N EW E XPONENTIAL C UMULATIVE H AZARD M ETHOD FOR G ENERATING C ONTINUOUS F AMILY D ISTRIBUTIONS

The article aims to expand the use of the cumulative hazard function and cumulative generalized exponential distribution of Gupta and Kunda (1999) for introducing a new method for generating families from continuous distributions called the Exponential Cumulative Hazard (ECH) method. It's providing some of well-known methods and distributions embedded within the proposed method. New 5-parameter uniform distributions with 3-shape parameters and bathtub hazard function are introduced as a practical example to support the proposed method. Finally, application on real data-set is provided.


Introduction
There is an interest in presenting a new generalized method for some of the previously presented methods and distributions. The purpose of the new proposed method in this article is to expand the use of the cumulative hazard function and the cumulative generalized exponential distribution of (Gupta and Kunda (1999)). It's called the Exponential Cumulative Hazard (ECH) method. Kupta and Kunda relied on the Gompertz-Verhulst distribution function to find a generalized distribution of the exponential distribution. Depending on Kupta and Kunda's method and the cumulative hazard function, the new method will be described in details as follows : the cumulative generalized exponential distribution of (Gupta and Kunda (1999)) given by Replacing the random variable t in (1) by the cumulative hazard function (2), the cumulative distribution function (cdf) of the new proposed method will be expressed as Note that the proposed method to generate various statistical distributions provides more options for analyzing data restricted to the time interval, while the  

Journal of Alasmarya University: Basic and Applied Sciences
The rest of this article is organized as follows : the generalization of generalized some of well-known distributions with sub-methods are presented in Section 2. Generalization of generalized some of well-known distributions with sub-models are presented in Section 3. New 5-parameter uniform distributions with some mathematical properties are provided in Section 4. Real data is used to illustrate the usefulness of the 5-parameter uniform distributions for modeling lifetime data is provided in Section 5.

Generalization of Generalized
Parameters cdf generalized exponential distributions of (Gupta and Kunda (1999))

New Generalization
Base cdf New cdf

New Generalization
Base cdf New cdf generalized type-II Topp-Leone-G family of distributions Parameters cdf type-II Topp-Leone-G family of distributions of (Elgarhy et al.

New Generalization
Base cdf New cdf

Well-known Sub Method
Parameters cdf generalized Kumaraswamy-G family of distributions of (Zohdy et al.

Well-known Sub Models
Parameters cdf

Well-known Sub Models
Parameters cdf inverted exponential distribution of (Lin, Duran, and Lewis (1989))

Well-known Sub Models
Parameters cdf inverted Kumaraswamy distribution of (Abd Al-Fattah, El-Helbawy, and Al-Dayian

Five Parameter Uniform Distributions
Let ) Q(t be the cdf of the proposed method and  . It is noted that the pdf takes many shapes, and this gives a good advantage to the distribution in that it is a more flexible and more suitable distribution for many life phenomena. The plots of the cdf and the reliability indicate increasing cdf with decreasing reliability function, see Figure 2 and The moment generating function   g t m of a r.v t distributed as 5parameter uniform distributions is

Journal of Alasmarya University: Basic and Applied Sciences
For a random sample . By taking the partial derivative of lnL with respect to  , the closed form of ˆ is gotten as Unfortunately, to get  and  , the computation of lnL cannot be performed by solving the following two normal equations for  , . Therefore, they will be solving numerically.

Journal of Alasmarya University: Basic and Applied Sciences
The elements of the Fisher information matrix (FI) for the MLE can be obtained as the expectations of the negative of the 2 nd partial derivatives, and the asymptotic variance-covariance matrix for the MLE is defined as the inverse of the Fisher information matrix. The 2 nd partial derivatives with respect to   , and  are 2 n 2 ) ; i

Application
The data in the table below are 55 smiling times, in seconds, of an eightweek-old baby. The smiling times, in seconds, follow 5-parameter uniform distributions between 0.7 and 22.8 seconds. 10.4 19.6 18.8 13.9 17.8 16.8 21.6 17.9 12.5 11.1 4.9 8 From the data of this example, the distribution parameters with mean squared error (MSE) and 99% lower bound (L.C.I) and upper bound confidence interval (U.C.I) were estimated. As well as, mean, median, variance, coefficient of skewness, and coefficient of kurtosis of the data were calculated, see table 1 and table 2.

Conclusions
Depending on Kupta and Kunda's method and the cumulative hazard function, the new method called the Exponential Cumulative Hazard (ECH) method is introduced. It's providing some of well-known methods and distributions embedded within the proposed method. This method is considered one of the most important methods that enables us to generate methods of some of previously proposed methods and families from continuous distributions, such as generalization of inverted exponential distribution of (Lin, Duran, and Lewis (1989)), generalized Weibull distribution and exponentiated Weibull of (Mudholkar and Srivastava (1993)), generalized exponentiated distribution of (Ramesh, Pushpa, and Rameshwar (1998), generalized exponential distributions of (Gupta and Kunda (1999)), generalized Rayleigh distribution of (Kundu and Raqab (2005)), Kumaraswamy Weibull distribution of (Cordeiro, Ortega, and Nadarajah (2010), 4-parameter exponentiated modified Weibull distribution of (Elbatal (2011)), exponential generalized distributions of (Cordeiro, Ortega, and Daniel (2013)), inverse generalized Weibull ( ‫المجلد‬ 5 ‫العدد‬ ) 1 ‫(يونيو‬ 2020 ) Volume (5) Issue 1 (June 2020) distribution and generalized inverse generalized Weibull distribution of (Kanchan, Neetu, and Suresh(2014)), Kumaraswamy inverse exponential distribution of (Oguntunde, Babatunde and Ogunmola (2014)), generalization of 5-parameter exponentiated generalized modified Weibull distribution of (Gokarna and Ibrahim (2015)), generalization of inverted Kumaraswamy distribution of (Abd Al-Fattah, El-Helbawy, and Al-Dayian (2017)), generalization of type-II Topp-Leone-G family of distributions of (Elgarhy et al. (2018)), 4-parameter exponential distributions of (Bukoye and Oyeyemi (2018)), and generalization of generalized Kumaraswamy-G family of distributions of (Zohdy et al. (2019)). Finally, 5-parameter uniform distributions with some mathematical properties with application on real data-set are provided. The new distribution takes many shapes with hazard function modeled as the bathtub curve of three life stages, and this gives a good advantage to the distribution in that it is a more flexible and more suitable distribution for many life phenomena.