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Estimating the Prediction Mean Squared Error in Gaussian Stochastic Processes with Exponential Correlation Structure

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  • Markus Abt

Abstract

Given one or more realizations from the finite dimensional marginal distribution of a stochastic process, we consider the problem of estimating the squared prediction error when predicting the process at unobserved locations. An approximation taking into account the additional variability due to estimating parameters involved in the correlation structure was developed by Kackar & Harville (1984) and was revisited by Harville & Jeske (1992) as well as Zimmerman & Cressie (1992). The present paper discusses an extension of these methods. The approaches will be compared via an extensive simulation study for models with and without random error term. Effects due to the designs used for prediction and for model fitting as well as due to the strength of the correlation between neighbouring observations of the stochastic process are investigated. The results show that considering the additional variability in the predictor due to estimating the covariance structure is of great importance and should not be neglected in practical applications.

Suggested Citation

  • Markus Abt, 1999. "Estimating the Prediction Mean Squared Error in Gaussian Stochastic Processes with Exponential Correlation Structure," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(4), pages 563-578, December.
    • Handle: RePEc:bla:scjsta:v:26:y:1999:i:4:p:563-578
    DOI: 10.1111/1467-9469.00168
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    Cited by:

    1. Jack Kleijnen & Wim Beers & Inneke Nieuwenhuyse, 2012. "Expected improvement in efficient global optimization through bootstrapped kriging," Journal of Global Optimization, Springer, vol. 54(1), pages 59-73, September.
    2. Kleijnen, Jack P.C. & van Beers, W.C.M. & van Nieuwenhuyse, I., 2011. "Expected Improvement in Efficient Global Optimization Through Bootstrapped Kriging - Replaces CentER DP 2010-62," Other publications TiSEM d3b15c46-27c4-493e-8c53-9, Tilburg University, School of Economics and Management.
    3. Kęstutis Dučinskas & Lina Dreižienė, 2018. "Risks of Classification of the Gaussian Markov Random Field Observations," Journal of Classification, Springer;The Classification Society, vol. 35(3), pages 422-436, October.
    4. David Ginsbourger & Delphine Dupuy & Anca Badea & Laurent Carraro & Olivier Roustant, 2009. "A note on the choice and the estimation of Kriging models for the analysis of deterministic computer experiments," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(2), pages 115-131, March.
    5. Dučinskas, Kęstutis & Dreižienė, Lina & Zikarienė, Eglė, 2015. "Multiclass classification of the scalar Gaussian random field observation with known spatial correlation function," Statistics & Probability Letters, Elsevier, vol. 98(C), pages 107-114.
    6. Bachoc, François, 2014. "Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 1-35.

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