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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, March 18, 2003}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Wei Liem Loh}
\centerline{\sl National University of Singapore}
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\centerline{\bf Fixed-Domain Asymptotics for Gaussian Random Fields with 
Matern-Type Covariance}
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In the modeling of computer experiments, it has become rather common
practice to approximate the deterministic response as a realization of a 
stochastic process. In this regard, Jerry Sacks, et. al. 
(\sl{Statist. Sci.}, 1989) proposed modeling using a Gaussian random 
field $X(x), x\in [0,1]^d$, with a multiplicative covariance function 

$$
  {\rm Cov}(X(x), X(y)) = \sigma^2 \prod_{t=1}^d 
  exp(-\theta_t |x_t-y_t|^{\gamma}) , \quad
  x= (x_1,\cdots, x_d)', y=(y_1,\cdots,y_d)',
$$

where $\gamma\in (0,2], \theta_1,\cdots, \theta_d$ and $\sigma^2$ are
strictly positive parameters. 

Michael Stein (\sl{Statist. Sci.}, 1989) observed that the above
Gaussian model may have some undesirable properties. In particular for 
$\gamma\in (0,2)$, the Gaussian random field with this covariance function 
will not be mean square differentiable. However for the case $\gamma=2$, 
it is infinitely mean square differentiable. Not allowing for processes 
that are differentiable but not infinitely differentiable may be 
unnecessarily restrictive. Stein further suggested using a Gaussian random 
field model, $X(x)$, $x\in [0,1]^d$, with the multiplicative 
Mat\'{e}rn-type covariance function
$$
{\textrm Cov}(X(x),X(y)) =  \prod_{t=1}^d \frac{\pi^{1/2} \phi}{2^{\alpha-1} 
\Gamma (\alpha+1/2) \theta_t^{2\alpha}} (\theta_t |x_t-y_t|)^{\alpha}
K_{\alpha} (\theta_t |x_t-y_t|)$$

where $\alpha, \phi, \theta_1,\cdots, \theta_d$ are positive constants
and $K_{\alpha}$ is the modified Bessel function of the second kind.
The interesting point is that $X$ will be $m$ times mean square
differentiable if and only if $\alpha> m$.

In this talk, we shall focus on some fixed-domain asymptotic results for
Gaussian random fields with $\alpha = 3/2$ multiplicative 
Mat\'{e}rn-type covariance functions.

For a PDF version of the above abstract, please see the attached file.

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