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Accurate multivariate estimation using triple sampling. (English) Zbl 0722.62041
With \(\underset{\tilde{}} X_ 1,\underset{\tilde{}} X_ 2,..\). i.i.d. random p-vectors with unknown mean \({\underset{\tilde{}} \theta}\) and positive-definite covariance matrix \({\underset{\tilde{}} \Sigma}\), it is desired to determine a sample size \(\tau\) such that the resulting estimator \({\hat \theta}{}_{\tau}\) of \(\theta\) has accuracy A and confidence \(\gamma\) : P(\({\hat \theta}\)-\(\theta\in A)\geq \gamma\). Under these conditions a sequential or step-sequential procedure will be necessary. Restricting data collection to stages, double-sampling procedures with pilot sample size m will be relatively inefficient if m is small relative to N.
A triple-sampling procedure is proposed, that achieves finite regret and second-order asymptotic efficiency. The method consists in revising the sample size estimate coming from the first (pilot) sample, after collecting a fraction of the additional observations prescribed under double sampling. Double sampling, triple sampling and purely sequential sampling are compared in terms of regrets, asymptotic variances of stopping times, approximate distributions of stopping times, and coverage probabilities.

MSC:
62H12 Estimation in multivariate analysis
62L12 Sequential estimation
62F25 Parametric tolerance and confidence regions
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