Multiple crossing sequential fixed-size confidence region methodologies for a multivariate normal mean vector

The asymptotically efficient and asymptotically consistent purely sequential procedure of Mukhopadhyay and Al-Mousawi (1986) is customarily used to construct a confidence region $R$ for the mean vector $\mu$ of $N_p(\mu ,{\sigma }^{2}H)$. This procedure does not have the exact consistency property. $H_{p\times p}$ is assumed known and positive definite with ${\sigma }^2$ unknown. The maximum diameter of $R$ and the confidence coefficient are prefixed.

A purely sequential sampling strategy is proposed allowing sampling until sample size crosses the boundary multiple times. We ascertain asymptotic efficiency and asymptotic consistency properties (Theorem 3.1). Its ability to nearly achieve required coverage probability without significant over-sampling is demonstrated with simulations. A truncation technique plus fine-tuning of the multiple crossing rule are proposed to increase practicality. Two real data illustrations are highlighted.