The Exponentiated Generalized Topp Leone-G Family of Distributions: Properties and Applications

In this paper, we propose a new class of continuous distributions called the exponentiated generalized Topp Leone-G family that extends the Topp Leone-G family introduced by Al-Shomrani et al. (2016). We derive explicit expressions for certain mathematical properties of the new family such as; ordinary and incomplete moments, generating functions, reliability analysis, Lorenz and Bonferroni curves, Rényi entropy, stress strength model, moment of residual and reversed residual life, order statistics, extreme values and characterizations. We discuss the maximum likelihood estimates and the observed information matrix for the model parameters. Two real data sets are used to illustrate the flexibility of the new family.


Introduction
There has be an increase interest in constructing new generated families of univariate continuous distributions by adding additional shape parameter(s) to a baseline model due to the desirable properties of the new models. Some of the well-known generated families are the following: exponentiated-G by Gupta et al. (1998), beta-G by Eugene et al.(2002), Kumaraswamy-G by Cordeiro where, 0 a  andare two additional shape parameters. The rest of this paper is as follows. In Section 2, we define the EGTL-G and obtain some associated reliability functions. In Section 3, the asymptotic of the EGTL-G are investigated. The expansion of EGTL -G is discussed in Section 4. In Section 5, some special models corresponding to EGTL-G are introduced. In Section 6, some statistical properties of the EGTL-G are discussed. Characterizations for the new family are presented in Section 7. In Section 8, the maximum likelihood estimates and the observed information matrix are obtained for the parameters of EGTL-G. A simulation study is conducted in Section 9. In Section 10, two applications for EGTL-G are presented. Some concluding remarks are given in the last Section.

The Exponentiated Generalized Topp Leone-G Family
In this section, we define the exponentiated generalized Topp leone-G family of distributions and discuss some of the reliability functions.
The cdf of the EGTL-G family can be obtained by using The pdf corresponding to (5) is ( ) The EGTL-G has the following sub-families: *If 1  = , then the EGTL-G class reduces to the EG-G family.

Asymptotics
The asymptotics of cdf, pdf and hrf of EGTL-G as x → − are given by ( ) The asymptotics of equations cfd, pdf and hrf of EGTL-G as x → are given by Gx as x →. These results show the effect of the parameters on the tails of EGTL-G.

Expansion for Density and Distribution Functions
We can expand the EGTL-G family as mixture representation of the exponentiated-G family of distributions. For 1   and b a positive real non-integer, we have the series representation Therefore using (12) in (5), the cdf of EGTL-G family can be expressed as follows: where, Likewise, we can express the pdf of EGTL-G family using (11) (14) where, 1 is the exponentiated-G distribution with power parameter 1 m + .

The EGTL-G Sub-Models
In this section, we introduce three special models of the EGTL-G family.

The EGTL-Weibull (EGTLW) Model
Suppose the cdf and pdf of the Weibull distribution are the following respectively. Then, the cdf and pdf of EGTL-Weibull (EGTLW) distribution are, respectively, given by

The EGTL-Lomax (EGTLLx) Model
Consider the cdf and pdf of the lomax distribution Moreover, For 1 ab ==, then the EGTLLx distribution is reduced to the TL-lomax (TLLx) distribution. For 1 ab = = = , the EGTLLx distribution is reduced to the Exp-Lomax (ELx) distribution. The plots of the density and hazard functions are given in Figure 2. The shape of the density is skeweed, reversed J shapes and near symmetric, while the hazard function introduces increasing, decreasing, and upside down bathtub shapes.

The EGTL-Quasi Lindley (EGTLQL) Model
The cdf and pdf of the quasi-Lindley distribution are (

Statistical Properties
In this section, we study some statistical properties of the EGTL-G family such as: ordinary and incomplete moments, generating function, Lorenz and Bonferroni curves, Rényi entropy, stress strength model, moment of residual and reversed residual life, order statistics and extreme values.

Moments and Generating Functions
Suppose X is a random variable with EGTL-G distribution, then the ordinary moments, say , r  is given by From (16), the measures of skewness and kurtosis of the EGTL-G distribution can be obtained as respectively. Figure (1) shows the behavior of skewness and kurtosis of EGTL-Lx distribution.

Suppose
X is a random variable with EGTL-G distribution, then the th r incomplete moment, denoted by ( ), ( ) Therefore, these quantities for the EGTL-G family are obtained from

Rényi Entropy
The concept of entropy has been applied in different fields such as statistics, queuing theory and reliability estimation. The Rényi entropy is defined as Consequently, the Rényi entropy for the EGTL-G family is given by

Stress Strength Model
The stress strength model is a common criterion used in different applications in engineering and physics. Let 1 X and 2 X be two independent random variables with EGTL-G ( 1 1 1 , , , ab ) and EGTL-G ( 2 2 2 , , , ab ) distributions. Then, the stress strength model is given by

Moment of Residual and Reversed Residual Life
The moment of residual and reversed residual life uniquely determine () Fx. The th n moment of the residual life, say ( ), Consequently, () n mt for the EGTL-G family is given by , , , 0 0 The th n moment of the reversed residual life, say

Order Statistics
Order statistics play an important role in probability and statistics. Let 1, 2: : ,... where, Moreover, the th r moment of th k order statistic for EGTL-G family is given by First, Suppose that G belongs to the max domain of attraction of the Gumbel extreme value distribution. Then by Leadbetter et al. (1987), there must exist a strictly positive function, say () ht , such that  characterizations of the EGTL-G distribution. These characterizations are based on: (i) a simple relationship between two truncated moments; (ii) the hazard function; (iii) the reverse (or reversed) hazard function and (iv) conditional expectation of a function of the random variable. It should be mentioned that for characterization (i) the cdf is not required to have a closed form. We present our characterizations (i)-(iv)in four subsections.

Characterizations based on two truncated moments
In this subsection we present characterizations of the EGTL-G distribution in terms of a simple relationship between two truncated moments. The first characterization result employs a theorem due to Glanzel (1987), see Theorem 1 below. Note that the result holds also when the interval H is not closed. Moreover, as mentioned above, it could be also applied when the cdf F does not have a closed form. As Shown in Glanzel (1990), this characterization is stable in the sense of weak convergence.
 Assume that for . x The random variable X has pdf (6) if and only if the function  defined in Theorem 1 has the form Let X be a random variable with pdf (6), then Conversely, if  is given as above, then ( ) The general solution of the differential equation in Corollary 7.1 is where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 7.1 with 0. D = However, it should be also noted that there are triplets 12 ( , , ) qq satisfying the conditions of Theorem 1.

Characterization based on hazard function
It is known that the hazard function, ,  .stat.oper.res. Vol.XV No.1 2019 pp1-24 14 the EGTL-G distribution, for 1, b = in terms of the hazard function, which is not of the above trivial form.

Proposition 7.2. Let :
X → be a continuous random variable. For 1, b = the pdf of X is (6) if and only if its hazard function () F hx satisfies the differential equation X has pdf (6), for 1, b = then clearly the above differential equation holds. Now, if the differential equation holds, then

Characterization in terms of the reverse (or reversed) hazard function
The reverse hazard function, ,    which is the reverse hazard function of the EGTL-G distribution.

Characterizations Based on Conditional Expectation
The following propositions have already appeared in Hamedani (2013), so we will just state them here which can be used to characterize the EGTL-G distribution.
be a continuous random variable with cdf .  In this section, we describe the maximum likelihood estimates (MLEs) for the model parameters of the EGTL-G family. Letbe an independent random sample 12

Estimation of Parameters
   =     whose elements are given in Appendix A.

Simulation Study
In this section, the maximum likelihood estimators for the parameters of EGTLLx density function have been assessed by simulating: ( , , , , ) (0.5, 2,1.5, 2,1.5).    Table 1. It is observed, from Table 1, reveals how the biases, mean squared errors vary with respect to .
n As expected, the Biases and MSEs of the estimated parameters converge to zero while n growing.

Applications
In this section, we provide two application to real data to illustrate the applicability of the EGTL-G family. We focus on the EGTLLx distribution introduced in Subsection 6.2. We have used data from Nigm et al. (2003) and is about ordered failure of components. The data is given as follows:   models for the first data set and second data set are presented in Tables 2, 3, 4       The values in Tables 3 and 5 indicate that the EGTLLx model has the lowest values for A*, W*, KS and largest P-values among all fitted models (for the two real data sets). So, the EGTLLx models could be chosen as the best models. The estimated pdfs and cdfs plots are displayed in Figures (5) and (6). It is clear from Figures (5) and (6) that the new EGTLLx distribution provides the best fits to both data sets.