1. Introduction

Although the statistical distributions used to describe and interpret the phenomena, there are continuous motivations for developing these distributions to become more flexible or more fitting for specific real data sets. These new statistical distributions are called exponentiated distributions. The idea of exponentiated distributions were utilized to create new distributions. Cordeiro & Castro (2011) [5] extended many known distributions as normal, Weibull, gamma, Gumbel, and inverse Gaussian distributions. They expressed the ordinary moments of these new family of generalized distributions as linear functions of probability weighted moments of the parent distribution. Mudholkar & Srivastava (1994) [11] proposed the exponentiated Weibull distribution to analyze bathtub failure data. Gupta et al (1998) [8] introduced a class of exponentiated distributions which is based on cumulative distribution function as follows:

(1.1)

where is the cumulative distribution function of the random variable and c is an additional shape parameter.

The Weibull and exponential distributions are the most widely used in the reliability and survival studies. This is due to their simplicity and easy mathematical manipulations. In additions, the exponential distribution is one of the members for Weibull- family as well. Let be a random variable taken from exponential distribution with probability density function pdf and cumulative distribution function

(CDF) respectively, and . Also, let be a random variable taken from Weibull distribution with pdf is . Then, the pdf of Weibull-Exponential distribution is

(1.2)

For the exponentiated T-X distribution, Alzaatreh et al (2013) [2] proposed a new method for generating many new distributions. It is called, the T -X family of distributions. It has a connection between the hazard functions and each generated distribution as a weighted hazard function of the random variable X. Alzaatreh et al (2013) [2] founded several known continuous distributions to be special cases of these new distributions. Akinsete et al (2008) [1] studied a four-parameter beta-Pareto distribution. They discussed various properties of the distribution and they founded the distribution to be unimodal and has either a unimodal or a decreasing hazard rate. Also, they obtained the mean, mean deviation, variance, skewness, kurtosis entropies, and they used the method of maximum likelihood to estimate the parameters. Carl et al (2013) [4] used five general methods of combination and variations of two historical periods. These combinations for generating statistical distribution approaches are the method of generating skew distributions, the method of adding parameters, beta generated method, transformed-transformer method and composite method.

The paper is organized as follows. In Section 2, The GWED will be defined. The properties for the new distribution as moments, limiting behavior, quantile function, Shannon’s entropy, skewness and kurtosis are discussed in section 3. Finally, comparing of results for the GWED with other distributions via real data sets.

2. The GWED

Alzaghal A. et al (2013) [3] presented the pdf of the Exponentiated T-X distributions as follows

(2.1)

therefore, the pdf of the GWED is

(2.2)

And the CDF is

(2.3)

where are the shape parameters and are the scale parameters respectively.

In (2.3), note that, if , the pdf of the GWED reduces to the exponential distribution with parameter . Also, when and the GWED reduces to the Weibull distribution with parameters . The exponentiated exponential distribution produced when as well.

Figure (1). Different forms for pdf of the GWED with various values of the parameters respectively

Figure (2). Different forms for hazard function of the GWED with various values of the parameters respectively

The survival function and the hazard function of the GWED respectively take the following forms:

(2.4)

(2.5)

3. Properties of the GWED

There are some relations between the GWED and another distributions. These relations can be obtained by using the transform of variables. For example, let be an Exponentiated Weibull random variable with parameters , then, using this transform , the result is the pdf for the GWED. Also, if be a standard exponential random variable with parameters , then, using as a transform, the result is the pdf for the GWED. And using the transform be a random variable for the Exponentiated Frechet distribution to get on the pdf of the GWED.

3.1. Limiting Behaviors of PDF and Hazard Function

The limiting behaviors of the pdf for the GWED is given by the following lemmas.

Lemma 1: The limit of GWED density is zero when goes to infinity, and when , the limit is given by:

(3.1.1)

Proof. A result in (3.1.1) can be proofed as follows :

(3.1.2)

When goes to zero, the limit of the quantity between brackets

equals one. So, (3.1.2) reduces to:

(3.1.3)

If , then, (3.1.3) goes to zero, if , it goes to infinity and if , it reduces to .

Lemma 2: The limiting of the hazard function for the GWED when goes to infinity is:

(3.1.4)

Proof. The result comes by using a similar proof as in lemma 1.

3.2. Quantile Function and Shannon Entropy

Alzaghal et al (2013) [3] concluded the quantile function and Shannon entropy for the Exponentiated T-X distributions by using the CDF for the X and T distributions as follows:

(3.2.1)

And

(3.2.2)

So, the and of the GWED respectively are:

(3.2.3)

(3.2.4)

And the quartiles of the GWED can be obtained by setting in (3.2.3). Also, based on quartiles, the skewness and kurtosis for the GWED respectively are:

(3.2.3)

(3.2.4)

3.3. Moments

The moments of the GWED can be obtained via the moment generating function which can be derived as follows:

(3.3.1)

Let , then, , and the integral in (3.3.1) can be written as:

(3.3.2)

where .

The result in (3.3.3) came by using the following series expansions:

,

and

.

Figure (3). Skewness and Kurtosis of the GWED

Now, taking the derivative of (3.3.2) and replacing to obtain the moments. Therefore,

(3.3.3)

The mean of the GWED is:

(3.3.4)

And the variance of the GWED is:

(3.3.5)

Table (1) shows that the mean and variance of GWED are increasing when and c increase. And are fixed the mean of GWED is increase while the mean and variance of GWED are decrease.

Table (1). Mean and variance for the GWED various values      and       is fixed

4. Application

Park et al (1964) [14] and Park (1954) [13] presented three frequency distributions related to adult numbers of Tribolium Confusum, Tribolium Castaneum Cultured at 24℃ and Tribolium Confusum strain. These data sets can be fitted via different distributions. For example, Exponentiated Weibull, Generalized Weibull, Lagrange-Gamma and Weibull-Exponential distributions which were presented by Mudholkar et al (1995) [11], Mudholkar et al (1996) [12], Famoya & Govindarajulu (1998) [6] and Alzaatreh et al (2013) [2] respectively. The main objective of this application is to compare the previously mentioned distributions with the GWED to provide an adequate distribution that best fits data.

Tables (2), (3) and (4) show relatively better fit of data for the GWED than for the other four distributions and corresponding p-value and skewness are the highest for the GWED among all the distribution considered.

Table (2). Calculated       Values for Tribolium Castaneum Cultured at      C

Table (3). Calculated       Values for Tribolium Castaneum Cultured at 24℃

Table (4). Calculated       Values for Tribolium Castaneum Cultured at 24℃

5. Conclusions

In this paper, The Generalized Weibull-Exponential Distribution has been defined and studied. various properties of GWED including moments, variance, quantal function, Shannon entropy, skewness and kurtosis have been obtained. Three real data were fitted for the GWED and compared with four known distributions. The results showed that the GWED is a relatively better model to fit data than the other four distributions and the GWED characterized with highest p-value and skewness among the other.

In simulation study, the mean and variance of GWED are increasing when the parameters and c increase. And are fixed the mean of GWED is increase while the mean and variance of GWED are decrease.