DUS-neutrosophic multivariate inverse Weibull distribution: properties and applications

The existing DUS-multivariate inverse Weibull distribution under classical statistics can be applied when all observations in the data are imprecise. In this paper, we introduce DUS-neutrosophic multivariate inverse Weibull distribution that can be used when the observations in the data are imprecise or in intervals. We derive some statistical properties and functions of DUS-neutrosophic multivariate inverse Weibull distribution. We also discuss the maximum likelihood estimation method for estimating the parameters. Monte-Carlo simulation study is performed to study the behavior of maximum likelihood estimates. We compare the efficiency of the proposed DUS-neutrosophic multivariate inverse Weibull distribution with the existing distributions under classical statistics. From the comparison, it is found that the proposed DUS-neutrosophic multivariate inverse Weibull distribution provides smaller values of Akaike’s information criteria and Bayesian information criteria than the existing distributions under classical statistics. The proposed study can be extended for other statistical distributions as future research.


Introduction
Multivariate distributions are very important in many applications in real life also it is important in studying multivariate analysis, regression analysis, and so on. Many methods are used to construct the bivariate and multivariate distributions such as various copula functions see [2][3][4]14]. In this study, we are interested in multivariate inverse Weibull distribution. The main aim of choosing this distribution is the inverse Weibull distribution is a lifetime probability distribution that can be used in reliability, engineering, and biological studies for more details see [16,24]. Inverse Weibull distribution is used also as a stress-strength model see [11]. Definition 1 If a random variable Y has Weibull distribution, then the random variable X In the context of Multivariate inverse Weibull which introduced by Al-Hussaini and Ateya [3] and denoted as MIWD(α i , β i , γ ) using copula approach as follows.
In recent years, neutrosophic statistics is appear and are defined as an extension of classical statistics and one deal with set values instead of crisp values. Ashbacher [5] introduced fuzzy logic and neutrosophy, for more details on neutrosophic statistics, see [25,26,28]. Many authors interested in neutrosophic probability distribution and are shown that it is better than the classical statistics in real life problems such as [25] introduced the neutrosophic Weibull distribution, [22,23] introduced neutrosophic beta distribution, Patro and Smarandache [19] introduced neutrosophic normal distribution and neutrosophic binomial distribution. These new distributions allowed us to solve more problems depending on indeterminacy which is ignored in classical statistics. Some authors constructed the new distributions using DUStransformation which is introduced by [15] and used it to construct a new distribution bounded within [0,1].
Definition 5 Let f (x), F(x) and h(x) denote, respectively, probability density function, the cumulative distribution function, and the hazard rate function of baseline distribution. Then the DUS-family are given by where x ∈ D and D is a domain of the baseline distribution.
Since multivariate analysis is one of the most useful methods to determine relationships and analyze patterns among large sets of data. It is particularly effective in minimizing bias if a structured study design is employed. However, the complexity of the technique makes it a less sought-out model for novice research enthusiasts. Therefore, although the process of designing the study and interpretation of results is a tedious one, the techniques stand out in finding the relationships in complex situations. Since the multivariate analysis depends on the multivariate distribution, then we need more multivariate distributions also for indeterminacies in real data, then we need to construct neutrosophic multivariate distributions. A rich literature on DUS-transformation is available under classical statistics. The existing distributions cannot be applied when the data have imprecise observations. To overcome this issue, we will introduce DUS-MIWD which can be applied when the data has imprecise observations. By exploring the literature and according to the best of our knowledge; there is no work on DUS-MIWD under neutrosophic statistics. To fill this gap, we will propose DUS-MIWD under neutrosophic statistics. Then our aim in this study is to extend the use of DUS-transformation to construct new distribution called DUS-neutrosophic multivariate inverse Weibull distribution and denoted by DUS-NMIWD. This paper is organized as follows: the pdf, CDF, and hazard function of our new distribution are introduced, some statistical properties of our new distribution are obtained, an estimation of distribution parameters is obtained, a simulation study was performed to discuss the distribution parameters is introduced, the real example is introduced .

DUS-neutrosophic multivariate inverse Weibull
Suppose that I N ∈ (I L , I U ) is an indeterminacy interval N is neutrosophic statistical number and let X sm N X sm L + X smU I N , be a random vector of random variables follow neutrosophic multivariate inverse Weibull X sm N be a random vector with m-variables and S-observations. If X N ∈ [n L , n U ] are observations of neutrosophic from neutrosophic variable P N ∈ [P L , P U ], then The form of neutosophic X N X L + X N I N , I N ∈ [I L , I U ] and in multivariate case X m N X mL + X m N I N . Therefore, we consider neutrosophic multivariate inverse Weibull distribution of a random vector X N (X 1N . . . X m N ) with neutrosophic joint probability distribution function (njpdf) as The corresponding neutrosophic joint cumulative distribution (njCDF) is Using Eqs. (8) and (9) in Dus-transformation then we get DUS-neutrosophic multivariate inverse Weibull distribution (DUS-NMIWD) with njpdf, njCDF, and NJhf as follow, respectively (Figs. 3,4,5). and, Shape parameters 1 2 Shape parameters 1 2 , , < 1 > 1

Statistical properties of DUS-NMIWD
In this section, we obtained some important statistical properties of DUS-NMIWD, when m 2, is called DUSneutrosophic bivariate inverse Weibull distribution and is denoted by DUS-NBIWD. The neutrosophic joint probability density function, the neutrosophic joint cumulative distribution function, the neutrosophic joint hazard function, the neutrosophic marginal distribution, the neutrosophic product moments, the neutrosophic moment generating function, the neutrosophic conditional distribution and neutrosophic reliability function.

The neutrosophic marginal distribution
The neutrosophic marginal distributions of X 1N and X 2N are getting as follows where, i, j 1, 2 and i j Then the neutrosophic marginal distributions of X 1N and X 2N are And,

Neutrosophic product moments
If the random vector (x 1N , x 2N ) is distributed as DUS-NBIWD, then the rth and sth moments about the origin is given by

Neutrosophic moment generating function
If the random vector (x 1N , x 2N ) is distributed as DUS-NBIWD, then the moment generating function is defined by

Neutrosophic conditional distribution
The conditional probability distribution of X 1N given X 2N is given as follows Also, we can get the conditional probability distribution of X 2N given X 1N is given as follows

Maximum likelihood estimation method
To estimate the parameters of our new distribution we use the maximum likelihood estimation method which introduced by Elaal and Jarwan [8] for all joint models. Now, the maximum likelihood estimates of the vector (α 1 , α 2 , β 1 , β 2 , γ ) is given as follows,

Comparative studies using real data
To show the performance of our new distribution DUS-NMIWD (bivariate case), we compare it and other distributions i.e., DUS-MIWD, NMIWD and MIWD (bivariate case) on real data which represent the data of the quality control division for quality characteristics of glass production see [27]. There are two quality characteristics such as cutter line X 1N and Edge distortion X 2N . The target of the cutter is 115 mm, and the Edge distortion is 40 mm. The data is shown in Table 4 as follows. Figure 6 shows the architectural diagram of the proposed algorithm in this section. In this section, we use two statistical criteria to test the performance of our new distribution. AIC (Akaike's Information Criteria) and BIC (Bayesian Information Criteria), where, where, : the vector of distribution parameters, L : the likelihood function, k: the number of estimates, n: the data size. The small value of AIC and BIC means good-fit distribution. Table 5 shows the result of the comparison between our new distribution and other distributions under classical statistics. From Table 5, we get the AIC (1175.6, 1341.29) and BIC (1175.81, 1342.62) so it is smaller than other distributions therefore, the DUS-NMIW is the best for data than the other distributions under classical statistics.

Limitations
As mentioned earlier that the neutrosophic statistics is an extension of classical statistics and applied when the observations in the data are imprecise. The proposed DUS-NMIWD has advantages over the existing DUS-MIWD under classical statistics in terms of AIC and BIC. From Table 5, it is clear that the proposed DUS-NMIWD provides smaller values of AIC and BIC as compared to the existing distributions under classical statistics. Therefore, for the interval data, the use of the proposed DUS-NMIWD can be recommended over the existing distributions under classical statistics. The proposed DUS-NMIWD under neutrosophic statistics has some limitations such as it cannot be applied when the multivariate data has no imprecise observations. In addition, the proposed DUS-NMIWD can be applied only when the quality of characteristics follows the normal distribution.

Conclusions
The main aim of this paper was to introduce the new distribution is called DUS-neutrosophic multivariate inverse Weibull distribution. The proposed DUS-NMIWD was the generalization of DUS-MIWD under classical statistics. The proposed DUS-NMIWD reduces to the existing DUS-MIWD when no indeterminate observations are found in the data. We derived some properties of the proposed distribution. The simulation study and the analysis of real examples showed that the proposed DUS-NMIWD performs better than the existing DUS-MIWD under uncertain environments. The proposed DUS-NMIWD can be applied to reliability, and survival analysis, in many engineering fields and medicine when the data have imprecise observations. The proposed DUS-NMIWD can be extended for non-normal distributions as future research. The proposed distribution can be used in a designing control chart for future research.