The New Extended Flexible Weibull Distribution and Its Applications

The present article considers a new function to propose a new lifetime distribution. The new distribution is introduced by mixing up a linear system of the two logarithms of cumulative hazard functions. The proposed model is called new extended flexible Weibull distribution and is able to model lifetime with bathtub shaped failure rates and offers greater flexibility. Therefore, it can be quite valuable to use an alternative model to other existing lifetime distributions, where, modeling of real data sets with bathtub shaped failure rates are of interest. A brief description of the statistical properties along with estimation of the parameters through maximum likelihood procedure are discussed. The potentiality of the proposed model is showed by discussing two real data sets. For these data sets, the proposed model outclasses the Flexible Weibull Extension, Inverse Flexible Weibull Extension and Modified Weibull distributions.


Introduction
In the practice of analyzing real phenomena of nature one frequently uses the Rayleigh, Exponential, or Weibull distributions. These models possess numerous desirable properties and nice interpretations of its parameters enabling them to be utilized frequently. Between these lifetime models, Weibull distribution is the most prominent distribution for modeling real phenomena of nature. The Weibull model was originally introduced by Weibull [19], a Swedish physicist, and utilized it to represent the distribution of breaking strength of materials. Weibull model also has the shape and scale parameters, offering characteristics of both the exponential and Rayleigh distributions. In recent past years, the Weibull model becoming very popular in modeling lifetime data, because in the presence of censoring which makes it much easier to handle, at least numerically. The cumulative distribution function (CDF) of the Weibull model is given by.
The Weibull model is very useful in modeling real phenomena exhibiting monotonic failure rates. But, the Weibull model is inappropriate to use in modeling data having non-monotonic failure rates. Among non-monotonic failure rate function, the bathtub shaped failure rate is very useful and has a number of applications in the literature. For example, in bio-analysis the human mortality rate and in reliability engineering the lifecycle of electronic components is observed to have a bathtub shaped failure rate function. Due to practical utility in bio and reliability disciplines, numerous generalizations of Weibull distribution have been proposed in the literature aiming to improve its characteristics and to model real world scenario with nonmonotonic failure rate functions. These generalizations, including a statistical model with bathtub failure rate studied by Xie and Lai [20], Sarhan and Zaindin [17], Beta-Weibull (BW) distribution of Famoye et al. [11], Kumaraswamy Weibull (KW) distribution proposed by Cordeiro et al. [10], Generalized modified Weibull (GMW) distribution proposed by Carrasco et al. [9], Exponentiated modified Weibull extension (EMWEx) distribution introduced by Sarhan and Apaloo [16], Flexible Weibull (FWEx) distribution of Bebbington et al. [8], Generalized Flexible Weibull Extension (GFWEx) distribution studied by Ahmad and Iqbal [1], other extensions of Weibull model proposed by  are [2], [3], [4], [5] and [6], respectively.
For a concise review of these distributions one may call to Pham and Lai [15] and Murthy et al. [14]. These distributions have numerous applications including reliability analysis, clinical studies, applied statistics and life testing experiments etc. Gurvich et al. [12] proposed a new class of aging distributions defined by the CDF given by.
By substituting (8) in (5), one can easily get the CDF of the new extended flexible Weibull (NEx-FW) distribution. The suggested model is capable of modeling data with bathtub failure rate. The present article is designed as: Section 2, offers the definition and graphical display of the new model. Section 3, contains the basic statiatical properties. Section 4, 5 and 6, derives the moment generating function, probability generating function and factorial moment generating function of the NEx-FW distribution. Section 7 and 8, contains the estimation of the parameters and density functions of the order statistics. Section 9, offers the analysis to real data sets. Finally, section 10, contains concluding remarks.

New Extended Flexible Weibull Distribution
The CDF of the NEx-FW distribution is given by The probability distribution function (PDF) corresponding to (9) is given by

Basic Properties
This section of the paper covers the basic statistical properties of the NEx-FW distribution.

Quantile and Median
The expression for the th q quantile q z of the NEx-FW distribution is given by Using 0.50, q = in (11), one can easily obtain the median of the NEx-FW distribution. Also, putting 0.25, q = and 0.75, q = in (11), one may get the 1 st and 3 rd quartiles of the NEx-FW distribution, respectively.

Generation of Random Numbers
The formula for generating random numbers from NEx-FW distribution can be derived as The expression for generating random numbers from the proposed distribution is not closed form. Therefore, the random numbers from the proposed model can be generated using computer software.

Moments
Using the definition of gamma function (Zwillinger [21]) in the following form,  Using the above definition of gamma function in (12), and finally, one may get

Moment Generating Function
If Z~ NEx-FW ( ) By using (13), in (14), one may have the proof of the NEx-FW distribution.

Probability Generating Function
The probability generating function (PGF) of NEx-FW distribution is derived On substituting (13), in (15), one may get the expression for the PGF of NEx-FW distribution.

Factorial Moment Generating Function
The factorial moment generating function (FMGF) of NEx-FW distribution can be derived as By substituting (13), in (16), result in proof of the FMGF of NEx-FW distribution.

Estimation
This section of the article, concern with estimation of the model parameters through maximum likelihood (ML) procedure. Let 1 By attaining the partial derivatives of the expression in (17) on parameter, and then equating to zero, one may have . Then, the PDF of ( ) So, the PDF of smallest order statistic is Also, the PDF of largest order statistic is

Applications
In this section, two real life application are presented. The result of the goodness of fit of the suggested model is compared with three other well-known competing lifetime models. The investigative tools such as Akaike's Information

Example 1
The first data set obtained from Tahir et al. [18]

Example 2
The second data set denotes the failure times of a sample of 30 devices taken from Khan and Jan [13].

Conclusion
In this paper, a new lifetime distribution entitled New Extended Flexible Weibull Distribution is introduced by taking into account a linear system of the two logarithms of cumulative hazard functions. The suggested model offers greater distribution flexibility and is able to model lifetime data with bathtub shaped failure rates. A concise explanation of the mathematical properties of the proposed model, with estimation of parameters using maximum likelihood procedure are discussed. The proposed modal is illustrated by means of analyzing two real data sets, and the goodness fit of the proposed model is compared with that of three other existing lifetime distributions. Analyzing these two data sets, it is observed that the new model provides best fit than the competitive models. It is hoped that the New Extended Flexible Weibull distribution will serve as one of the most useful lifetime model and will attract a wide range of practical applications in the field of bio-medical and reliability engineering.