This paper examines various estimation methods for the parameters of the transmuted inverse Rayleigh distribution (TIRD) using both ranked set sampling (RSS) and simple random sampling (SRS) designs. The parameters are estimated using maximum likelihood estimation, ordinary and weighted least squares, and the maximum product of spacings. Additionally, five goodness-of-fit estimators are evaluated: Anderson-Darling (AD), right-tail AD, left-tail AD, left-tail second-order, and the Cramér-von Mises estimator. A comprehensive simulation study is conducted to assess the performance of these estimators while ensuring an equal number of observations across both sampling designs. Furthermore, an analysis of a real COVID-19 dataset belonging to the Netherlands of 30 days, which is fitted both numerically and graphically to the TIRD, demonstrates the practical applicability of the proposed estimation methods. The results show that RSS-based estimators consistently outperform their SRS counterparts in terms of mean squared error, bias, and mean absolute relative error across all methods. The findings highlight the advantages of RSS for parameter estimation in the TIRD, demonstrating its superiority over SRS for statistical inference. In particular, RSS proves to be more effective when dealing with small sample sizes.

Ranked set sampling is a well-known and efficient method compared to simple random sampling for estimating population parameters. In this study, we focus on the challenge of estimating the scale parameter of the primary variable $Z$ using a multistage ranked set sample obtained by ordering the marginal observations of an auxiliary variable $W$, where the pair $(W,Z)$ follows the Farlie–Gumbel–Morgenstern bivariate Bilal distribution. Assuming that the dependence parameter $\varphi$ is known, we introduce the best linear unbiased estimator for the scale parameter of the primary variable, utilizing a multistage ranked set sample. We also compare the efficiency of the proposed estimator with that of the maximum likelihood estimator based on the same number of measured units. It is found that the suggested estimators are more efficient than the classical estimators considered in this study.

The unit new X-Lindley distribution, which is a novel one-parameter distribution on the unit interval, is presented in this study. It was developed by altering the new X-Lindley distribution using the exponential transformation. This new one-parameter distribution's fundamental features, including moments, incomplete moments, Lorenz and Bonferroni curves, Gini index, mode, extropy, Havrda and Charvat entropy, Rényi entropy, and Tsallis entropy, are explored. Additionally, it has bathtub-shaped hazard rate functions and monotonically increasing hazard rate functions with a single parameter. The new distribution is therefore sufficiently rich to model real data. Also, different estimation methods, such as maximum likelihood, least-squares, and weighted least-squares, are used to estimate the parameters of this model, and using a simulation research, their respective performances are evaluated. Finally, two real-life datasets are used to demonstrate the suggested model's competency.

This research extends traditional statistical distribution theory, which often neglects issues such as ambiguity, imprecision, or indeterminacy. The primary aim is to develop the neutrosophic moment exponential distribution as a refined version of the moment exponential distribution, specifically to tackle situations involving uncertainty. The study derives the proposed model's quantile function, Mills ratio, and elasticity, as well as its mean, variance, $r^{\text{th}}$ moment, index of dispersion, and moment-generating function. It also establishes expressions for the survival function, hazard rate function, cumulative hazard function, and mean residual life function, which are visually explored through graphs. Furthermore, the research calculates information measures including extropy, weighted extropy, cumulative residual extropy, Shannon entropy, and Rényi entropy. The parameters of the proposed model are determined using maximum likelihood estimation, followed by a simulation study and an illustration of the distribution of the order statistics. Finally, the practical superiority of the proposed distribution over several existing models in the literature is demonstrated using a child mortality rate dataset.