Constructing the D-optimal four-by-four ranked set sampling design for location-scale estimation
摘要
Ranked set sampling design (RSSD) enhances estimation efficiency when precise measurement is costly but ranking is inexpensive. Although optimal RSSD theory is well-developed, a critical practical gap remains: determining specific, implementable D-optimal designs for standard set sizes, especially for multi-parameter estimation. Existing work often defaults to balanced designs or provides only theoretical frameworks. This paper fills that gap by explicitly constructing the D-optimal four-by-four RSSD for normal location and scale parameters under the D-optimality criterion, which minimizes the determinant of the covariance matrix. This criterion is particularly suited for multi-parameter estimation because it minimizes the generalized variance, thereby reducing overall joint estimation uncertainty and yielding the most precise simultaneous inference for both location and scale parameters. The resulting design is counterintuitive: it allocates all measurements exclusively to the extreme order statistics in a (2, 0, 0, 2) pattern. Replicating this D-optimal design over multiple cycles, we derive corresponding best linear invariant estimators (BLIEs). Theoretical and simulation analyses demonstrate substantial efficiency gains: under perfect ranking, BLIEs achieve up to 48.9% higher efficiency than balanced RSSD for moderate samples, with gains increasing with cycle count. Remarkably, under imperfect ranking, efficiency improvements can exceed 200% for small samples. In a real ecological application to tree height data, the proposed D-optimal estimators maintain significant advantages under both perfect and imperfect ranking conditions. Moreover, they outperform the best linear unbiased estimators (BLUEs) under the same D-optimal design, offering superior precision without additional sampling cost. This work bridges optimality theory with practice, delivering a ready-to-use, highly efficient inferential procedure for normal location-scale estimation under RSSD.