On Clarification to the Previously Published Research Articles

In the recent literature, attempts have been made to propose new statistical distributions for modeling real phenomena of nature by adding one or more additional shape parameter (s) to the distribution of baseline random variable. The major contribution of these distributions are to obtain monotonic and non-monotonic shaped failure rates. This short article, offers a clarification to the research articles proposed by Al-Kadim and Boshi [4], El-Bassiouny et al. [9] and El-Desouky et al. [10]. A brief discussion on the properties of this general class is given. A future research motivation on this subject is also provided.


Introduction
In the outlook of reliability modeling, some of the prominent interrelationships between the various functions such as CDF, PDF, HF, CHF and SF, for a continuous lifetime random variable, can be summarized as In term of CHF the CDF of a random variable can be written in the form provided in (5). It is to be noted that all the cumulative hazard functions must fulfill the following two conditions: In this short scenario, it will be showed that how (5) eases the introduction of exponential-type lifetime distributions. In the recent literature, many researchers have proposed new lifetime distributions based on the traditional exponential distribution. These distributions published in different journals are either not new, or description of the model suggested by Gurvich et al. [13]. A lifetime distribution proposed by Gurvich et al. [13] has the CDF given by A number of research articles have been proposed by using the approach provided in (6). In the recent literature, attention has been focused to introduce new family of distributions by introducing new additional parameter (s). The introduction of additional parameter (s) into model offers a more flexible distribution. The most well-known family of distributions are: beta-G by Eugene et al. [11], Log-Gamma-G Type-2 of Amini et al. [7], Gamma-G Type-1 proposed by Zografos and Balakrishnan [20], Gamma-G Type-2 due to Risti´c and Balakrishnan [16] Gamma-G Type-3 of Torabi and Montazeri [17], Exponentiated T-X by Alzaghal et al. [6], Logistic-G studied by Torabi and Montazeri [18] and Weibull-G family of Bourguinion et al. [8]. Recently, Alzaatreh et al. [6] introduced T-X family of distributions defined by ( ) (7) where ( ) f y stands for the PDF of a random variable say Y,  satisfies the settings given below.
(c) Table 1 shows the  functions for some parametric models of the T-X family.
 functions for special models of the T-X family.

Range of T Members of T-X family
( ) , proposed Gamma-G Type-1 by changing the upper limit of integral in (7) with (7), can be re-written as ( ) ( ) Similarly, Amini et al. [7] studied Log-Gamma-G Type-1 family of distributions by replacing the upper limit of integral in (8) Recently, using the expression in (8), Ahmad and Iqbal [1] proposed the GFWEx distribution. Al-Kadim and Boshi [4], introduced a new life time model entitled EP distribution by replacing the upper limit of integral in (7), with ( ) , and ( ) f y with the density function of the exponential distribution. So the expression provided in (7), becomes.
On solving, one might get the following expression ( ) Where, ( ) F z is a monotonically increasing function of z.
Using the expression given in (9), El-Bassiouny et al. [9], proposed a new lifetime distribution titled as EL distribution. Also, using the same technique El-Desouky et al. [10], introduced another lifetime model named as The EFWEx distribution. By comparing (9) to (7) or (8), it is noted that the class characterized by (9) is a familiar general result in literature. Also, comparing (9) to (5), it is instantly noted that ( ) ( ) . But, the quantity used in (9) or (10), is ( ) does not meet the conditions, stated in (a)-(e). Therefore, it is observed that the function suggested in (9), is recognized by the common audience of journals. So, the claim of work provided in (9), is therefore, incorrect in the view of above discussion. For further detail about such class of distributions, one may call to Pham and Lai [15]. The key aims of the newly proposed distributions are to develop statistical models capable of modeling real phenomena with monotonic and non-monotonic shaped failure rates.

Characteristics of Hazard Rates
A lifetime model can be classified based on its shape of If the HF of a distribution increases over time then it is a very good model for describing the lifecycle of a machine's component. For example, the failure rate of a machine's component in its fifteenth year of working may be many times greater than its failure rate during the very first month of service. A distribution with increasing HF is said to be a lighter tailed distribution.
(g) Monotonically decreasing (non-increasing) shape over time, if the first derivative of ( ) h z with respect to z yields a negative value .
z ∀ If the HF of a distribution decreases over time, then it is considered a very good candidate model for modeling skewed data. A decreasing HF can be obtained by improving the system. For example, if a machine's component is defective, then by removing the defects, the lifetime of machine may be increased, resulting in a decreasing failure rate. A distribution with decreasing HF is said to be a heavier tailed distribution.
(h) Constant shape over time, if the value of the first derivative of ( ) h z with respect to z is zero . z ∀ If the HF of a distribution is constant (neither increasing nor decreasing) then the distribution is said to be memory less, such as exponential distribution (in the continuous class of distributions) and geometric distribution (in the discrete class of distributions). There are attractive and wide ranges of applications of constant hazard function in reliability engineering. It reveals that the past information is entirely ignored and only the current information is utilized to take decision about the future event. For example, if a machine performed up to a specific time units say t, then the probability of performing up to another time units say 0 t is entirely independent of the past information that how long the machine has been performing. Alternatively, we can say that the hazard function will be constant, if the survival time is distributed exponentially. A distribution with a constant HF is said to have medium tails. The bathtub shaped HF initially decreases, followed by a less or more constant pattern (known as useful life period), then increases (known as wearing out period). A bathtub HF is very useful for describing the behavior of human mortality, where initially, during infant period (0-6 months after birth) the hazard rate is very high, then slowly decreases and followed up by less or more constant period (between 25-45 years of age), and then increases (in older ages 70 years and above). (k) Modified unimodal shaped, if the HF initially, has unimodal shape and then increasing. A distribution is said to have modified unimodal (also called modified upside down bathtub) failure rate function, if it's ( ) h z gradually increases in the initial phase, then declines and finally again increases. The death rate of cancer patients is observed to have unimodal or modified unimodal shapes, where initially the failure rate is very high, after surgical removal the failure declines and finally again increases. The figure 1 & 2, displays monotonic and nonmonotonic HF.