Abstract
There is a need to comprehend real-world problems that are marked by ambiguity and inflexibility. By taking into account the indeterminacies and inconsistencies, DUS transformation has been taken to Neutrosophic Weibull distribution and DUS-Neutrosophic Weibull distribution is proposed. The probability density function is unimodal and decreasing in nature. Several statistical properties have been studied. The parameters of the proposed distribution are estimated using the maximum likelihood method. The proposed distribution has been validated on a real data set. The estimates are found to be more accurate than the classical distributions.
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Introduction
In the classical theory of statistics, the data are formed in crisp numbers and analyzed. Since one cannot always compute or offer exact values for statistical traits in real life, it is quite desirable to approximate them. Making the transition from classical to neutrosophic statistics is one technique; depending on the types of indeterminacies, however, numerous approaches may be available. Several advances have been made in recent years to model such inaccurate situations using fuzzy logic and neutrosophy [5, 9, 34, 36, 39, 40]. In addition, according to neutrosophic logic, the parameters take undetermined values which arise while working with statistical data, particularly when working with vague and inaccurate statistical data [5]. Smarandache [33, 36] proposed that a logical statement can be represented in a 3D Neutrosophic Space, where each dimension of the space represents the statement's Truth (T), Falsehood (F), and Indeterminacy (I). Several studies have been conducted using neutrosophic probability distributions to calculate indeterminacy in real-world scenarios, with better results than classical statistics. The concepts of probability distribution in crisp logic were generalized to neutrosophic logic with applications. In terms of neutrosophic statistics, Alhasan and Smarandache [3] proposed the Neutrosophic Weibull distribution. Similarly, distributions like neutrosophic-Poisson, neutrosophic-Uniform, neutrosophic-Exponential are available in the literature. A neutrosophic-Beta distribution for the data is in interval form was developed by Sherwani et al. [32]. Patro and Samarande [28] introduced Neutrosophic Binomial and Neutrosophic Normal distributions. These distributions are provided with an additional room in the relevant area, allowing it to solve more problems that were previously neglected in classical statistics due to indeterminacy and aberrant values [34]. Neutrosophic time series prediction [30] and modelling were also examined in a variety of scenarios, including neutrosophic moving averages, neutrosophic logarithmic models, neutrosophic linear models, and so on, see Guan et al. [13]. Neutrosophic logic has successfully solved a variety of decision-making challenges, such as examining green credit ratings and personnel selection, see Nabeeh et Al. [24,25,26]. Aslam et al. [2, 4, 6,7,8] extended the idea of control charts and sampling plans with the presence of indeterminacy and presented various neutrosophic control charts and sampling plans, as well as discussed applying the neutrosophic theory to engineering systems.
The Weibull model is one among the few essential distributions used in reliability theory that are crucial to understanding dependability applications. Many of the analyses using the Weibull model are now available across the engineering literature [18, 22, 27]. Fisher and Tippett [31] proposed a basic theory of the Weibull distribution as one of three types of extreme value distributions. Grabner et al. [12] used the complex combination of Weibull density for wind speed data. The effect of the shape and scale parameters on distribution characteristics such as the reliability, hazard rate etc. is a key feature of the Weibull distribution. The discrete additive Weibull distribution is proposed by Unnikrishnan et al. [31]. An extended version and modified version of Weibull distribution were studied in reliability engineering systems by Yong [39] and Hong [15], respectively. Kundu and Ganguly [20] initiated a cumulative exposure model using Weibull distribution. Wirwicki [38] developed a two-parameter Weibull distribution and studied the strength of ceramic materials like Zirconium dioxide and Wais [37] applied Weibull distribution in the wind sector. Hangal [14] studied the use of generalized Weibull distribution in public health and disease modelling. Quinn and Quinn [29] have reviewed Weibull statistics in reporting strengths of dental materials. Lai et al. [21] prepared a monograph on the Weibull distribution and its extensions that cover practically every aspect.
The paper proposed a new life distribution using the DUS transformation of Neutrosophic Weibull distribution. In the literature of statistics, there are various DUS transformed distributions proposed with some well-known baseline distributions. An approach for proposing new distributions with the DUS transform is found to be effective for modelling. A DUS transformation method has been used to analyze patient information on bladder cancer by Kumar et al. [19]. Kavya and Manoharan [17] introduced the Generalized DUS (GDUS) transformation of Weibull distribution. Deepthi and Chacko [10] studied the properties of DUS-Lomax distribution for survival data analysis. Tripathi et al. [36] studied the inferences for the DUS-Exponential Distribution based on upper record values. Maurya et al. [23] worked with Lindley distribution. AbuJarad et al. [1] developed Bayesian survival analysis on generalized DUS-exponential. Gauthami and Chacko [11] proposed an upside-down failure rate model using DUS-inverse Weibull distribution. Karakaya et al. [16] developed DUS-Kumaraswamy distribution with applications in biomedical and epidemiological research.
The paper is organized into seven sections. The next section describes the probability density (pdf) function, cumulative density function (cdf), and hazard rate function. The statistical properties are studied in the subsequent section followed by which the parameter estimation using the method of maximum likelihood is discussed. Then a simulation study to check the flexibility of the estimates is explained. In the penultimate section, real-life data analysis is presented and a conclusion is given in the final section.
DUS-neutrosophic Weibull
Let \(f\left( x \right)\) and \(F\left( X \right)\) be the pdf and cdf of the baseline distribution respectively, we consider the DUS transformed distribution with pdf
The cdf of DUS transformation is given as
The hazard rate function is
Suppose \(I_{N} \in \left( {I_{L} ,I_{U} } \right)\) is an indeterminacy interval, where N is the neutrosophic statistical number and let \(X_{N} = X_{L} + X_{U} I_{N}\) be a random variable following neutrosophic Weibull distribution with scale parameter, \(\alpha\) and shape parameter,\(\beta\). The neutrosophic distribution tends to the classical distribution when \(I_{N} = 0\). Therefore, consider neutrosophic Weibull distribution of a random variable (r.v) \(X_{N}\) with neutrosophic probability distribution function (npdf) as
The corresponding neutrosophic cumulative distribution (ncdf) is
Using Eqs. (1), (2), and (3), DUS-neutrosophic Weibull (DUS-NW) distribution is proposed below with two parameters such as \(\alpha\), the scale parameter and \(\beta\), the shape parameter. The npdf and ncdf of the proposed distribution, respectively, are
and
Also, the hazard rate function is
From Eq. (4), the DUS-neutrosophic Weibull becomes the DUS-Weibull model when \(I_{N} = 0\) and it is a special case of a generalized DUS-exponential model studied by Kavya and Manoharan [17].
Shape of the density curve
From Fig. 1, it can observe the following properties of the npdf of \(DUS - NW\left( {\alpha ,\beta } \right)\).
-
(i)
\(\alpha <1, \beta \ge 1\), \(g(x)\) is decreasing
-
(ii)
\(\alpha \ge 1, \beta >1\), \(g(x)\) is unimodal
-
(iii)
\(\alpha <1, \beta <1\), \(g(x)\) is decreasing
-
(iv)
\(\alpha \ge 1, \beta \le 1\), \(g(x)\) is decreasing
Figure 2 represents the plot of ncdf of \(DUS - NW\left( {\alpha ,\beta } \right)\).
The plot of hazard function rate is shown in Fig. 3 which has upside-down bathtub shaped and decreasing shapes.
Statistical properties of DUS-neutrosophic Weibull
The main statistical properties like moments, generating functions, quantile function, order statistics and entropy have been studied in this section.
Moments
The \(r^{th}\) moment of the distribution, for \(r \in N,\mu^{\prime}_{r}\) is given as
Using exponential expansion \(e^{x} = \sum\nolimits_{i = 0}^{\infty } {\frac{{x^{i} }}{i!}}\) and binomial series of expansion \(\left( {1 - y} \right)^{b} = \sum\nolimits_{i = 0}^{\infty } {\left( { - 1} \right)} \left( {\mathop {}\limits_{i}^{b} } \right)y^{i}\), Eq. (9) becomes
The higher order moments can be found similarly. Hence, the mean and variance is given by:
Moment generating function
As \({X}_{N}\sim DUS-NW\left(\alpha ,\beta \right)\), the moment generating function is given as
Characteristic function
The characteristic function of \({X}_{N}\sim DUS-NW\left(\alpha ,\beta \right)\) is given as
Cumulant generating function
The cumulant generating function \({X}_{N}\sim DUS-NW\left(\alpha ,\beta \right)\) is as follows:
Quantile function
The quantile function specifies the value of a random variable so that the probability of the variable being less than or equal to that value equals the specified probability. Suppose is \({X}_{N}\sim DUS-NW\left(\alpha ,\beta \right)\) a neutrosophic random variable with a distribution function \(F\left( {X_{N} } \right)\) such that
Hence, the quantile function is given by
While setting \(p = \frac{1}{2}\) in Eq. (16), we get the median of \(DUS - NW\left( {\alpha ,\beta } \right)\) as
Order statistics
Order statistics are highly pertinent in reliability theory and survival analysis because of the significance of the hazard rate function in these domains. Let \(X_{N1} ,X_{N2} ,...X_{Nn}\) be the n independent and identically distributed (i.i.d) random variable with corresponding order statistics \(X_{N\left( 1 \right)} ,X_{N\left( 2 \right)} ,...X_{N\left( n \right)}\) derived from DUS-NW with density function \(g\left( {x_{N} } \right)\) and distribution function \(G\left( {x_{N} } \right)\). Hence the \(r^{th}\) order statistics for pdf and the cdf is given as
Entropy
The entropy function is used to investigate system reliability and repairability in many engineering systems. Renyi (1961) introduced the quantum form of the classical Rényi entropy. Let \({X}_{N}\sim DUS-NW\left(\alpha ,\beta \right)\) and \(\gamma\) is the order \(\left( {\gamma \ne 1,\gamma \ge 0} \right)\).
where \(\tau_{\gamma } \left( \gamma \right)\) is a non-decreasing function of \(\gamma\).
Estimation of parameters
The method of the maximum likelihood function for parameter estimation is discussed in this section.
Maximum likelihood function
For a given n observation \(x_{N1} ,x_{N2} ,....x_{Nn}\), the likelihood function of \({X}_{N}\sim DUS-NW\left(\alpha ,\beta \right)\) is given as
The corresponding log-likelihood is obtained as
The maximum likelihood estimator (MLE) for the parameters \(\alpha \) and \(\beta \) can be obtained by maximizing the log-likelihood function. The derivatives corresponding to the parameters \(\alpha \) and \(\beta \) are given by
With the difficulty in the computation of the non-linear equations, Eqs. (23) and (24) can be solved by the inversion method using statistical software R. This yields the ML estimates of \(\alpha \) and \(\beta \) for a chosen initial value.
Simulation studies
To evaluate the performance of MLE of the DUS-NW, an extensive examination has been done. This is compared with the flexibility of the proposed distribution with two indeterminacy measures, i.e. \({I}_{N}=(0.2, 0.5)\) and \({I}_{N}=(0.6, 0.8)\) with classical DUS-Weibull distribution \(({I}_{N}=0)\). Bias and mean square error (MSE) are calculated for different values \(\alpha \) and \(\beta \) for assorted sample sizes \(n=100, 250, 500, 1000\). The results are given below.
The samples are generated using the inversion method for different values of parameters and for two interval measures with \({I}_{N}=(0.2, 0.5)\) and \({I}_{N}=(0.6, 0.8)\). From Tables 1, 2, 3, it can be observed that the Bias and MSE results of DUS-NW (\(\alpha ,\beta )\) are lower compared to classical DUS-Weibull distribution for increasing values of sample size with the different parameter sets for \(\alpha \) and \(\beta \). Here the indeterminacy parameter \({I}_{N}\) plays a crucial role in reducing the error for the estimates.
Application
In this section, the parameters of the proposed distribution is computed using real-life data examples, and the goodness of fit of the proposed distribution is verified using the AIC (Akaike's Information criteria) and BIC (Bayesian Information criteria) values. The real-life neutrosophic data (see Table 4) of population density of some villages in the USA are considered, see Albassam et al. [2].
The MLE approaches are used to determine the parameters of the DUS-NW(\(\alpha ,\beta )\) distribution, and the goodness of fit is used to assess the performance of the models. The estimates are shown in Table 5.
In terms of neutrosophy, it can be demonstrated that given results of DUS-NW(\(\alpha ,\beta )\) in terms of imprecision and uncertainty are more accurate and flexible, rather than ignoring imprecision and uncertainty that of DUS-Weibull. As a result, when the data have an interval measure \(({I}_{N})\) and contains some indeterminacy, the recommended DUS-NW((\(\alpha ,\beta )\) distribution has more accurate results than the standard DUS-Weibull distribution. Moreover, DUS-NW(\(\alpha ,\beta )\) has a better-fit and contending results among all other distributions mentioned in the study.
Conclusion
Unlike classical probability distributions, which accept a specific value, neutrosophic probability distributions adopt an intrusive approach to explaining and solving a wide range of real-world situations. This explains why neutrosophic statistics take into account aberrant and imprecise values that classical logic ignores. The Weibull distribution has a wide range of applications in several disciplines such as engineering system, reliability, and sampling plans so that by considering the interval form of data that have imprecisions and arises in many real-life scenarios, thus a DUS-transformation on Neutrosophic Weibull distribution have been proposed. It has been established with several properties of the proposed distribution such as shape, the measure of spread r-th moment, hazard rate function, generating functions, and order statistics. An MLE approach was performed for estimating the parameters. The efficiency of the distribution was investigated by a simulation study. The flexibility of the proposed distribution is driven with an application to a real-life data set with comparison to the classical distributions and Neutrosophic Weibull. It has been concluded that the proposed DUS-NW offers reliable results even by dealing with data's indeterminacy and uncertainty. The study can be further extended to neutrosophic multivariate statistics considering many parameters. Furthermore, future research can be made to plithogenic statistics [35] which is a generalization of classical multivariate statistics.
Data availability
The data are given in the paper.
References
AbuJarad MH, Khan AA, AbuJarad ES (2020) Bayesian survival analysis of generalized DUS exponential distribution. Austrian J Stat 49(5):80–95
Albassam M, Khan N, Aslam M (2020) The W/S test for data having neutrosophic numbers: An application to USA village population. Complexity
Alhasan KFH, Smarandache F (2019) Neutrosophic Weibull distribution and Neutrosophic Family Weibull Distribution. Infinite Study
Al-Marshadi AH, Shafqat A, Aslam M, Alharbey A (2021) Performance of a new time-truncated control chart for Weibull distribution under uncertainty. Int J Comput Intell Syst 14(1):1256–1262
Ashbacher C (2014) Introduction to Neutrosophic logic. Infinite Study.
Aslam M, Arif OH (2018) Testing of grouped product for the Weibull distribution using neutrosophic statistics. Symmetry 10(9):403
Aslam M (2019) Attribute control chart using the repetitive sampling under neutrosophic system. IEEE Access 7:15367–15374
Aslam M, Bantan RA, Khan N (2019) Monitoring the process based on belief statistic for neutrosophic gamma distributed product. Processes 7(4):209
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96
Deepthi KS, Chacko VM (2020) An Upside-Down Bathtub-Shaped Failure Rate Model Using a DUS Transformation of Lomax Distribution. Stochastic Models in Reliability Engineering. CRC Press, Boca Raton, pp 81–100
Gauthami P, Chacko VM (2021) Dus transformation of inverse Weibull distribution: an upside-down failure rate model. Reliab Theory Appl 16(2(62)):58–71
Gräbner J, Jahn J (2017) Optimization of the distribution of wind speeds using convexly combined Weibull densities. Renew Wind Water Solar 4(1):1–8
Guan H, Dai Z, Guan S, Zhao A (2019) A neutrosophic forecasting model for time series based on first-order state and information entropy of high-order fluctuation. Entropy 21(5):455
Hanagal DD (2017) Frailty models in public health. Handbook of statistics, vol 37. Elsevier, Amsterdam, pp 209–247
HONG J (2009) Modified Weibull distributions in reliability engineering
Karakaya K, Kinaci İ, Coşkun KUŞ, Akdoğan Y (2021) On the DUS-Kumaraswamy Distribution. Istatistik J Turk Stat Assoc 13(1):29–38
Kavya P, Manoharan M (2020) On a Generalized Lifetime Model Using DUS Transformation. Applied Probability and Stochastic Processes. Springer, Singapore, pp 281–291
Kumar AR, Krishnan V (2017) A Study on System Reliability in Weibull Distribution. Int J Innov Res Electr Electron Instrum Control Eng
Kumar D, Singh U, Singh SK (2015) A method of proposing new distribution and its application to Bladder cancer patients data. J Stat Appl Pro Lett 2(3):235–245
Kundu D, Ganguly A (2017) Analysis of step-stress models: existing results and some recent developments. Academic Press, Cambridge
Lai CD, Murthy DN, Xie M (2006) Weibull distributions and their applications. Springer Handbooks. Springer, New York, pp 63–78
Martz HF (2003) Reliability Theory
Maurya SK, Kaushik A, Singh SK, Singh U (2017) A new class of exponential transformed Lindley distribution and its application to yarn data. Int J Stat Econ 18(2):135–151
Nabeeh NA, Smarandache F, Abdel-Basset M, El-Ghareeb HA, Aboelfetouh A (2017) An Integrated Neutrosophic-TOPSIS Approach and its Application to Personnel Selection: A New Trend in Brain Processing and Analysis. IEEE Access, pp 29734–29744
Nabeeh NA, Abdel-Basset M, Soliman G (2021) A model for evaluating green credit rating and its impact on sustainability performance. J Clean Prod 280(1):124299
Nabeeh NA, Abdel-Basset M, El-Ghareeb HA, Aboelfetouh A (2019) Neutrosophic Multi-Criteria Decision Making Approach for IoT-Based Enterprises. IEEE Access
Nair U, Sankaran PG, Balakrishnan N (2018) Reliability modelling and analysis in discrete time. Academic Press, Cambridge
Patro SK, Smarandache F (2016) The Neutrosophic Statistical Distribution, More Problems, More Solutions. Infinite Study
Quinn JB, Quinn GD (2010) A practical and systematic review of Weibull statistics for reporting strengths of dental materials. Dent Mater 26(2):135–147
Alhabib R, Salama AA (2020) Using moving averages to pave the neutrosophic time series. Int J Neutrosophic Sci 3(1):14–20
Fisher RA, Tippett LMC (1928) Limiting forms of frequency distribution of the largest or smallest member of a sample. Proc Camb Philos Soc 24:180–190
Sherwani RAK, Naeem M, Aslam M, Raza MA, Abid M, Abbas S (2021) Neutrosophic beta distribution with properties and applications. Neutrosophic Sets Syst 41:209–214
Smarandache F (2005) Neutrosophic set-a generalization of the intuitionistic fuzzy set. Int J Pure Appl Math 24(3):287
Smarandache F (2014) Introduction to Neutrosophic Statistics. Sitech & Education Publishing, Craiova
Smarandache F (2021) Plithogenic probability & statistics are generalizations of multivariate probability & statistics. Neutrosophic Sets Syst 43:280–289
Tripathi A, Singh U, Singh SK (2019) Inferences for the dus-exponential distribution based on upper record values. Ann Data Sci 1–17
Wais P (2017) A review of Weibull functions in wind sector. Renew Sustain Energy Rev 70:1099–1107
Wirwicki M (2018) Two-parametric analysis of the Weibull distribution strength of advanced ceramics materials. In: E3S Web of conferences, Vol. 49. EDP Sciences, p 00130
Yong T (2004) Extended Weibull distributions in reliability engineering
Zeina MB, Hatip A (2021) Neutrosophic Random Variables. Neutrosophic Sets Syst 39(1)
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Nayana, B.M., Anakha, K.K., Chacko, V.M. et al. A new neutrosophic model using DUS-Weibull transformation with application. Complex Intell. Syst. 8, 4079–4088 (2022). https://doi.org/10.1007/s40747-022-00698-6
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DOI: https://doi.org/10.1007/s40747-022-00698-6