Improve estimation when multiple nonzero initial coordinates are available

Improve the estimators of the SPDE parameters theta and mu_0 in the case where multiple distinct index pairs (l_1,l_2) in mathbb{N}^2 satisfy the nonzero-coordinate condition [A2], namely langle X_0,e_{l_1,l_2}rangleneq 0.

Background

The estimation procedure for theta_0 and mu_0 selects one index pair (l_1,l_2) for which the initial coordinate langle X_0,e_{l_1,l_2}rangle is nonzero, as guaranteed by condition [A2]. The selected coordinate process is used to construct quasi-maximum likelihood estimators of the corresponding OrnsteinUhlenbeck drift and noise-scale parameters, which are then transformed into estimators of theta_0 and mu_0.

The paper also considers the stronger condition [A2]' under which two distinct coordinate pairs have nonzero initial coordinates. Using both coordinate pairs permits an alternative estimator of theta_2 and consequently weighted estimators of the full parameter vector. However, the authors do not develop an improved general procedure for the situation in which more than one pair satisfies [A2]; they explicitly identify improving the estimators in that setting as future work.