Fixed Precision Estimate of Mean of a Gaussian Sequence with Unknown Covariance Structure

Abstract

Let C(δ), δ>0, be the class of sequences of covariance matrices K = (Kt(u, v), u, v=1,...,t)t=1 2,... such that $${{\sum\limits_{u=1}^{t}{\sum\limits_{v=1}^{t}{K}}}_{t}}\left( u,v \right)=0\left( {{t}^{2-\sigma }} \right)ast\to \infty $$ Consider discrete-time stochastic process Xt=μ + ξt, t=1,2,..., where μ is an (unknown) constant and (ξ t) is a Gaussian sequence such that Eξt=0 and K=(Eξuξv, u,v=1,...,t)t=1,2.. is an(unknown) covariance structure of (ξt). Let P μ,K)be the probability measure induced by (Xt).Suppose that for every k=1,2,... we can observe simultaneously k independent copies of (Xt).The result is: for every δ>0, e>0 and γ Є(0,1) there exist a sequential estimate \({{\overset{\wedge }{\mathop{\left( {{\mu }_{t}} \right)}}\,}_{t}}=1,2\ldots \) and a finite stopping rule τ such that $${\text{P}}_{{\text{(}}\mu {\text{,K)}}} \{ |\hat \mu _\tau - \mu | > \varepsilon \} < \gamma \,for\,all\,\mu \in R^1 \,and\,K \in C(\delta ).$$