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 \(X=\frac{1}{Y}\) follows the inverse Weibull distribution denoted as \(\mathrm{IWD}(\alpha ,\beta )\). Then the probability density function (pdf) and cumulative distribution function (CDF) of \(X\) are defined as follow, respectively.

$$f\left(x;\alpha ,\beta \right)=\alpha \beta {x}^{-\alpha -1} {\mathrm{e}}^{- \beta {x}^{- \alpha }},$$
(1)
$$F\left(x;\alpha ,\beta \right)={\mathrm{e}}^{- \beta {x}^{- \alpha }},$$
(2)

where, \(x, \alpha , \beta >0\).

Copula function is a more suitable approach to get bivariate and multivariate distributions for more details see [18] (Fig. 1).

Fig. 1
figure 1

The pdf of inverse Weibull distribution for shape parameter \(\beta =2\) and different scale parameter \(\alpha \)

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Definition 2

Let a random vector \(({X}_{1}\dots {X}_{d})\) and the marginal CDFs denoted by \({F}_{i}\left(x\right)=P[{X}_{i}<x]\) for \(i=1\dots d.\) Then using probability integral transformation (PIT) for each component, the distribution of the random vector \(\left({U}_{1}\dots {U}_{d}\right)=({F}_{1}\left({X}_{1}\right)\dots {F}_{d}\left({X}_{d}\right))\) belongs to the (\(\mathrm{unif}{(\mathrm{0,1})}^{d})\) family distribution and the copula related to \(({X}_{1}\dots {X}_{d})\) is defined as the joint CDF of \(\left({U}_{1}\dots {U}_{d}\right)\),i.e.

$$C\left({u}_{1},\dots ,{u}_{d}\right)=P[{U}_{1}\le {u}_{1}\dots {U}_{d}\le {u}_{d}]$$

With \(\left({u}_{1}\dots {u}_{d}\right)\in {[\mathrm{0,1}]}^{d}\).

Definition 3

\(\mathrm{C}:{[\mathrm{0,1}]}^{d}\to [\mathrm{0,1}]\) is a d-dimensional copula if it’s a CDF with \(C\left({u}_{1}\dots .{u}_{i-1},0,{u}_{i+1}\dots {u}_{d}\right)=0\), \(C\left(1\dots u\dots .1\right)=u,\) with \(\left({u}_{1},\dots ,{u}_{d}\right)\in {[\mathrm{0,1}]}^{d}\) and \(u\in \left[\mathrm{0,1}\right].\)

In the context of Multivariate inverse Weibull which introduced by Al-Hussaini and Ateya [3] and denoted as \(\mathrm{MIWD}({\alpha }_{i},{\beta }_{i},\gamma )\) using copula approach as follows.

Definition 4

Let \({\varvec{X}}=({X}_{1}\dots {X}_{m})\) is a random vector of random variables such that \({X}_{i}\sim \mathrm{IWD}({\alpha }_{i},{\beta }_{i})\),\(i=1\dots m\). Then the CDF and pdf of \({\varvec{X}}\) are defined as

$$F\left({\varvec{X}};\boldsymbol{\alpha },{\varvec{\beta}},\upgamma \right)={\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{i}^{-{\alpha }_{i}}}\right]}^{-\gamma },$$
(3)
$$\begin{aligned}f\left({\varvec{X}};\boldsymbol{\alpha },{\varvec{\beta}},\upgamma \right)&=\frac{\Gamma (\gamma +m)}{\Gamma (\gamma )} \left(\prod_{i=1}^{m}\frac{{\beta }_{i }{\alpha }_{i}}{\gamma } {x}_{i}^{-{\alpha }_{i}-1}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{i}^{-{\alpha }_{i}}}\right)\\&\quad \times{\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{i}^{-{\alpha }_{i}}}\right]}^{-\gamma -m},\end{aligned}$$
(4)

where \(\gamma >0, \boldsymbol{\alpha }=({\alpha }_{1}\dots {\mathrm{\alpha }}_{{\varvec{m}}})\) and \({\varvec{\beta}}=({\beta }_{1}\dots {\upbeta }_{{\varvec{m}}})\) (Fig. 2).

Fig. 2
figure 2

The pdf and CDF of multivariate inverse Weibull distribution \(m=2\)

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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 DUS-transformation which is introduced by [15] and used it to construct a new distribution bounded within [0,1].

Definition 5

Let \(f\left(x\right)\), \(F(x)\) 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

$${f}_{\mathrm{DUS}}\left(x\right)=\frac{1}{e-1} f\left(x\right){e}^{F(x)},$$
(5)
$${F}_{\mathrm{DUS}}\left(x\right)=\frac{1}{e-1} ({e}^{F\left(x\right)}-1),$$
(6)
$${h}_{\mathrm{DUS}}\left(x\right)=\frac{1}{e-{e}^{F\left(x\right)}} f\left(x\right){e}^{F(x)},$$
(7)

where \(x\in D\) and \(D\) is a domain of the baseline distribution.

Many authors used the DUS-transformation to construct new distributions such as [15] introduced DUS-transformation with an exponential distribution, Deepthi and Chacko [6] introduced DUS-transformation with the Lomax distribution and Karakaya [12] introduced DUS-transformation with the Kumaraswamy distribution. Nayana et al. [17] introduced DUS-transformation with neutrosophic Weibull distribution. More details about the applications of neutrosophic statistics can be seen [1, 7, 10, 13, 20,21,22,23]. The idea for improving the quality aspect can be seen in Ghosh et al. [9].

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 \({I}_{N}\in ({I}_{L},{I}_{U})\) is an indeterminacy interval \(N\) is neutrosophic statistical number and let \({X}_{smN}={X}_{smL}+{X}_{smU}{I}_{N}\), be a random vector of random variables follow neutrosophic multivariate inverse Weibull \({X}_{smN}\) be a random vector with m-variables and S-observations. If \({X}_{N}\in [{n}_{L},{n}_{U}]\) are observations of neutrosophic from neutrosophic variable \({P}_{N}\in [{P}_{L},{P}_{U}]\), then

$${X}_{N}\in \left(\left(\begin{array}{ccc}{X}_{11L}& {X}_{1mL}& {X}_{1PL}\\ \begin{array}{c}\dots \\ {X}_{s1L}\\ \dots \end{array}& \begin{array}{c}\dots \\ {X}_{smL}\\ \dots \end{array}& \begin{array}{c}\dots \\ {X}_{sPL}\\ \dots \end{array}\\ {X}_{n1L}& {X}_{nmL}& {X}_{nPL}\end{array}\right),\left(\begin{array}{ccc}{X}_{11U}& {X}_{1mU}& {X}_{1PU}\\ \begin{array}{c}\dots \\ {X}_{s1U}\\ \dots \end{array}& \begin{array}{c}\dots \\ {X}_{smU}\\ \dots \end{array}& \begin{array}{c}\dots \\ {X}_{sPU}\\ \dots \end{array}\\ {X}_{n1U}& {X}_{nmU}& {X}_{nPU}\end{array}\right)\right).$$

The form of neutosophic \({X}_{N}={X}_{L}+{X}_{N} {I}_{N}\), \({I}_{N}\in [{I}_{L},{I}_{U}]\) and in multivariate case \({X}_{mN}={X}_{mL}+{X}_{mN} {I}_{N}\). Therefore, we consider neutrosophic multivariate inverse Weibull distribution of a random vector \({{\varvec{X}}}_{{\varvec{N}}}=({X}_{1N}\dots {X}_{mN})\) with neutrosophic joint probability distribution function (njpdf) as

$$\begin{aligned}f({{\varvec{X}}}_{{\varvec{N}}};\boldsymbol{\alpha },{\varvec{\beta}},\gamma )&=\frac{(1+{I}_{N})\Gamma (\gamma +m)}{\Gamma (\gamma )} \\& \left(\prod_{i=1}^{m}\frac{{\beta }_{i }{\alpha }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}-1}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right)\\&\times{\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right]}^{-\gamma -m}\end{aligned}$$
(8)

The corresponding neutrosophic joint cumulative distribution (njCDF) is

$$F\left({{\varvec{X}}}_{{\varvec{N}}};\boldsymbol{\alpha },{\varvec{\beta}},\upgamma \right)=(1+{I}_{N}){\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right]}^{-\gamma }$$
(9)

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).

$$\begin{aligned} & {f}_{\mathrm{DUS}}\left({{\varvec{X}}}_{{\varvec{N}}};\boldsymbol{\alpha },{\varvec{\beta}},\gamma \right)\\ &\quad =\frac{1}{e-1} \left(\frac{\left(1+{I}_{N}\right)\Gamma \left(\gamma +m\right)}{\Gamma \left(\gamma \right)}\right.\\ &\quad\quad \left. \left(\prod_{i=1}^{m}\frac{{\beta }_{i }{\alpha }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}-1}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right)\right. \\ &\quad\quad \times\left.{\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right]}^{-\gamma -m}\right)\\ &\quad\quad {e}^{\left(1+{I}_{N}\right){\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right]}^{-\gamma }}\end{aligned}$$
(10)
$$ \begin{aligned} & {F}_{\mathrm{DUS}}\left({{\varvec{X}}}_{{\varvec{N}}};\boldsymbol{\alpha },{\varvec{\beta}},\gamma \right)\\ & \quad =\frac{1}{e-1} \left({e}^{(1+{I}_{N}){[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}]}^{-\gamma }}-1\right) \end{aligned}$$
(11)

and,

Fig. 3
figure 3

The njpdf of DUS-NMIWD when \(m=2\)

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Fig. 4
figure 4

The njCDF of DUS-NMIWD when \(m=2\)

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Fig. 5
figure 5

The njhf of DUS-NMIWD when \(m=2\)

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$$\begin{aligned} & {h}_{\mathrm{DUS}}\left({{\varvec{X}}}_{{\varvec{N}}};\boldsymbol{\alpha },{\varvec{\beta}},\gamma \right)\\ &\quad =\frac{1}{e-{e}^{\left(1+{I}_{N}\right){\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right]}^{-\gamma }}}\\ &\qquad \left( \frac{\left(1+{I}_{N}\right)\Gamma \left(\gamma +m\right)}{\Gamma \left(\gamma \right)} \left(\prod_{i=1}^{m}\frac{{\beta }_{i }{\alpha }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}-1}\right.\right.\\& \qquad \left.\left.{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right){\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right]}^{-\gamma -m}\right)\\ & \qquad {e}^{(1+{I}_{N}){\left[1-m+\sum_{i=1}^{m}{e}^{\frac{{\beta }_{i}}{\gamma } {x}_{iN}^{-{\alpha }_{i}}}\right]}^{-\gamma }} \end{aligned}$$
(12)

Statistical properties of DUS-NMIWD

In this section, we obtained some important statistical properties of DUS-NMIWD, when \(m=2\), is called DUS-neutrosophic 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 joint probability density function

$$\begin{aligned}&{f}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)\\& \quad =\frac{(1+{I}_{k}){\alpha }_{1}{\beta }_{1}{\beta }_{2}{\alpha }_{2}\Gamma [2+\gamma ]{x}_{1N}^{-1-{\alpha }_{1}}{x}_{2N}^{-1-{\alpha }_{2}}}{\left(-1+e\right)\gamma\Gamma \left[\gamma \right]}\\& \quad *{e}^{{\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-\gamma }\left(1+{I}_{k}\right)+\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }+\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\\& \quad \times {\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-2-\gamma },\end{aligned}$$
(13)

where,\({{ \alpha }_{1},\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma >0\), \({x}_{1N},{x}_{2N}>0,{x}_{1N},{x}_{2N}\in \left[{X}_{mL},{X}_{mU}\right], m=\mathrm{1,2}\) and \({I}_{k}\in \left[{I}_{L},{I}_{U}\right].\)

The neutrosophic joint cumulative distribution function

$$ F_{{{\text{DUS}}}} \left( {x_{{1N}} ,x_{{2N}} ;\alpha _{1} ,\alpha _{2} ,\beta _{1} ,\beta _{2} ,\gamma } \right) = ~\frac{1}{{ - 1 + {\text{e}}}}~\left( { - 1 + {\text{e}}^{{\left( { - 1 + {\text{e}}^{{\frac{{\beta _{1} x_{{1N}}^{{ - \alpha _{1} }} }}{\gamma }}} + {\text{e}}^{{\frac{{\beta _{2} x_{{2N}}^{{ - \alpha _{2} }} }}{\gamma }}} } \right)^{{ - \gamma }} \left( {1 + I_{k} } \right)}} } \right) $$
(14)

The neutrosophic joint hazard function

$$\begin{aligned}&{h}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)\\&= \frac{(1+{I}_{k}){\alpha }_{1}{\beta }_{1}{\beta }_{2}{\alpha }_{2}\Gamma [2+\gamma ]{x}_{1N}^{-1-{\alpha }_{1}}{x}_{2N}^{-1-{\alpha }_{2}}}{\left(e-{e}^{{\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_4{2}}}{\gamma }}\right)}^{-\gamma }(1+{I}_{k})}\right)\gamma \Gamma [\gamma ]}\\&\times {e}^{{\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-\gamma }(1+{I}_{k})+\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }+\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\\&\times{\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-2-\gamma }\end{aligned}$$
(15)

The neutrosophic marginal distribution

The neutrosophic marginal distributions of \({X}_{1N}\) and \({X}_{2N}\) are getting as follows

$${f}_{\mathrm{DUS}}\left({x}_{iN},;{\alpha }_{i},{\beta }_{i},\gamma \right)=\int_{0}^{\infty }{f}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right) \mathrm{d}{x}_{jN}$$

where, \(i,j=\mathrm{1,2}\) and \(i\ne j\)

Then the neutrosophic marginal distributions of \({X}_{1N}\) and \({X}_{2N}\) are

$$\begin{aligned}&{f}_{\mathrm{DUS}}\left({x}_{1N},;{\alpha }_{1},{\beta }_{1},\gamma \right)\\&\quad =\left(\frac{{\left(-{e}^{-{x}_{1N}^{-{\alpha }_{1}}{\beta }_{1}}\left(1+{I}_{k}\right)\right)}^{-1/\gamma }{x}_{1N}^{-1-{\alpha }_{1}}{\beta }_{1}{\alpha }_{1}\Gamma \left[2+\gamma \right]}{\left(-1+e\right){\gamma }^{2}\Gamma \left[\gamma \right]}\right.\\& \qquad \times \left({e}^{{e}^{-{x}_{1N}^{-{\alpha }_{1}}{\beta }_{1}}\left(1+{I}_{k}\right)}{\left(-{e}^{-{x}_{1N}^{-{\alpha }_{1}}{\beta }_{1}}\left(1+{I}_{k}\right)\right)}^{\frac{1}{\gamma }}\gamma \right.\\& \qquad \left.\left.+\Gamma \left[\frac{1}{\gamma },-{e}^{-{x}_{1N}^{-{\alpha }_{1}}{\beta }_{1}}\left(1+{I}_{k}\right)\right]\right)\right)\\& \qquad -\frac{{e}^{\frac{{x}_{1N}^{-{\alpha }_{1}}{\beta }_{1}}{\gamma }}{(-1-{I}_{k})}^{-1/\gamma }{x}_{1N}^{-1-{\alpha }_{1}}{\beta }_{1}{\alpha }_{1}\Gamma [\frac{1}{\gamma }]\Gamma [2+\gamma ]}{(-1+e){\gamma }^{2}\Gamma [\gamma ]}\end{aligned}$$
(16)

And,

$$ \begin{aligned}&{f}_{DUS}\left({x}_{2N},;{\alpha }_{2},{\beta }_{2},\gamma \right)\\& \quad =\left(\frac{{\left(-{e}^{-{x}_{2N}^{-{\alpha }_{2}}{\beta }_{2}}\left(1+{I}_{k}\right)\right)}^{-1/\gamma }{x}_{2N}^{-1-{\alpha }_{2}}{\beta }_{2}{\alpha }_{2}\Gamma \left[2+\gamma \right]}{\left(-1+e\right){\gamma }^{2}\Gamma \left[\gamma \right]}\right.\\& \qquad \times \left({e}^{{e}^{-{x}_{2N}^{-{\alpha }_{2}}{\beta }_{2}}\left(1+{I}_{k}\right)}{\left(-{e}^{-{x}_{2N}^{-{\alpha }_{2}}{\beta }_{2}}\left(1+{I}_{k}\right)\right)}^{\frac{1}{\gamma }}\gamma \right.\\& \qquad \left.\left.+\Gamma \left[\frac{1}{\gamma },-{e}^{-{x}_{2N}^{-{\alpha }_{2}}{\beta }_{2}}\left(1+{I}_{k}\right)\right]\right)\right)\\& \qquad -\frac{{e}^{\frac{{x}_{2N}^{-{\alpha }_{2}}{\beta }_{2}}{\gamma }}{(-1-{I}_{k})}^{-1/\gamma }{x}_{2N}^{-1-{\alpha }_{2}}{\beta }_{2}{\alpha }_{2}\Gamma [\frac{1}{\gamma }]\Gamma [2+\gamma ]}{(-1+e){\gamma }^{2}\Gamma [\gamma ]}\end{aligned}$$
(17)

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

$$\begin{aligned}&{\mu }_{rs}^{\backslash }={\int }_{0}^{\infty }{\int }_{0}^{\infty }{x}_{1N}^{r} {x}_{2N}^{s}{f}_{\mathrm{DUS}}\\& \quad \left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)\mathrm{d}{x}_{1N} \mathrm{d}{x}_{2N}\\&=\frac{(1+{I}_{k}){\alpha }_{1}{\beta }_{1}{\beta }_{2}{\alpha }_{2}\Gamma [2+\gamma ]}{\left(-1+e\right)\gamma\Gamma \left[\gamma \right]}A,\end{aligned}$$
(18)
$$\begin{aligned}A&=\int_{0}^{\infty }\int_{0}^{\infty }{x}_{1N}^{r-1-{\alpha }_{1}}{x}_{2N}^{s-1-{\alpha }_{2}}\\&\times {e}^{{\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-\gamma }\left(1+{I}_{k}\right)+\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }+\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\\&\times {\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-2-\gamma } \mathrm{d}{x}_{1N} \mathrm{d}{x}_{2N}\end{aligned}$$

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

$$\begin{aligned}{M}_{\left({(X}_{1N},{X}_{2N}\right)}\left({t}_{1},{t}_{2}\right)&=E\left[{e}^{{t}_{1}{X}_{1N}} {e}^{{t}_{2}{X}_{2N}}\right]\\&=\frac{\left(1+{I}_{k}\right){\alpha }_{1}{\beta }_{1}{\beta }_{2}{\alpha }_{2}\Gamma \left[2+\gamma \right]}{\left(-1+e\right)\gamma\Gamma \left[\gamma \right]}B,\end{aligned}$$
(19)
$$\begin{aligned}B&=\int_{0}^{\infty }\int_{0}^{\infty }{x}_{1N}^{-1-{\alpha }_{1}}{x}_{2N}^{-1-{\alpha }_{2}}{\mathrm{e}}^{{t}_{1}{x}_{1N}} {\mathrm{e}}^{{t}_{2}{x}_{2N}}\\&\times {\mathrm{e}}^{{\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-\gamma }\left(1+{I}_{k}\right)+\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }+\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\\&\times {\left(-1+{\mathrm{e}}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{\mathrm{e}}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-2-\gamma } \mathrm{d}{x}_{1N} \mathrm{d}{x}_{2N}\end{aligned}$$

Neutrosophic conditional distribution

The conditional probability distribution of \({X}_{1N}\) given \({X}_{2N}\) is given as follows

$$ \begin{aligned}\mathrm{f}\left({X}_{1N}\left|{X}_{2N}\right.\right)&=\frac{{f}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)}{{f}_{\mathrm{DUS}}\left({x}_{2N},;{\alpha }_{2},{\beta }_{2},\gamma \right)}\\&=\frac{{e}^{\frac{{{x}_{1N}}^{-{\alpha }_{1}}{\beta }_{1}+{(-1+{e}^{\frac{{{x}_{1N}}^{-{\alpha }_{1}}{\beta }_{1}}{\gamma }}+{e}^{\frac{{{x}_{2N}}^{-{\alpha }_{2}}{\beta }_{2}}{\gamma }})}^{-\gamma }(1+{I}_{k})\gamma +{{x}_{2N}}^{-{\alpha }_{2}}{\beta }_{2}}{\gamma }}{(-1+{\mathrm{e}}^{\frac{{{x}_{1N}}^{-{\alpha }_{1}}{\beta }_{1}}{\gamma }}+{e}^{\frac{{{x}_{2N}}^{-{\alpha }_{2}}{\beta }_{2}}{\gamma }})}^{-2-\gamma }{(-1-{I}_{k})}^{1+\frac{1}{\gamma }}{{x}_{1N}}^{-1-{\alpha }_{1}}{\alpha }_{1}{\beta }_{1}\gamma }{-{(-1-{I}_{k})}^{\frac{1}{\gamma }}({e}^{{e}^{-{{x}_{2N}}^{-{\alpha }_{2}}{\beta }_{2}}(1+{I}_{k})}\gamma +\mathrm{ExpIntegralE}[\frac{-1+\gamma }{\gamma },-{e}^{-{{x}_{2N}}^{-{\alpha }_{2}}{\beta }_{2}}(1+{I}_{k})])+{e}^{\frac{{{x}_{2N}}^{-{\alpha }_{2}}{\beta }_{2}}{\gamma }}\Gamma [\frac{1}{\gamma }]}\end{aligned}$$
(20)

Also, we can get the conditional probability distribution of \({X}_{2N}\) given \({X}_{1N}\) is given as follows

$$\mathrm{f}\left({X}_{2N}\left|{X}_{1N}\right.\right)=\frac{{f}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)}{{f}_{\mathrm{DUS}}\left({x}_{1N},;{\alpha }_{2},{\beta }_{2},\gamma \right)}.$$

Neutrosophic reliability function

If the random vector \({(x}_{1N},{x}_{2N})\) is distributed as DUS-NBIWD, then the reliability function is denoted by \({R}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)\) and given by

$$\begin{aligned}&{R}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)\\&=1-{F}_{\mathrm{DUS}}\left({x}_{1N},{x}_{2N};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)\\&=1- \frac{1}{-1+e} \left(-1+{e}^{{\left(-1+{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}\right)}^{-\gamma }(1+{I}_{k})}\right)\end{aligned}$$
(21)

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 \(\Theta =({\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma )\) is given as follows,

  1. 1.

    Construct the likelihood function as

    $$L\left({X}_{1N},{X}_{2N}\left|\Theta \right.\right)=\prod_{i=1}^{n}{f}_{\mathrm{DUS}}\left({x}_{1Ni},{x}_{2Ni};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)$$
  2. 2.

    Take Log for likelihood function as

    $$\begin{aligned}&\mathrm{Log} L({X}_{1N},{X}_{2N}|\Theta )\\&=\sum_{i=1}^{n}\mathrm{Log}([({x}_{1Ni},{x}_{2Ni};{\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma ))]\\&=(n+1)(\mathrm{Log}[(1+{I}_{k})])+n(\mathrm{Log}[{\alpha }_{1}]\\&\quad +\mathrm{Log}[{\beta }_{1}]+\mathrm{Log}[{\alpha }_{2}]+\mathrm{Log}[{\beta }_{2}]+\mathrm{Log}[\Gamma [2+\gamma ]])\\&\quad -n (\mathrm{Log}[e-1]+\mathrm{Log}[\gamma ]+\mathrm{Log}[\Gamma [\gamma ]])\\&\quad -(1+{\alpha }_{1})\sum_{i=1}^{n}{x}_{1Ni}-(1+{\alpha }_{2})\sum_{i=1}^{n}{x}_{2Ni}\\&\quad -\gamma \sum_{i=1}^{n}\left({e}^{\frac{{\beta }_{1}{x}_{1Ni}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2Ni}^{-{\alpha }_{2}}}{\gamma }}-1\right)\\&\quad +\sum_{i=1}^{n}\frac{{\beta }_{1}{x}_{1Ni}^{-{\alpha }_{1}}}{\gamma }+\sum_{i=1}^{n}\frac{{\beta }_{2}{x}_{2Ni}^{-{\alpha }_{2}}}{\gamma }\\&\quad -(2+\gamma )\sum_{i=1}^{n}\mathrm{Log}\left[{e}^{\frac{{\beta }_{1}{x}_{1N}^{-{\alpha }_{1}}}{\gamma }}+{e}^{\frac{{\beta }_{2}{x}_{2N}^{-{\alpha }_{2}}}{\gamma }}-1\right]\end{aligned}$$
  3. 3.

    Get the normal equations for \({(\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \))

    $$\begin{aligned}\frac{n}{{\widehat{\alpha }}_{1}}&-\sum_{i=1}^{n}{x}_{1Ni}+ \widehat{\gamma }\sum_{i=1}^{n}\frac{{e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}\mathrm{Log}\left[{x}_{1Ni}\right]}{\widehat{\gamma }}\\&-\sum_{i=1}^{n}\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}+\left(2+\widehat{\gamma }\right)\\&\times\sum_{i=1}^{n}\frac{{e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}\mathrm{Log}\left[{x}_{1Ni}\right]}{\left({e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}+{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}-1\right)\widehat{\gamma }}=0\end{aligned}$$
    $$\begin{aligned}\frac{n}{{\widehat{\alpha }}_{2}}&-\sum_{i=1}^{n}{x}_{2Ni}+ \widehat{\gamma }\sum_{i=1}^{n}\frac{{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}\mathrm{Log}\left[{x}_{2Ni}\right]}{\widehat{\gamma }}\\&-\sum_{i=1}^{n}\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}+\left(2+\widehat{\gamma }\right)\\&\times\sum_{i=1}^{n}\frac{{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}\mathrm{Log}\left[{x}_{2Ni}\right]}{\left({e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}+{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}-1\right)\widehat{\gamma }}=0\end{aligned}$$
    $$\begin{aligned}\frac{n}{{\widehat{\beta }}_{1}}&-\sum_{i=1}^{n}{e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}\\&+\sum_{i=1}^{n}\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}}{\widehat{\gamma }}-\left(2+\widehat{\gamma }\right)\\&\times\sum_{i=1}^{n}\frac{{e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}}{\left({e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}+{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}-1\right)\widehat{\gamma }}=0\end{aligned}$$
    $$\begin{aligned}\frac{n}{{\widehat{\beta }}_{2}}&-\sum_{i=1}^{n}{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}\\&+\sum_{i=1}^{n}\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}}{\widehat{\gamma }}-\left(2+\widehat{\gamma }\right)\\&\times\sum_{i=1}^{n}\frac{{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}}{\left({e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}+{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}-1\right)\widehat{\gamma }}=0\end{aligned}$$
    $$\begin{aligned}&n\mathrm{PolyGamma}\left[\mathrm{0,2}+\widehat{\gamma }\right]-n \mathrm{PolyGamma}\left[0,\widehat{\gamma }\right]\\&-\frac{{n}}{\widehat{\gamma }}+\widehat{\gamma }\sum_{i=1}^{n}\left(\frac{{e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{{\widehat{\gamma }}^{2}}+\frac{{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{{\widehat{\gamma }}^{2}}\right)\\&+\left(2+\widehat{\gamma }\right) \sum_{i=1}^{n}\frac{\left(\frac{{e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{{\widehat{\gamma }}^{2}}+\frac{{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{{\widehat{\gamma }}^{2}}\right)}{\left({e}^{\frac{{{x}_{1Ni}}^{-{\widehat{\alpha }}_{1}}{\widehat{\beta }}_{1}}{\widehat{\gamma }}}+{e}^{\frac{{{x}_{2Ni}}^{-{\widehat{\alpha }}_{2}}{\widehat{\beta }}_{2}}{\widehat{\gamma }}}-1\right)}=0\end{aligned}$$

We can not solve normal equations analytically, so we use a numerical method to get the maximum likelihood estimates for our distribution parameters.

Monte-Carlo simulation study

Monte-Carlo simulation study is used to show the performance of our new distribution when we compare the maximum likelihood estimates of our new distribution DUS-NMIWD with their corresponding maximum likelihood estimates of classical version DUS-MIWD according to two statistical criteria bias and MSE. To perform the Monte-Carlo simulation study we use the following steps

  1. 1.

    Assume the different initial values of our distribution parameters \(\left({\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2},\gamma \right)=\left(\mathrm{0.5,0.6,1.5,2.5,0.3}\right), \left(1.5,2.5,0.5,0.8,3\right), \left(1.5,2.5,2,\mathrm{3,0.5}\right).\)

  2. 2.

    For our new distribution we use two indeterminacy measures \({I}_{N}=(\mathrm{0.2,0.5})\) and\({I}_{N}=(\mathrm{0.6,0.8})\). Note when \({I}_{N}=0\), then we get the classical DUS-IWD.

  3. 3.

    Generate different sample sizes \(n=50, 100, 200, 500\). Using the different initial values of our distribution parameters which assumed in step one.

  4. 4.

    Find the maximum likelihood estimates of the parameters which denoted by \({(\widehat{\alpha }}_{1},{\widehat{\alpha }}_{2},{\widehat{\beta }}_{1},{\widehat{\beta }}_{2},\widehat{\gamma })\).

  5. 5.

    Calculate bias and mean square error.

Tables 1 and 2 show the results of the simulation study for our new distribution. Table 3 shows the result of the simulation study for the classical version of our distribution. From Tables 1, 2, 3 we get the bias and Mean Square Error (MSE) for our new distribution DUS-NMIWD is smaller than their corresponding for the classical version DUS-MIWD. So, we can decide whether the new distribution is better than the classical version.

Table 1 Bias and MSE for \({(\widehat{\alpha }}_{1},{\widehat{\alpha }}_{2},{\widehat{\beta }}_{1},{\widehat{\beta }}_{2},\widehat{\gamma })\) when \({I}_{N}=(\mathrm{0.2,0.5})\)
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Table 2 Bias and MSE for \({(\widehat{\alpha }}_{1},{\widehat{\alpha }}_{2},{\widehat{\beta }}_{1},{\widehat{\beta }}_{2},\widehat{\gamma })\) when \({I}_{N}=(\mathrm{0.6,0.8})\)
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Table 3 Bias and MSE for \({(\widehat{\alpha }}_{1},{\widehat{\alpha }}_{2},{\widehat{\beta }}_{1},{\widehat{\beta }}_{2},\widehat{\gamma })\) when \({I}_{N}=0\)
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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.

Table 4 The neutrosophic data of cutter line and edge distortion
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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,

$$\mathrm{AIC}=-2 \mathrm{Log} L\left(\underline{\Theta }\right)+2k,$$
$$\mathrm{BIC}=-2\mathrm{ Log} L\left(\underline{\Theta }\right)+kLog[n]$$

where, \(\underline{\Theta }\): the vector of distribution parameters, \(L\left(\underline{\Theta }\right)\): 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.

Fig. 6
figure 6

Architectural diagram of the proposed algorithm

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Table 5 The result of comparison between our new distribution and other distributions
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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.