This study reports on the asymptotic behavior of the maximum likelihood function, encountered when constructing Kriging approximations using the Gaussian correlation function. Of specific interest is a maximum likelihood function that decreases continuously as the correlation function hyper-parameters approach zero. Since the global minimizer of the maximum likelihood function is an asymptote in this case, it is unclear if maximum likelihood estimation (MLE) remains valid. Numerical ill-conditioning of the correlation matrix also occurs in this case. Analytical and numerical examples are presented that demonstrates the validity of MLE, provided that arbitrary precision arithmetic is used. A recent result that claims the MLE function always approaches infinity as the hyper-parameters approach zero is also disproved.
Reference:
Kok, S. The asymptotic behaviour of the maximum likelihood function of Kriging approximations using the Gaussian correlation function. EngOpt 2012 - International Conference on Engineering Optimization, Rio de Janeiro, Brazil, 1-5 July 2012
Kok, S. (2012). The asymptotic behaviour of the maximum likelihood function of Kriging approximations using the Gaussian correlation function. http://hdl.handle.net/10204/6033
Kok, S. "The asymptotic behaviour of the maximum likelihood function of Kriging approximations using the Gaussian correlation function." (2012): http://hdl.handle.net/10204/6033
Kok S, The asymptotic behaviour of the maximum likelihood function of Kriging approximations using the Gaussian correlation function; 2012. http://hdl.handle.net/10204/6033 .