As mentioned in the introduction, the reliability of a system can be improved by reducing the failure rate for some of the system’s components by a factor ߩ א ሺ 0 ǡ 1 ሻ . For the **exponentiated** **Weibull** **distribution**, reducing only the scale parameter reduces the failure rate. Here, we consider reducing a set A of the system’s components by a factor ߩ , in order to reduce the failure rate (hazard function) for the whole system. This is a logical procedure for the **exponentiated** **Weibull** **distribution**.

as given in (32) is plotted in Figure 1. It can be approximated by normal **distribution** function as mentioned in Subsection 4.1. Also the 95% , approximate maximum likelihood estimation (AMLE) confidence intervals, Bootstrap confidence intervals and approximate credible intervals based on the MCMC samples, the results are given in Table 1. Figures 2 and 3 plot the MCMC output of α and β , using 10000 MCMC samples (dashed line represent means and red lines represent lower and upper bounds of 95% probability intervals.). The plot of histogram of α and β generated by MCMC method are given in Figures 4 and 5. This was done with 1000 bootstrap sample and 10000 MCMC sample and discard the first 1000 values as `burn-in'.

**distribution** of Mudholkar & Srivastava (1993) with identical parameters. We choose this **distribution** because it includes all common shapes of hazard function and because its hazard and reliability are elementary functions. In particular, it includes the monotone hazard function of the **Weibull** distribu- tion but also permits bathtub and inverted bathtub hazard functions. Special cases of the **exponentiated** **Weibull** **distribution** include the **Weibull**, **exponentiated** exponential and Burr-type X distributions men-

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Proposed by the Swedish engineer and mathematician Ernst Hjalmar Waloddi **Weibull** (1887- 1979), the **Weibull** **distribution** is a probability **distribution** that is widely used to model lifetime data. Because of its flexibility, some modifications of the **Weibull** **distribution** have been made from several researches in order to best adjust the non-monotonic shapes. This paper gives a study on the performance of two specific modifications of the **Weibull** **distribution** which are the **exponentiated** **Weibull** **distribution** and the additive **Weibull** **distribution**.

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DOI: 10.4236/ojs.2017.76067 957 Open Journal of Statistics used as a probabilistic model in studies on lifetimes. Mudholkar and Srivastava (1993) [3] introduced the **exponentiated** **Weibull** **distribution** to analyse bathtub failure rate data which cannot be handled well by the regular **Weibull** for monotonicity of its hazard rate. Also Zhang and Xie (2011) [4] worked on bathtub failure data using the truncated **Weibull** **distribution**. Soumaya and Soufiane (2014) [5] have given estimation of the parameters of the **exponentiated** **Weibull** **distribution** and the additive **Weibull** **distribution**, which are two specific generalizations of the **Weibull** **distribution**.

Mahdavi and Kundu (2017) introduced a family for generating univariate distributions called the alpha power transformation. They studied as a special case the properties of the alpha power trans- formed exponential **distribution**. We provide some mathematical properties of this **distribution** and define a four-parameter lifetime model called the alpha power **exponentiated** **Weibull** **distribution**. It generalizes some well-known lifetime models such as the **exponentiated** exponential, **exponentiated** Rayleigh, **exponentiated** **Weibull** and **Weibull** distributions. The importance of the new **distribution** comes from its ability to model monotone and non-monotone failure rate functions, which are quite common in reliability studies. We derive some basic properties of the proposed **distribution** including quantile and generating functions, moments and order statistics. The maximum likelihood method is used to estimate the model parameters. Simulation results investigate the performance of the estimates. We illustrate the importance of the proposed **distribution** over the McDonald **Weibull**, beta **Weibull**, modified **Weibull**, transmuted **Weibull** and **exponentiated** **Weibull** distributions by means of two real data sets.

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defined by Nadarajah [14]. Usually, is defined on the positive side of the real line and so one would hope that models on the basis of the **distribution** of would have greater applicability. Nadarajah [14] introduced five **exponentiated** Pareto distributions and derived several of their properties including the moment generating function, expectation, variance, skewness, kurtosis, Shannon entropy, and the Rényi entropy. Note that another type of **exponentiated** Pareto **distribution** was considered by Shawky and Abu-Zinadah [15] and characterized using record values. Shawky and Abu- Zinadah [16] derived the maximum likelihood estimation of the different parameters of an **exponentiated** Pareto **distribution**. Also, they considered five other estimation procedures and compared them. Afify [2] obtained Bayes and classical estimators for a two parameter **exponentiated** Pareto **distribution** for when samples are available from complete, type I and type II censoring schemes. He proposed Bayes estimators under a squared error loss function as well as under a LINEX loss function using priors of non- informative type for the parameters.

In order to assess the behavior of estimates derived by the method of MLE from TEMP **distribution**, a small scaled experiment is carried out based on simula- tions study. Performance of MLE is evaluated on the basis of mean square errors (MSEs). For this we generate size n = 100, 200, 300, 400 and 500 samples from Equation (5.3) and results are achieved by 1000 simulations. Statistical software R is used to develop the empirical results.

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The parameters are estimated by maximizing the likelihood function, and the standard deviation for 95% confidence interval are approximated using the inverse of the negative Hessian of the log-likelihood function. The results of the fitted trivariate **Weibull** model are in Table 1.

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Reference [3] obtained asymptotic results for the MLE of sur- vival probabilities, and when possible compared them with the MLE based on the **Weibull**, Rayleigh, and exponential distri- butions. In contrast, the main emphasis in this paper is on the graphical aspects of the model, such as the WPP plot.

Mixtures of life distributions occur when two different causes of failure are present each with the same parametric form of life **distribution**. Finite mixture of distributions have been used as models throughout the history of modern statistics. There are several areas of applications of finite mixture models. For example, in biology it is often required to measure certain charac- teristics in natural populations of particular species. Samples of individuals are taken from the natural habitat of the species and the characteristics under investigation is recorded for each in- dividual in sample. The **distribution** of many such characteristics may very greatly with the age of the individuals and age is frequently difficult to ascertain in samples from wide populations. Consequently the biologist observing the population as a whole is dealing with a mixture of distributions, where mixing is over a parameter depending on the unobservable variate age. For examples, see Titterington et al. [2].

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Theorem G. Let ( Ω , , P ) be a given probability space and let H = [ ] a b , be an interval for some a < b ( a = −∞ , = b ∞ mightaswellbeallowed . ) Let X : Ω → H be a continuous random variable with the **distribution** function F and let g and h be two real functions defined on H such that

We shall consider the **exponentiated** **Weibull** model, which includes as special case the **Weibull** and exponential models. The **Exponentiated** **Weibull** family EW [introduced by Mudholkar and Srivastava (1993) as extention of the **Weibull** family] contains distributions with bathtub shaped and unimodal failure rates besides a broader class of monoton failure rates. Applications of the **exponentiated** models have been carried out by some authors as Bain (1974); Gore et al. (1986); and Mudholkar and Hutson (1996). Some statistical properties of this **distribution** (EW) are discussed by Singh et al. (2002) obtained Bayes estimators for the **distribution** parameters, reliability function and hazard function with type II censored sample under squared error loss function as well as under LINEX loss function. Nassar and Eissa (2004) obtained Bayes estimators of the two parameters EW **distribution**, reliability and failure rate functions using Bayes approximation form due to Lindley (1980) under the squared error loss and LINEX loss functions. Elshahat (2006), derived Bayes estimators for the two unknown shape parameters of the EW based on progressive type I interval censored sample. Salem and Abo-Kasem (2011) derived Bayes estimators for the two unknown shape parameters of the EW based on progressive hybrid censored sample. Approximate Bayes estimators for the two unknown shape parameters are drived by Elshahat (2008) based on Lindley (1980) and tierny and kadane (1986) and approximate credible intervals for the unknown parameters are obtained with progressive interval censoring. Ashour and afifiy (2008) derived maximum likelihood estimators of the parameters for EW with type II progressive interval censoring with random removals and their asymptotic variances. Elshahat and Mahmoud (2016) obtained maximum likelihood estimators of the parameters of the mixture of **exponentiated** **Weibull** **distribution**, reliability and hazard functions from type II censored samples.

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A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. Several properties of the T-Cauchy{Y} family are studied including moments and Shannon ’ s entropy. Some members of the T-Cauchy{Y} family are presented. A member of the T-Cauchy{Y} family, the gamma- Cauchy{exponential} **distribution**, is studied in detail. This **distribution** is interesting as it consists of **exponentiated** Cauchy **distribution** and distributions of record values of Cauchy **distribution** as special cases. Various properties of the gamma- Cauchy{exponential} **distribution** are studied, including mode, moments and Shannon’s entropy. Unlike the Cauchy **distribution**, the gamma-Cauchy{exponential} **distribution** can be right-skewed or left-skewed. Also, the moments of the gamma- Cauchy{exponential} **distribution** exist under certain restrictions on the parameters. In particular, the r-th moment for the gamma- Cauchy{exponential} **distribution** ex- ists if and only if α, β − 1 > r and this is not the case for the Cauchy **distribution**. The flexibility of the gamma-Cauchy{exponential} **distribution** and the existence of the moments in some cases make this **distribution** as an alternate to the Cauchy **distribution** in situations where the Cauchy **distribution** may not provide an ad- equate fit.

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generalized inverse Gaussian **distribution**. Cordeiro et al. (2013) proposed the beta **exponentiated** **Weibull** **distribution**, whereas Adepoju, K.A Chukwu A.U and Shittu, O.I (2014) defined and studied the **exponentiated** nakagami **distribution**. Oguntunde P. E (2015) introduced the **exponentiated** weighted exponential **distribution**. Recently, Fatima and Ahmad (2016) considered the characterization and bayesian estimation of Minimax **distribution**. Here, in the same way, we have generalized the Minimax **distribution**.

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Although the generalization of **distribution** functions given in (1.2) has attracted number of researchers, it still involves the complexity of incomplete beta function ratio. Some researchers have suggested to use other bounded distributions on (0,1) to obtain the generalization of any parent cumulative **distribution** function. One such **distribution** is the Kumaraswamy (1980) **distribution** having density and **distribution** function as:

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This article defined a New **Weibull**-Pareto **Distribution** (NWPD) and studied various properties of the **distribution**. The moments, deviations from the mean and median, mode, survival function, hazard function and the maximum likelihood estimates of the parameters, have been investigated. The application of the new **distribution** has also been demonstrated with real life data. The results, compared with other known distributions, revealed that the NWPD provides a better fit for modeling real life data.

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This paper is outlined as follows. In Section 2, we define the cumulative **distribution** function, density func- tion, reliability function and hazard function of the Odd Generalized exponential-Gompertz (OGE-G) distribu- tion. In Section 3, we introduce the statistical properties include, the quantile function, the mode, the median and the moments. Section 4 discusses the **distribution** of the order statistics for (OGE-G) **distribution**. Moreover, maximum likelihood estimation of the parameters is determined in Section 5. Finally, an application of OGE-G using a real data set is presented in Section 6.

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In this paper, a new four-parameter lifetime **distribution**, called the **exponentiated** Lomax geometric (ELG) is introduced. The new lifetime **distribution** contains the Lomax geometric and **exponentiated** Pareto geometric as new sub-models. Explicit algebraic formulas of probability density function, survival and hazard functions are derived. Various structural properties of the new model are derived including; quantile function, Re'nyi entropy, moments, probability weighted moments, order statistic, Lorenz and Bonferroni curves. The estimation of the model parameters is performed by maximum likelihood method and inference for a large sample is discussed. The flexibility and potentiality of the new model in comparison with some other distributions are shown via an application to a real data set. We hope that the new model will be an adequate model for applications in various studies.

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From Table 1, it can be concluded that for fixed , the mean, the second moment and the variance are increasing functions of , while the skewness and the kurtosis are decreasing functions of . Also, for fixed , the mean, the second moment and the variance are increasing functions of , while the skewness and the kurtosis are decreasing functions of . Table 1 also shows that the GGW **distribution** is right skewed. Over-dispersion in a **distribution** is a situation in which the variance exceeds the mean, under-dispersion is the opposite, and equi-dispersion occurs when the variance is equal to the mean. From Table 1, the GGW **distribution** satisfies the over-dispersion property for almost all values of the parameters.

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