<p>Accurate estimation of location and scale parameters is fundamental across environmental monitoring, quality control, and medical research. Traditional methods often require large sample sizes for precise estimation, which can be prohibitively expensive in practice. The moving extreme ranked set sampling design (MERSSD) offers a cost-effective alternative where only extreme units are measured after ranking, but existing estimation methods under MERSSD rely primarily on best linear unbiased estimators (BLUEs). While BLUEs guarantee unbiasedness, they may sacrifice efficiency, particularly in small samples-a common scenario in MERSSD applications. To address this gap, we develop a unified framework of best linear invariant estimators (BLIEs) for location and scale parameters under MERSSD across three scenarios: known scale, known location, and both unknown. Theoretically, we prove that the proposed BLIE for the location parameter matches the BLUE when the scale is known, but strictly outperforms it in all other cases, achieving uniformly smaller mean squared error (MSE). Extensive numerical simulations across various distributions confirm efficiency gains of up to 112% compared to BLUEs under perfect ranking, with the most pronounced improvements in small-sample settings. Under imperfect ranking, efficiency gains can exceed 273%, demonstrating robustness to ranking errors. The practical utility of the proposed estimators is illustrated using a real ecological dataset. This work provides a statistically superior and computationally tractable estimation approach under MERSSD, with direct implications for fields where measurement costs are high but ranking is feasible.</p>