The ranked set sampling (RSS) procedure was initially established by McIntyre (1952) for estimating the mean of forage and pasture yield as more precise than simple random sampling (SRS). Recently, Aldrabseh and Ismail (2023) suggested the except extreme RSS (EERSS) approach as a modification to RSS for estimating the population mean. In this paper, a new estimator of the population variance is proposed using the EERSS method. The mean squared error and bias equations of the new estimator are derived. When the underlying distribution is non-symmetric, a simulation study is conducted to evaluate the suggested estimator relative to SRS and RSS, based on the same number of measured units, in terms of the relative precision and bias values for several sample sizes. For symmetric distributions, the exact values of the bias and relative precision of the EERSS variance estimator are evaluated. Two real datasets are utilized to illustrate the performance of the suggested variance estimator. It is found that the EERSS variance estimator is more efficient than the SRS estimator and more precise than RSS in most cases, especially for small set sizes.

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.

We develop the theory of Difference Lindelöf perfect functions. Through difference covers, we provide intrinsic characterizations; prove stability under composition, subspace restriction, and suitable products; and obtain preservation theorems. Under standard separation axioms, properties such as D-countable compactness, regularity, paracompactness, and the closedness of projections transfer along D-Lindelöf perfect maps. We also connect the framework to statistics. Uses include decision regions expressed as differences of open sets and parameter screening, with visualizations of countable subcovers and their pushforwards. The results point to practical countable cores for learning and inference and suggest extensions to bitopological and fuzzy contexts.

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.