## Maximum likelihood estimation under a spatial sampling scheme.(English)Zbl 0896.62029

Summary: Motivated by the modelling of computer experiments, Z. Ying [ibid. 21, No. 3, 1567-1590 (1993; Zbl 0797.62019)] considers estimation of the parameter $$(\lambda, \mu, \sigma^2)$$ based on a matrix-valued observation $$X= (X_{i,k})$$ (where $$i= 1,\dots,m$$ and $$k= 1,\dots,n$$) from a multivariate normal distribution with mean zero and covariances given by $\text{cov} (X_{i,k}, X_{j,l})= \sigma^2\exp (-\lambda| u_i-u_j|- \mu| v_k-v_l |).$ The grids $$0\leq u_1< u_2<\cdots <u_m\leq 1$$ and $$0\leq v_1< v_2<\cdots<v_n\leq 1$$ are known to the experimenter and the parameters $$\lambda$$ and $$\mu$$ are positive. Suppose that the grids become dense in $$[0,1]$$ in such a way that $\max| u_{i+1}-u_i|= o(m^{-1/2}); \qquad \max| v_{k+1}-v_k|= o(n^{-1/2}).$ Under this condition, Ying establishes asymptotic normality of the maximum likelihood estimator $$(\widehat{\lambda}, \widehat{\mu}, \widehat{\sigma}^2)$$ as $$m,n\to \infty$$ in such a way that $$n/m\to \rho\in (0,\infty)$$.
Somewhat surprisingly the “usual” theory concerning asymptotic efficiency of the maximum likelihood estimator does not apply. However, Ying conjectures that the maximum likelihood estimator is nevertheless asymptotically efficient. In this note we show this to be true.

### MSC:

 62F12 Asymptotic properties of parametric estimators 62M30 Inference from spatial processes 60G60 Random fields

Zbl 0797.62019
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