Fixed-domain asymptotics for a subclass of Matern-type Gaussian random fields

Stein [Statist Sci. 4 (1989) 432-433] proposed the Matern-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.

[1]  Leon M. Hall,et al.  Special Functions , 1998 .

[2]  Jerome Sacks,et al.  Designs for Computer Experiments , 1989 .

[3]  W. Welch,et al.  Fisher information and maximum‐likelihood estimation of covariance parameters in Gaussian stochastic processes , 1998 .

[4]  Stephen Wolfram,et al.  The Mathematica book (3rd ed.) , 1996 .

[5]  A. V. D. Vaart.  Maximum likelihood estimation under a spatial sampling scheme , 1996 .

[6]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[7]  Wei-Liem Loh,et al.  Estimating structured correlation matrices in smooth Gaussian random field models , 2000 .

[8]  M. Crowder.  Maximum Likelihood Estimation for Dependent Observations , 1976 .

[9]  A. M. Mathai,et al.  Quadratic forms in random variables : theory and applications , 1992 .

[10]  Zhiliang Ying,et al.  Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process , 1991 .

[11]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

[12]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[13]  Thomas J. Santner,et al.  Sequential design of computer experiments to minimize integrated response functions , 2000 .

[14]  Haotian Hang,et al.  Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics , 2004 .

[15]  C. O'Connor.  An introduction to multivariate statistical analysis: 2nd edn. by T. W. Anderson. 675 pp. Wiley, New York (1984) , 1987 .

[16]  Z. Ying.  Maximum likelihood estimation of parameters under a spatial sampling scheme , 1993 .