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

van der Vaart, Aad

Ann. Stat. 24, No. 5, 2049-2057 (1996).

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

Keywords:

local asymptotic normality; maximum likelihood estimator; asymptotic efficiency

Citations:

Zbl 0797.62019