# Top PDF The Beta Generalized Inverse Weibull Geometric Distribution ### The Beta Generalized Inverse Weibull Geometric Distribution

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. ### Bayesian and Non Bayesian Estimation of the Inverse Weibull Model Based on Generalized Order Statistics

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. ### The Kumaraswamy–Inverse Weibull Distribution

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. ### The Burr X Exponentiated Weibull Model: Characterizations,Mathematical Properties and Applications to Failure and Survival Times Data

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. ### On the Alpha Power Inverse Weibull Distribution

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 . 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 . 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) . 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. If the random variable 𝑌has a Weibull distribution, then the random variable X=𝑌 −1 has an ### Relations for Moments of Dual Generalized Order Statistics for a New Inverse Kumaraswamy Distribution

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 ### Transmuted Modified Inverse Weibull distribution: Properties and application

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. ### Transmuted Weibull-Inverse Exponential Distribution with Applications to Medical Science and Engineering

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 . Because of its inverted bathtub failure rate, it is widely competent model for the exponential distribution. ### Application of Hypergeometric Series in the Inverse Moments of Special Discrete Distribution*

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. ### A New Extension of Lindley Geometric Distribution and its Applications

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. ### Characterizations of Kumaraswamy-Laplace, McDonald Inverse Weibull and New Generalized Exponential Distributions

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, ### The Transmuted Geometric-Weibull distribution: Properties, Characterizations and Regression Models

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. ### Transmuted New Generalized Inverse Weibull Distribution

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) ### On Transmuted Flexible Weibull Extension Distribution with Applications to Different Lifetime Data Sets

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 , beta inverse Weibull (BIW) distribution proposed by Hanook et al.  and beta modified Weibull (BMW) distribution due to Silva et al. . 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 ### The McDonald’s Inverse Weibull Distribution

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: ### Transmuted Exponentiated Gumbel Distribution (TEGD) and its Application to Water Quality Data

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. ### The Generalized Transmuted Weibull Distribution for Lifetime Data

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): ### Odds Generalized Exponential-Inverse Weibull Distribution: Properties & Estimation

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.  