimation of non-stationary spatial phenomena, Advanced Geostatistics in the Mining Industry (eds. M.Guarascio, M.David and C.Huijbregts), 49–68, Reidel, Dordrecht.
Dubrule, O. (1983). Two methods with different objectives: splines and kriging, Math. Geol., 15, 245–247.
Eaton, M. L. (1985). The Gauss-Markov theorem in multivariate analysis, Multivariate Analysis-VI (ed. P. R.Krishnaiah), 177–201, Elsevier, Amsterdam.
Fuller, W. A. and Hasza, D. P. (1981). Properties of predictors for autoregressive time series, J. Amer. Statist. Assoc., 76, 155–161.
Goldberger, A. S. (1962). Best linear unbiased prediction in the generalized regression model, J. Amer. Statist. Assoc., 57, 369–375.
Harville, D. A. (1985). Decomposition of prediction error, J. Amer. Statist. Assoc., 80, 132–138.
Harville, D. A. and Jeske, D. R. (1992). Mean squared error of estimation or prediction under a general linear model, J. Amer. Statist. Assoc., 87 (to appear).
Journel, A. G. and Huijbregts, C. J. (1978). Mining Geostatistics, Academic Press, London.
Kackar, R. N. and Harville, D. A. (1984). Approximations for standard errors of estimators of fixed and random effects in mixed linear models, J. Amer. Statist. Assoc., 79, 853–862.
Khatri, C. G. and Shah, K. R. (1981). On the unbiased estimation of fixed effects in a mixed model for growth curves, Comm. Statist. A—Theory Methods, 10, 401–406.
Kitanidis, P. K. (1983). Statistical estimation of polynomial generalized covariance functions and hydrologic applications, Water Resources Research, 19, 909–921.
Kitanidis, P. K. (1985). Minimum variance unbiased quadratic estimation of covariances of regionalized variables, Math. Geol., 17, 195–208.
Mardia, K. V. and Marshall, R. J. (1984). Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika, 71, 135–146.
Marshall, R. J. and Mardia, K. V. (1985). Minimum norm quadratic estimation of components of spatial covariances, Math. Geol., 17, 517–525.
Matheron, G. (1973). The intrinsic random functions and their applications, Adv. in Appl. Probab., 5, 439–468.
Prasad, N. G. N. and Rao, J. N. K. (1986). On the estimation of mean square error of small area predictors, Proceedings of the Section on Survey Research Methods, American Statistical Association, 108–116, Washington, D.C.
Reinsel, G. C. (1980). Asymptotic properties of prediction errors for the multivariate autoregressive model using estimated parameters, J. Roy. Statist. Soc. Ser. B, 42, 328–333.
Reinsel, G. C. (1984). Estimation and prediction in a multivariate random effects generalized linear model, J. Amer. Statist. Assoc., 79, 406–414.
Rothenberg, T. J. (1984). Hypothesis testing in linear models when the error covariance matrix is nonscalar, Econometrica, 52, 827–842.
Starks, T. H. and Fang, J. H. (1982). The effect of drift on the experimental semivariogram, J. Internat. Assoc. Math. Geol., 14, 309–320.
Toyooka, Y. (1982). Prediction error in a linear model with estimated parameters, Biometrika, 69, 453–459.
Wolfe, D. A. (1973). Some general results about uncorrelated statistics, J. Amer. Statist. Assoc., 68, 1013–1018.
Yamamoto, T. (1976). Asymptotic mean square prediction error for an autoregressive model with estimated coefficients, Appl. Statist., 25, 123–127.
Yfantis, E. A., Flatman, G. T. and Behar, J. V. (1987). Efficiency of kriging estimation for square, triangular, and hexagonal grids, Math. Geol., 19, 183–205.
Zimmerman, D. L. and Cressie, N. (1989). Improved estimation of the kriging variance, Tech. Report, No. 161, Department of Statistics, University of Iowa.
Zimmerman, D. L. and Zimmerman, M. B. (1991). A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors, Technometrics, 33, 77–91.
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This research was partially supported by a University of Iowa Old Gold Fellowship (Zimmerman) and by the NSF under grant DMS-8703083 (Cressie).
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Zimmerman, D.L., Cressie, N. Mean squared prediction error in the spatial linear model with estimated covariance parameters. Ann Inst Stat Math 44, 27–43 (1992). https://doi.org/10.1007/BF00048668
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DOI: https://doi.org/10.1007/BF00048668