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 = (K_t(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 X_t=μ + ξ_t, t=1,2,..., where μ is an (unknown) constant and (ξ _t) is a Gaussian sequence such that Eξ_t=0 and K=(Eξ_uξ_v, u,v=1,...,t)_t=1,2.. is an(unknown) covariance structure of (ξ_t). Let P _μ,K)be the probability measure induced by (X_t).Suppose that for every k=1,2,... we can observe simultaneously k independent copies of (X_t).The result is: for every δ>0, ε>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

$${\text{P}}_{{\text{(}}\mu {\text{,K)}}} \{ |\hat \mu _\tau - \mu | > \varepsilon \} < \gamma \,for\,all\,\mu \in R^1 \,and\,K \in C(\delta ).$$