Any multiresponse estimation experiment requires a decision about the number of observations to be taken. If the covariance is unknown, no fixed-sample-size procedure can guarantee that the joint confidence region will have an assigned shape and level. Double-sampling procedures use a preliminary sample of size $m$ to determine the minimum number of additional observations needed to achieve a prescribed accuracy and coverage probability for the parameter estimates. The triple-sampling procedures of this paper, less sensitive to the choice of $m$, revise the sample size estimate after collecting a fraction of the additional observations prescribed under double sampling. Second-order asymptotic results relying on conditional inference show that triple sampling is asymptotically consistent; in addition, the regret for triple sampling is a bounded function of the covariance structure and is independent of $m$.
"Accurate Multivariate Estimation Using Triple Sampling." Ann. Statist. 18 (4) 1615 - 1633, December, 1990. https://doi.org/10.1214/aos/1176347869