The exponential distribution has always been prominent in various disciplines because of its wide range of applications. In this work, a generalization of the classical exponential distribution under a neutrosophic environment is scarcely presented. The mathematical properties of the neutrosophic exponential model are described in detail. The estimation of a neutrosophic parameter by the method of maximum likelihood is discussed and illustrated with examples. The suggested neutrosophic exponential distribution (NED) model involves the interval time it takes for certain particular events to occur. Thus, the proposed model may be the most widely used statistical distribution for the reliability problems. For conceptual understanding, a wide range of applications of the NED in reliability engineering is given, which indicates the circumstances under which the distribution is suitable. Furthermore, a simulation study has been conducted to assess the performance of the estimated neutrosophic parameter. Simulated results show that imprecise data with a larger sample size efficiently estimate the unknown neutrosophic parameter. Finally, a complex dataset on remission periods of cancer patients has been analyzed to identify the importance of the proposed model for real‐world case studies.

The fundamentals of neutrosophic statistics provide a new basis for working with indeterminate data problems. In this study, the notion of the neutrosophic Rayleigh distribution $\left ({{RD}_{N} }\right)$ has been introduced. The neutrosphic extension of the classical Rayleigh model with several application areas is highlighted. The major characteristics of the proposed distribution are described in a way that suggested model can be utilized in different situations involving undetermined, vague and fuzzy data. The usage of proposed distribution notably in the domain of statistical process control $\left ({SPC }\right)$ is considered. The classical structure of $V_{SQR}$ -chart is not capable of capturing uncertainty on studied variables. The mathematical structure of the $V_{NR}$ -chart based on the proposed neutrosophic distribution has been developed. The neutrosphic parameters of the proposed $V_{NR}$ -chart with other related performance metrics such as neutrosophy run length $\left ({{ARL}_{N} }\right)$ and neutrosophy power curve $\left ({{PC}_{N} }\right)$ are established. The proposed chart’s performance in a neutrosophic environment is also evaluated to the existing model. Results from this comparative analysis reveal that the suggested $\mathrm {V}_{\mathrm {NR}}$ -chart outperforms its current equivalent in terms of neutrosophic statistical power. Finally, a charting structure of proposed design for service life of ball bearings data is considered with a view to support implementation procedure of the proposed neutrosophic design in real-world scenarios.

In the absence of a correct distribution theory for complex data, neutrosophic algebra can be very useful in quantifying uncertainty. In applied data analysis, implementation of existing gamma distribution becomes inadequate for some applications when dealing with an imprecise, uncertain, or vague dataset. Most existing works have explored distributional properties of the gamma distribution under the assumption that data do not have any kind of indeterminacy. Yet, analytical properties of the gamma model for the more realistic setting when data involved uncertainties remain largely underdeveloped. This paper fills such a gap and develops the notion of neutrosophic gamma distribution (NGD). The proposed distribution represents a generalized structure of the existing gamma distribution. The basic distributional properties, including moments, shape coefficients, and moment generating function (MGF), are established. Several examples are considered to emphasize the relevance of the proposed NGD for dealing with circumstances with inadequate or ambiguous knowledge about the distributional characteristics. The estimation framework for treating vague parameters of the NGD is developed. The Monte Carlo simulation is implemented to examine the performance of the proposed model. The proposed model is applied to a real dataset for the purpose of dealing with inaccurate and vague statistical data. Results show that the NGD has better flexibility in handling real data over the conventional gamma distribution.

Process monitoring is typically performed with quality control charts. The normality assumption is often taken into consideration for the development of control charts. However, this assumption may not hold in many real-life situations. One such circumstance is utilizing the V _SQR chart to track the process variability of Rayleigh distributed processes. The structure of the V _SQR chart follows the basic design of the Shewhart-type chart, which is not sensitive mainly for smaller to moderate shifts in monitoring parameter. In this paper, an enhanced approach, namely Rayleigh exponential weighted moving average (REWMA) is introduced. The suggested REWMA scheme aims to significantly improve the detection capabilities of the traditional V _SQR chart. Designated limits, along with charting parameters, are evaluated for different sample sizes. The effectiveness of the suggested REWMA control chart is assessed in terms of run length distributional parameters. Moreover, the Monte Carlo simulations are made to compare the run length properties of the proposed REWMA chart with the existing competitor design. The comparative assessment demonstrates that the suggested offers a considerable enhancement relative to the competing design. An application of the REWMA chart on simulated data also reveals that the proposed chart is highly sensitive to smaller and persistent shifts in the scaling parameter of Rayleigh distribution. Finally, an example from real-life has been presented to illustrate the importance of the suggested chart.

In recent years, many supplementary designs have been developed incorporating the assumption that data follow the particular non-normal distribution. The VR-control chart is one such design proposed to monitor the parameter σ2 of the single parameter Rayleigh distributed data. Commonly, in authentic situations, practitioners need to estimate the scale parameter σ in the observed processes instead of σ2. However, the positive square root of the VR-statistic used in the existing design of VR-control chart is not an unbiased estimator of σ and thus could not be practiced to monitor the scale parameter σ of the Rayleigh distributed process. A new structure of the VR-control chart namely VSQR for monitoring the scale parameter of the Rayleigh distributed data has been originally developed in this study. The statistical basis of this newly VSQR design in terms of average run length (ARL), characteristic function and power curve have been derived. The analytical results are utilized further to determine the parameters of VSQR-chart and in comparing the performance of the proposed control chart with existing competitors. Comparative results illustrate the effectiveness of the proposed design in view of statistical power. Finally, the computational procedure of this newly VSQR-chart has been demonstrated using simulated data and real data on the breaking strength of carbon fibers.

The Pareto distribution is widely used to model industrial, biological, engineering, and other various types of data. A new generalized model, namely the neutrosophic Pareto distribution (NPD), is developed in this article. The proposed model is a neutrosophic variant of the classical Pareto distribution, potentially useful for analyzing vague, unclear, indeterminate, or imprecise data. The structure form of the proposed distribution is skewed to the right and determined to be unimodal. Several characteristics of the NPD are investigated under the neutrosophic framework. The expressions for basic properties such as mean, variance, raw moments, and shape coefficients are obtained. The maximum likelihood approach is presented for estimating the imprecise distributional parameters of the proposed model. The extended notions of the NPD are explained with various key functions in the domain of applied statistical methods. Finally, the practical benefits of NPD are proven by analyzing two real datasets.