Sequential Estimation of the Mean Vector of a Multivariate Linear Process

Sequential procedures are proposed to estimate the unknown mean vector of a multivariate linear process of the form Xt − μ = ∑∞j = 0AjZt − j, where the Zt are i.i.d. (0, Σ) with unknown covariance matrix Σ. The proposed point estimation is asymptotically risk efficient in the sense of Starr (1966, Ann. Math. Statist.37 1173-1185). The fixed accuracy confidence set procedure is asymptotically efficient with prescribed coverage probability in the sense of Chow and Robbins (1965, Ann. Math. Statist.36 457-462). A random central limit theorem for this process, under a mild summability condition on the coefficient matrices Aj, is also obtained.