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Fixed-domain asymptotics for a subclass of Matérn-type Gaussian random fields. (English) Zbl 1086.62111

Summary: M. L. Stein [Stat. Sci. 4, 432–433 (1989)] proposed the Matérn-type Gaussian random fields as a very flexible class of models for computer experiments. This article considers a subclass of these models that are exactly once mean square differentiable. In particular, the likelihood function is determined in closed form, and under mild conditions the sieve maximum likelihood estimators for the parameters of the covariance function are shown to be weakly consistent with respect to fixed-domain asymptotics.

MSC:

62M40 Random fields; image analysis
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics

Software:

Mathematica
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References:

[1] Abt, M. and Welch, W. J. (1998). Fisher information and maximum likelihood estimation of covariance parameters in Gaussian stochastic processes. Canad. J. Statist. 26 127–137. · Zbl 0899.62124
[2] Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis , 2nd ed. Wiley, New York. · Zbl 0651.62041
[3] Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions . Cambridge Univ. Press. · Zbl 0920.33001
[4] Crowder, M. J. (1976). Maximum likelihood estimation for dependent observations. J. Roy. Statist. Soc. Ser. B 38 45–53. · Zbl 0324.62023
[5] Loh, W.-L. (2003). Fixed-domain asymptotics for a subclass of Matérn-type Gaussian random fields. Available at www.stat.nus.edu.sg/ wloh/randomfield.pdf.
[6] Loh, W.-L. and Lam, T.-K. (2000). Estimating structured correlation matrices in smooth Gaussian random field models. Ann. Statist. 28 880–904. · Zbl 1105.62376
[7] Matérn, B. (1986). Spatial Variation , 2nd ed. Lecture Notes in Statist. 36 . Springer, New York. · Zbl 0608.62122
[8] Mathai, A. M. and Provost, S. B. (1992). Quadratic Forms in Random Variables : Theory and Applications. Dekker, New York. · Zbl 0792.62045
[9] Sacks, J., Schiller, S. B. and Welch, W. J. (1989). Designs for computer experiments. Technometrics 31 41–47.
[10] Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments (with discussion). Statist. Sci. 4 409–435. · Zbl 0955.62619
[11] Stein, M. L. (1989). Comment on “Design and analysis of computer experiments,” by Sacks et al. Statist. Sci. 4 432–433.
[12] Stein, M. L. (1999). Interpolation of Spatial Data : Some Theory for Kriging. Springer, New York. · Zbl 0924.62100
[13] van der Vaart, A. (1996). Maximum likelihood estimation under a spatial sampling scheme. Ann. Statist. 24 2049–2057. · Zbl 0896.62029
[14] Williams, B. J., Santner, T. J. and Notz, W. I. (2000). Sequential design of computer experiments to minimize integrated response functions. Statist. Sinica 10 1133–1152. · Zbl 0961.62069
[15] Wolfram, S. (1996). The Mathematica Book , 3rd ed. Cambridge Univ. Press. · Zbl 0878.65001
[16] Ying, Z. (1991). Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process. J. Multivariate Anal. 36 280–296. · Zbl 0733.62091
[17] Ying, Z. (1993). Maximum likelihood estimation of parameters under a spatial sampling scheme. Ann. Statist. 21 1567–1590. · Zbl 0797.62019
[18] Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99 250–261. · Zbl 1089.62538
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