The **inverse** **Weibull** (IW) **distribution** has many applications in the reliability engineering discipline and model degradation of mechanical components such as the dynamic components (pistons, crankshafts of diesel engines, etc). It provides a good fit to several data such as the times to breakdown of an insulating fluid, subject to the action of constant tension. Also, it can be used to model a variety of failure characteristics such as infant mortality, useful life and wear-out periods, applications in medicine and ecology, determining the cost effectiveness, maintenance periods of reliability centered maintenance activities. Keller et al. (1985) obtained the IW model by investigating failures of mechanical components subject to degradation. de Gusmão et al. (2011) introduced the three-parameter **generalized** IW (GIW) **distribution** with decreasing and unimodal failure rate.

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The concept of **generalized** order statistics has been introduced as a unified approach to a variety of models of ordered random variables with different interpretations. In this paper, we develop methodology for constructing inference based on n selected **generalized** order statistics (GOS) from **inverse** **Weibull** **distribution** (IWD), Bayesian and non-Bayesian approaches have been used to obtain the estimators of the parameters and reliability function. We have examined Bayes estimates under various losses such as the balanced squared error (balanced SEL) and balanced LINEX loss functions are considered. We show that Bayes estimate under balanced SEL and balanced LINEX loss functions are more general, which include the symmetric and asymmetric losses as special cases. This was done under assumption of discrete-con- tinuous mixture prior for the unknown model parameters. The parametric bootstrap method has been used to construct confidence interval for the parameters and reliability function. Progressively type-II censored and k-record values as a special case of GOS are considered. Finally a practical example using real data set was used for illustration.

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Eugene et al. (2002) has used the CDF of Normal **distribution** in (1.2) to propose the **Beta**–Normal **distribution**. The generalization given in (1.2) has been used by number of authors to propose new distributions. Some notable references include Nadarajah and Kotz (2004, 2005), Famoye, Lee and Olumolade (2005), Hanook, Shahbaz, Mohsin and Kibria (2012) and many others.

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In this article, we introduce a new three-parameter lifetime model called the Burr X exponentiated **Weibull** model. The major justification for the practicality of the new lifetime model is based on the wider use of the exponentiated **Weibull** and **Weibull** models. We are motivated to propose this new lifetime model because it exhibits increasing, decreasing, bathtub, J shaped and constant hazard rates. The new lifetime model can be viewed as a mixture of the exponentiated **Weibull** **distribution**. It can also be viewed as a suitable model for fitting the right skewed, symmetric, left skewed and unimodal data. We provide a comprehensive account of some of its statistical properties also some useful characterization results are presented. The maximum likelihood method is used to estimate the model parameters. We prove empirically the importance and flexibility of the new model in modeling two types of lifetime data. The proposed BrXEW lifetime model is a much better fit than the Poisson Topp Leone- **Weibull**, the Marshall Olkin extended-**Weibull**, gamma-**Weibull**, Kumaraswamy-**Weibull**, **Weibull**-Fréchet, **beta**-**Weibull**, transmuted modified-**Weibull**, Kumaraswamy transmuted- **Weibull**, modified **beta**-**Weibull**, Mcdonald-**Weibull** and transmuted exponentiated **generalized**-**Weibull** models, so the new lifetime model is a good alternative to these models in modeling aircraft windshield data. It is also a much better fit than the **Weibull**-**Weibull**, odd **Weibull**-**Weibull**, **Weibull** Log-**Weibull**, the gamma exponentiated-exponential and exponential exponential-**geometric** models, so it is a good alternative to these models in modeling the survival times of Guinea pigs. We hope that the new model will attract wider applications in reliability, engineering and other areas of research.

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Reliability is a human characteristic that has been in.ect for a long period of time. Reliability acts the probability of components parts and systems to perform the desired task for a specific period of time without failure in fixed environments with required confidence. The Exponential, Raylight, linear failure rate, **Weibull** and **inverse** **Weibull** are the most common life **distribution** in reliability and life testing [11]. Exponential **distribution** has constant failure rate **distribution**, Raylight **distribution** has increasing failure rate **distribution**, and there are distributions that have increasing or decreasing failure rate introduced by Nassar and Abo-kasem [10]. The hazard rate function can be of bath tube shape it plays a central role in the work of reliability engineers (the **Weibull** **distribution**, **Generalized** exponential, Rraylight **distribution**, **inverse** **Weibull**) [4]. The **inverse** **Weibull** was originally used in reliability industry, it is one of the most common life **distribution** in reliability engineering, the need for an extended form of the **distribution** grow in many applied fields Corderio and Lemonte[7]. If the random variable 𝑌has a **Weibull** **distribution**, then the random variable X=𝑌 −1 has an

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The Kumaraswamy **distribution** is a useful alternative to the **Beta** **distribution**. Cordeiro and Castro (2010) have proposed a family of distributions by using Kumaraswamy **distribution** and is named as the Kumaraswamy family of distributions. Recently, AL-Fattah et al. (2017) have proposed an Inverted Kumaraswamy **distribution** by transforming the **distribution** (4) as Y = X −1 − 1. The density function of the **inverse** Kumaraswamy **distribution**, proposed by AL-Fattah et al. (2017) is

where G t ( ) is the cdf of the baseline model. Elbatal (2013) introduced the transmuted modified **inverse** **Weibull** **distribution** by using the quadratic rank transmutation map technique pioneered by Shaw et al. (2009). The article is organized as follows, Section 2 presents the flexibility of the transmuted modified **inverse** **Weibull** **distribution** and special sub-models. Section 3 demonstrates a range of mathematical properties, which includes quantile functions, mean deviation, entropy, mean, **geometric** mean and harmonic mean. The maximum likelihood estimates (MLEs) and the asymptotic confidence intervals of the unknown parameters are presented in Section 4. In Section 5, a real lifetime dataset is analyzed to show the flexibility of the transmuted modified **inverse** **Weibull** **distribution**. Finally, some concluding remarks are given in Section 6.

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The one parameter exponential **distribution** is continuous analogue of the **geometric** **distribution**. This **distribution** gives us the description of the time between the events in a Poisson process. This model is widely applicable in life testing and is well-known for its memory less property. Due to its constant failure rate, this probability model is inappropriate for the analysis of the data with bathtub failure rates and inverted bathtub failure rates. In order to overcome such shortcomings and improve the flexibility and competence of the model, the one parameter inverted exponential **distribution** was studied by Keller and Kamath [1]. Because of its inverted bathtub failure rate, it is widely competent model for the exponential **distribution**.

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In this paper, we use the **generalized** hypergeometric series method the high- order **inverse** moments and high-order **inverse** factorial moments of the ge- neralized **geometric** **distribution**, the Katz **distribution**, the Lagrangian Katz **distribution**, **generalized** Polya-Eggenberger **distribution** of the first kind and so on.

The distributions that have been explored based on the B-G class are: the **beta** normal (Eugene et al., 2002), the **beta** exponential (Nadarajah and Kotz, 2006), the **beta** **weibull** **geometric** (Cordeiro et al., 2013), the **beta** exponential-**geometric** (Bidram, 2012) and the **beta** **Weibull** **geometric** distributions (Bidram et al., 2013). The **beta** transmuted-H and **beta** WeibullG families due to Afify et al. (2017) and Yousof et al. (2017), respectively. In this article, we propose a new extension of the LGc **distribution** of Zakerzadeh and Mahmoudi (2012) by taking 𝐺(𝑥; 𝜙) in (3) to the CDF of the LGc **distribution**. The new model is referred to as the **beta** Lindley **geometric** (BLGc) **distribution**. We also study some of its mathematical properties and its applications to real data.

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We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see, Glänzel 1990), in particular, let us assume that there is a sequence {𝑋 𝑛 } of random variables with **distribution** functions {𝐹 𝑛 } such that the functions ℎ 𝑛 , 𝑔 𝑛 and 𝜉 𝑛 (𝑛 ∈ ℕ) satisfy the conditions of Theorem 1 and let ℎ 𝑛 → ℎ , 𝑔 𝑛 → 𝑔 for some continuously differentiable real functions ℎ and 𝑔 . Let, finally, 𝑋 be a random variable with **distribution** 𝐹 . Under the condition that ℎ 𝑛 (𝑋) and 𝑔 𝑛 (𝑋) are uniformly integrable and the family {𝐹 𝑛 } is relatively compact,

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In this section, a brief simulation study is conducted to examine the performance of the MLEs of TGW parameters. **Inverse** transform method is used to generate random observations from TGW **distribution**. We generate 1000 samples of size, n =50, 100, 500 and n=1000 of TGW **distribution**. The evaluation of estimates was based on the bias of the MLEs of the model parameters, the mean squared error (MSE) of the MLEs. The empirical study was conducted with software R and the results are given in Table 1. The values in Table 1 indicate that the estimates are quite stable and, more importantly, are close to nominal values when 𝑛 goes to infinity. It is observed from Table 1 that the biases and MSEs decreases as n increases. The simulation study shows that the maximum likelihood method is appropriate for estimating the parameters of TGW **distribution**. In fact, the MSEs of the parameters tend to be closer to the zero when n increases. This fact supports that the asymptotic normal **distribution** provides an adequate approximation to the finite sample **distribution** of the MLEs. The normal approximation can be improved by using bias adjustments to these estimators.

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The CDF given in equation (1) approches to the eleven lifetime distributions when its parameters change. Khan and King (2012) proposed the modified **Inverse** **Weibull** **distribution** and presented a comprehensive description of the mathematical properties of this model along with its reliability behavior. Using quadratic rank transmutation map (QRTM) technique, we introduce the transmuted the new **generalized** **inverse** **Weibull** (NGIW) **distribution** by introducing a new parameter λ that would offer more flexibility in the proposed model. Several distributions have been proposed under this methodology such as transmuted extreme value **distribution** (Gokarna and Chris, 2009) studied with application to climate data, the transmuted **Weibull** **distribution** (Gokarna and Chris, 2011) proposed with two applications, Gokarna (2013) proposed the transmuted Log- Logistic **distribution** and studied its various structural properties. Khan and King (2013) proposed the transmuted modified **Weibull** **distribution** as an important competitive model with eleven lifetime distributions as sub-models along with its theoretical properties. Khan and King (2013) studied the flexibility of the transmuted **generalized** **Inverse** **Weibull** **distribution** with application to reliability data. Merovci (2013) studied the transmuted rayleigh **distribution**. Elbatal et al. (2013a, 2013b) proposed and studied the transmuted additive **Weibull** and transmuted modified **inverse** **Weibull** distributions. Khan et al. (2014a, 2014b) proposed the transmuted **inverse** **Weibull** **distribution** and studied its various structural properties with an application to survival data. More recently, Khan and King (2015) explored the flexibility of the transmuted modified **Inverse** Rayleigh **distribution** using QRTM technique which extends the modified **Inverse** Rayleigh **distribution** with application to reliability data. A random variable X is said to have transmuted **distribution** if its cumulative **distribution** function (cdf) is given by 𝐹(𝑥) = (1 + 𝜆)𝐺(𝑥) − 𝜆𝐺(𝑥) 2 , |𝜆| ≤ 1 (3)

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The FWEx **distribution** has non-monotonic HF. As we discussed earlier that lots of **generalized** forms of **Weibull** **distribution** have been proposed that has non-monotonic failure rate. But, many of these **generalized** forms of **Weibull** model does not have closed form of its CDF, SF and HF for example, Gamma **Weibull** (GW) **distribution** pro- posed by Stacy [16], **beta** **inverse** **Weibull** (BIW) **distribution** proposed by Hanook et al. [9] and **beta** modified **Weibull** (BMW) **distribution** due to Silva et al. [18]. Due to incomplete form of CDF, the estimation difficulties have increased. To address some of the problems that have been occurred with some modified forms of **Weibull** model; we propose a new model by generalizing the FWEx **distribution** using quadratic rank trans- mutation map (QRTM). The new model may be named as transmuted flexible **Weibull** extension (TFWEx) **distribution**, and possess a closed form of CDF allowing a very simple expression for SF and HF. The proposed model provides greater flexibility and modeling real phenomena with in- creasing and modified unimodal shaped failure

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f x a b abx x x a b R (5) Jones(2009) has studied the properties of (5) and has shown that the **distribution** can be used as an alternate of **Beta** **distribution**. The density (5) has also provided basis for generalization of **distribution** on the lines of (4). Cordeiro & de Castro(2011) have used (5) to propose the Kumaraswamy **generalized** distributions having density function as:

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Now a days transmuted distributions and their mathematical properties are widely studied for applied sciences experimental data sets. Transmuted Rayleigh **Distribution** (Merovci, 2013), Transmuted **Inverse** Rayleigh **Distribution** (Ahmad et al., 2014), Transmuted **Generalized** **Inverse** **Weibull** **Distribution** (Khan and King, 2013), Transmuted Modified **Inverse** **Weibull** **Distribution** (Elbatal, 2013), Transmuted Log-logistic **Distribution** (Aryal, 2013), Transmuted Modified **Weibull** **Distribution** & Transmuted Lomax **Distribution** (Ashour and Eltehiwy, 2013), Transmuted Frechet **Distribution** (Mahmoud & Mandouh, 2013), Transmuted Pareto **Distribution** (Merovci & Puka, 2014), Transmuted **Generalized** Gamma **Distribution** (Lucena et al., 2015), Transmuted **Weibull** Lomax **Distribution** (Afify et al., 2015) are reported with their various structural properties including explicit expressions for the moments, quantiles, entropies, mean deviations and order statistics. All the above transmuted distributions are derived by using Quadratic Rank Transmutation Map(QRTM) studied by Shaw & Buckley (2007). Report reveals that some properties of these distributions along with their parameters are estimated by using maximum likelihood and Bayesian methods. Usefulness of some of these new distributions are also illustrated with experimental data sets.

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In this section, we provide an application of the GT-W **distribution** to show the importance of the new model. We now provide a data analysis in order to assess the goodness-of-fit of the proposed model. We will make the use of the data set on the remission times (in months) of a random sample of 128 bladder cancer patients (Lee and Wang, 2003) is given by: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39,10.34, 14.83, 34.26, 0.90 , 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69. These data were previously studied by Mead and Afify (2017) to fit the Kumaraswamy exponentiated Burr XII **distribution**. We compare the fits of the GT-W **distribution** with other competitive models, namely: the McDonald **Weibull** (McW) (Cordeiro et al., 2014), transmuted linear exponential (TLE) (Tian et al., 2014), transmuted modified **Weibull** (TMW) (Khan and King, 2013), modified **beta** **Weibull** (MBW) (Khan, 2015), transmuted additive **Weibull** **distribution** (TAW) (Elbatal and Aryal, 2013), exponentiated transmuted **generalized** Rayleigh (ETGR) (Afify et al., 2015) and **Weibull** (W) distributions with corresponding densities (for 𝑥 > 0):

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Extended and **generalized** forms of IW **distribution** are studied by some authors, among them; Khan (2010) introduced and studied the **beta** **inverse** **Weibull** **distribution**. de Gusmão et al. (2011) introduced three-parameter **inverse** **Weibull** **distribution**, called the **generalized** **inverse** **Weibull** **distribution**, with unimodal, increasing and decreasing failure rates. Khan and King (2012) proposed four-parameter modified **inverse** **Weibull** **distribution**. Shahbaz et al. (2012) suggested the Kumaraswamy **inverse** **Weibull** **distribution**. Elbatal and Muhammed (2014) introduced the exponentiated **generalized** **inverse** **Weibull** **distribution**. The **generalized** **inverse** **Weibull** **distribution** including the exponentiated or proportional reverse hazard and Kumaraswamy **generalized** **inverse** **Weibull** distributions have been suggested by Oluyede and Yang (2014). Pararai et al. (2014) introduced gamma-**inverse** **Weibull** **distribution** based on gamma generated family. Khan et al. (2014) studied characterizations of the transmuted **inverse** **Weibull** **distribution** with an application to bladder cancer remission time's data. Khan and King (2016) introduced the four-parameter new **generalized** **inverse** **Weibull** **distribution** and investigated its potential usefulness with application to reliability data from engineering studies. Rodrigues et al. (2016) introduced exponentiated Kumaraswamy **inverse** **Weibull** **distribution**. Okasha et al. (2017) introduced the Marshall–Olkin extended **inverse** **Weibull** **distribution**.

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leaded to the action of constant tension, see Nelson [23]. Calabria and Pulcini [16] dis- cussed the maximum likelihood and least squares estimation of its parameters. Calabria and Pulcini [17] considered Bayes 2-sample prediction of the **distribution**. Mahmoud et al. [10] discussed moments of order statistics of **Inverse** **Weibull** **distribution** and obtained BLUE (best linear unbiased estimator) for both location and scale parameters. Aleem and Pasha [11] derived single, product and ratio moment of **Inverse** **Weibull** **Distribution**. Aleem [12] worked on the product, ratio, and single moments of lower record values of **Inverse** **Weibull** **distribution**. Hanook et al. [21] derived **Beta** **Inverse** **Weibull** **distribution**. Shahbaz et al. [13] proposed the Kumaraswamy **Inverse** **Weibull** **distribution** using distri- bution function of kumaraswamy family of distributions.

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In this paper we have studied the **distribution** of r–th concomitant and joint **distribution** of r–th and s–th concomitant of **generalized** order statistics for a bivariate **Weibull** **distribution**. We have derived the expression for single and product moments. Numerical study has also been conducted to see the behavior of mean of concomitants for selected values of the parameters.