# Exponentiated Weibull Distribution

## Top PDF Exponentiated Weibull Distribution:

### Reliability equivalence factors for a series parallel system assuming an exponentiated weibull distribution

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.

### Bayesian Estimation Based on Record Values from Exponentiated Weibull Distribution: An Markov Chain Monte Carlo Approach

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'.

### Reliability equivalence factors for a series–parallel system of components with exponentiated Weibull lifetimes

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-

### Parameter Estimations for Some Modifications of the Weibull Distribution

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.

### Maximum Likelihood Estimation of the Parameters of Exponentiated Generalized Weibull Based on Progressive Type II Censored Data

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.

### The Alpha Power Transformation Family: Properties and Applications

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.

### Detecting Outliers in Exponentiated Pareto Distribution

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.

### Transmuted Exponentiated Moment Pareto Distribution

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.

### A MULTIVARIATE WEIBULL DISTRIBUTION

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.

### A modified Weibull distribution

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.

### Estimation of a nonlinear discriminant function from a mixture of two exponentiated - Weibull distributions

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].

### Characterizations of Exponentiated Distributions

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

### A Study on The Mixture of Exponentiated-Weibull Distribution

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.

### The generalized Cauchy family of distributions with applications

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.

### A Generalization of Minimax Distribution

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.

### The Kumaraswamy–Inverse Weibull Distribution

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:

### The New Weibull-Pareto Distribution

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.

### The Odd Generalized Exponential Gompertz Distribution

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.

### Exponentiated Lomax Geometric Distribution: Properties and Applications

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.

### A Generalized Gamma-Weibull Distribution: Model, Properties and Applications

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.