A New Five Parameter Lifetime Distribution: Properties and Application

This paper deals with a new generalization of the Weibull distribution. This distribution is called exponentiated exponentiated exponential-Weibull (EEE-W) distribution. Various structural properties of the new probabilistic model are considered, such as hazard rate function, moments, moment generating function, quantile function, skewness, kurtosis, Shannon entropy and Rényi entropy. The maximum likelihood estimates of its unknown parameters are obtained. Finally, areal data set is analyzed and it observed that the present distribution can provide a better fit than some other known distributions.


INTRODUCTION
The statistic literature is filed with hundred of continuous univariat distributions; see Johnson et. al., [1,2]. Recent developments focus on new techniques for building meaningful distributions, including the two piece approach introduced by Hansen [3], the perturbation approach of Azzalini and Capitanio [4] and the generator approach pioneered by Eugene et. al., [5]. Many subsequent articles apply these techniques to introduce a skew in to well-known symmetric distributions such as the Student t; see Aas and Haff [6]. Numerous classical distributions have been extensively used over the bast decades for modeling data in several areas such as engineering and actuarial. However in many applied areas there is a clear need for extended forms of these distributions. Some attempts have been made to define new families of probability distribution. One such example is a abroad family of univariate distributions generated from the Weibull distribution introduced by Gurvich et. al., [7].
The Weibull distribution is a very popular model and it has been extensively used over the past decades for modeling data in reliability engineering and bio-logical studies. It is generally adequate for modeling monotone hazard rates. The hazard rate function of the Weibull distribution can only be increasing, decreasing or constant. For many years, researchers have been developing various extensions forms of the Weibull distribution. For some extended forms of the Weibull distribution (see, for instance, Bebbington et. al., [8], Zhang and Xie [9], Xie et. al., [10], Carrasco et. al., [11], Sarhan and Zaindin [12] and Almalki and Yuan [13]. Also, Xie and Lai [14] proposed a four-parameter additive Weibull (AW) distribution based on combining the failure rates of tow Weibull distributions: one has a decreasing failure rate and the other one has an increasing failure rate.
In the last years, several ways of generating new probability distributions were developed and discussed. Eugene et. al., [5] proposed a general class of distributions based on the logit of a beta random variable by employing two parameters whose role is to introduce Skewness and to vary tail weights. An extension of the beta-generated method was proposed in Cordiro and de Castro [15] by using the Kumaraswamy distribution instead of beta distribution. Following Eugene et al [5], who defined the Beta Normal (BN) distribution. Nadaraga and Kotz [16] introduce the beta Gumbel distribution (BGa). Nadaraga and Gupta [17] defined the Beta Fréchet (BF) distribution. Nadaraga and Kotz [18] proposed the beta exponential (BE) distribution.
Recently, Alzaatreh et. al., [19] developed a new method to generate family of distributions and called it the family of distributions. This new class of distributions is defined as: Then according to the above definition, Alzaatreh et. al., [19] defined TX  family of distributions by The p. d. f. corresponding to (3) is given by and the corresponding p. d. f. is given by In this new class, the distribution of random variable t is the generator. The new family of distributions generated from equation (3) is called "Exponentiated distributions".We will defined the exponentiated exponentiated exponential-X (EEE-X) distribution from equation (3) Then the p. d. f. of the EEE -X family is given by : The rest of the article is organized as follows. We introduce the EEE-W distribution in Section 2. A range of mathematical properties are considered in Sections 3., quan-tile function, random number generating, skewness, Kurtosis, moment generating function and moments. Two popular entropies are investigated in section 4, namely Shannon entropy and Rényi entropy and we get some numerical values for each one. Estimation by the method of maximum likelihood is presented in Section 5. Finally, Application of the distribution to a real data set is provided in Section 6.

THE EEE-WEIBULL DISTRIBUTION:
If X~Weibull distribution with p. d. f. .
From (5) we obtain the c. d. f. of EEE-W distribution as: From Eqs. (9) and (10) we can define the hazard function of EEE-W distribution as follow: where,

STATISTICAL PROPERTIES OF EEE-W.
In this section, we obtain some statistical properties of the new model, including quantile function, random number generating, Skewness, Kurtosis, moment generating function and moments.

Quantile function and random number generating.
For a non-negative continuous random variable X that follows the EEE-W distribution, the quantile function q x is given by In particular, the distribution median is One can use (13) to generate random numbers when the parameters  ,  ,  ,  and c are known 3.

Skewness and kurtosis based on quantiles.
Skewness measures the degree of the long tail and Kurtosis is a measure of the degree of tail heaviness. Based on quantile function () Q  , Galton [23] and Moors [24] defined the Skewness and Kurtosis, respectively, as Therefore, Galton's Skewness and Moors' Kurtosis of the quantile function defined by (12) can be get easily.
Hence, the m.g.f. defined as

The moments.
The moments of EEE-W distribution can be given as follow By using Eqs. (15), (16) and (17), we have

Remark:
The mean and the variance of EEE-W are reported in Table 1

ENTROPIES.
Entropy is measure of the randomness of systems and it is widely used in areas like physics, molecular imaging of tumors and sparse kernel density estimation. Two popular entropy measures are the Shannon entropy (Shannon [25]) and Rényi entropy (Rényi

Shannon entropy.
The Shannon entropy for the ET-X family defined by Alzaghal et. al., [22] as Then, from Equation (21) the Shannon entropy of EEE-W distribution is defined by where, are the mean and Shannon entropy for the Weibull distribution, respectively.

Remark.
Some numerical values for the Shannon entropy are displayed at   c     , then the corresponding Rényi entropy is obtained as:

Remark.
Some numerical values for the Renyi entropy are displayed at Table 3. It can be observed that this entropy decreasing with increasing  and  and can have negative values.
On taking the partial derivatives of the log-likelihood in Equation (24) 1 (1 (1 ) ) Table 9 gives some descriptive statistics for dataset and it is noted that the dataset has negative Kurtosis.