This article introduces a new family of distributions through the transformation involving tangent functions, referred to as the New Logarithmic Tangent-U family. The newly introduced family may be written as the NLT-U family of distribution. An overview of fundamental characteristics inherent to the proposed family of distributions is provided. We derived a special sub-model of the NLT-U distributions family using the Weibull distribution as a baseline reference. This special model is termed the New Logarithmic Tangent-Weibull (NLT-Wei) distribution. The hazard function of the special model is very flexible; it captures increasing, decreasing, unimodal, increasing-decreasing-decreasing, and bathtub shapes. The Maximum Likelihood estimation scheme is employed to estimate the model parameters of the proposed family. To assess the stability of the parameter estimation procedure, we perform a Monte Carlo simulation analysis. The results (numerically as well as graphically) of these simulations demonstration that biases and mean square errors decrease as sample sizes grow, even with smaller samples. We thoroughly examine the significance of the NLT-Wei distribution and illustrate its application using three real-world data sets from the engineering field. The suitability and flexibility of the NLT-Wei distribution are compared against several other distributions, including the Exponentiated Weibull, Kumaraswamy Weibull, New Generalized Logarithmic Weibull, Weibull, and Flexible Weibull distributions. The paper ends with a concluding discussion.

Crystal violet (Cry) is an essential textile dye belonging to the triphenylmethane group, that is widely used in the textile industry. It is also applied for paper printing and Gram staining. Previously, it was significant as a topical antiseptic due to its antibacterial, antifungal, and anthelmintic properties. Despite its various applications, crystal violet has been recognized as a biohazard dye due to its toxic and carcinogenic properties. It persists in the environment with long-lasting effects and has detrimental impacts. In this research, water extract from Moringa oleifera leaves is employed as environmentally friendly methods to synthesize zinc oxide nanoparticles (Mo/ZnO-NPs), and characterized by TEM, EDX, FT-IR, and Zeta potential. Mo/ZnO-NPs exhibit a Zeta potential of − 21.9 mV, and X-ray diffraction (XRD) analysis confirms their crystallographic structure. The size of the biogenic Mo/ZnO-NPs ranges from 5.52 to 41.59 nm. This study was designed to estimate and maximize the ability of Mo/ZnO-NPs to remove crystal violet using Central Composite Design (CCD), considering pH (ranging from 3 to 11), incubation time (ranging from 30 to 150), nanoparticles concentrations (ranging from 0.2 to 1.8 mg/mL), and crystal violet concentrations (ranging from 25 to 125 ppm). The maximum percentage value of removal of crystal violet by Mo/ZnO-NPs was 97.26 with optimal conditions of pH 9, incubation time 120 min, Mo/ZnO-NPs 1.4 mg/mL, and crystal violet concentration of 50 ppm. The best-predicted conditions that caused the highest removal of crystal violet (97.8%) were determined using the desirability function as pH 10, incubation time of 140 min, Mo/ZnO-NPs concentrations of 1.3 mg/mL, and a concentration of crystal violet of 80 ppm. Under these optimal conditions, the maximum experimental crystal violet removal% by Mo/ZnO-NPs was (98.7%) was verified. Mo/ZnO-NPs synthesized by Moringa oleifera can be a promising candidate for the adsorption of crystal violet.

Ranked set sampling (RSS) has been proven an efficient alternative to simple random sampling (SRS). The use of auxiliary information also helps to improve the efficiency of the estimation procedures. Therefore, to accomplish higher efficiency and discuss the optimality issues, we proffer some optimal classes of estimators under RSS by employing multi-auxiliary information. It is seen that the ordinary mean estimator, traditional regression, and ratio estimators are the subsets of the proffered estimators. The expressions of the bias and mean square error are reported. An analytical comparison under some optimality conditions points out the ascendancy of the proffered classes of estimators over all reviewed works. The theoretical results have been furnished with computational study by employing some artificial and natural populations. The computational results show that the proffered estimators outperform the conventional estimators reviewed in this study. Furthermore, apposite advices are suggested to the survey persons.

The creating of novel models essentially stems from the requirement to appropriate describe survival cases. In this study, a novel lifetime model with two parameters is proposed and studied for modeling more types of data used in different study cases, including symmetric, asymmetric, skewed, and complex datasets. The proposed model is obtained by compounding the exponential and XLindley distributions, and it is regarded as a strong competitor for the widely applied symmetrical and non-symmetrical models. Several characteristics and statistical properties are investigated. The unknown parameters of the recommended model for the complete sample are estimated using two estimation methods; notably, maximum likelihood estimation and Bayes techniques based on several loss functions as well as an approximate tool are used to construct the confidence intervals for the unknown parameters of the suggested model. The estimation procedures are compared using a Monte Carlo simulation experiment to demonstrate their effectiveness. In the end, the applicability and flexibility of the recommended model are conducted using two real lifetime datasets. In our illustration, we compare the practicality of the recommended model with several well-known competing distributions.