Ranked set sampling (RSS) is a sampling design that combines random sampling with the judgment of researchers through preliminary ranking. The current study introduced a new generalization of RSS, called median-augmented ranked set sampling (MARSS), designed to further reduce the measurement cost and lessen the influence of outliers in estimating the population mean. The proposed MARSS estimator was compared with both simple random sampling (SRS) and RSS estimators. Its exact relative precision and bias were evaluated for a range of symmetric and skewed distributions under perfect ranking. A simulation study was also conducted to assess its performance under imperfect ranking, when using concomitant variables, in the presence of outliers, and when considering ranking cost efficiency. The variance and robustness were also interpreted in topological space. The theoretical results showed that the MARSS estimator was unbiased for symmetric distributions and achieved less variance than both RSS and SRS in unimodal symmetric distributions. Overall, MARSS is more precise than SRS and surpassed RSS in most scenarios, though some bias was observed for skewed distributions. Importantly, MARSS demonstrated a greater robustness to outliers than either SRS or RSS. Finally, the new sampling design was illustrated through an application to body health data analysis.

This paper presents a new generalized differential transform method (NGDTM) in the solution of fractional-order differential equations. The technique is based on the generalized Taylor formula and the Riemann–Liouville fractional derivative. Theorems of fundamental transformation are developed based on rigorous proofs, and the convergence and uniqueness of the solutions obtained are proven. A number of linear and nonlinear examples such as models of statistical relevance are provided to demonstrate the accuracy and efficiency of the proposed approach. Besides, the classical exponential distribution is derived using the proposed NGDTM, and a new fractional exponential distribution is proposed on the same basis with the use of the same framework. The findings reveal that the technique provides very good approximate solutions and, in few instances, the exact solution by only few iterations, thus validating it as a tool of fractional differential equations as well as use in statistics.