3 × 3 optimal ranked set sampling design with k cycles and best linear invariant estimators of the parameters for normal distribution

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. Cost effective sampling will be an important research problem. In this article, we find a 3 × 3 optimal ranked set sampling (RSS) design with $k$ cycles for the normal distribution $N(\mu ,{\sigma }^2)$ in which the location parameter $\mu$ and the scale parameter $\sigma$ are both unknown based on the D–optimal criterion in the experimental design. Then, the best linear invariant estimates (BLIEs) of $\mu$ and $\sigma$ from $N(\mu ,{\sigma }^2)$ and their properties are studied under this RSS design. The efficiency is compared by the determinant of the mean square error matrix. The theoretical results and numerical results show that the BLIEs under the optimal RSS are more effective than the BLIEs under the balanced RSS.