Type II General Exponential Class of Distributions

In this paper, a new class of continuous distributions with two extra positive parameters is introduced and is called the Type II General Exponential (TIIGE) distribution. Some special models are presented. Asymptotics, explicit expressions for the ordinary and incomplete moments, moment residual life, reversed residual life, quantile and generating functions and stress-strengh reliability function are derived. Characterizations of this family are obtained based on truncated moments, hazard function, conditional expectation of certain functions of the random variable are obtained. The performance of the maximum likelihood estimators in terms of biases, mean squared errors and confidence interval length is examined by means of a simulation study. Two real data sets are used to illustrate the application of the proposed class.

has cdf (3). An interpretation of the TIIGE family (3) can be given as follows. Let be a random variable describing a stochastic system by the cdf 1 − ̅ ( ) (for > 0). If the random variable represents the odds ratio, the risk that the system following the lifetime will not be working at time is given by (1 − ̅ ( ) )/ ̅ ( ) . If we are interested in modeling the randomness of the odds ratio by the Exponential pdf ( ) = e − t (for > 0), the cdf of is given by which is exactly the cdf (3) of the new family. The basic motivations for using the TIIGE-G family in practice are: (i) to make the kurtosis more flexible compared to the baseline model; (ii) to produce a skewness for symmetrical distributions; (iii) to construct heavy-tailed distributions that are not longertailed for modeling real data; (iv) to generate distributions with symmetric, left-skewed, right-skewed and reversed-J shaped; (v) to define special models with all types of the hrf; and (vi) to provide consistently better fits than other generated models under the same baseline distribution. Now, Several structural properties of the extended distributions may be easily explored using mixture forms of exponentiated-G ("Exp-G") models. In the following, we obtain expansions for ( ) and ( ). So here we provide a useful representation for (3). The cdf of the TIIGE family in (3) can be expressed as Expanding ( ) − and after some algebra we get The above equation can be expressed as where +1 = − and ( ) = ( ) ( ) −1 is the pdf of the Exp-G distribution with power parameter . The properties of Exp-G distributions have been studied by many authors in recent years, see Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) for exponentiated Weibull, Gupta et al. (1998) for exponentiated Pareto, Gupta and Kundu (1999) for exponentiated exponential, Nadarajah (2005) for the exponentiated-type distributions, Nadarajah and Kotz (2006) for exponentiated Gumbel, Shirke and Kakade (2006) for exponentiated log-normal and Nadarajah and Gupta (2007) for exponentiated gamma distributions, among others. The paper is unfolded as follows. In Section 2, we obtain some mathematical properties of the proposed model. In Section 3, we provide some useful characterizations of the new model. In Section 4, the model parameters are estimated by using maximum likelihood method and a simulation study is performed. Some special TIIGE models are given in Section 5. Two applications are given in Section 6 to illustrate the flexibility of the proposed model. Finally, Section 7offers some concluding remarks. We can evaluate the effect of parameters on tails of distribution using these equations.

Moments and generating function
The th moment of , say ′ , follows from (7) as ′ = ( ) = ∑ ∞ =0 +1 ( +1 ). , is given by The cumulants ( ) of follow recursively from The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. Here, we provide two formulas for the moment generating function (mgf) ( ) = (e ) of . Clearly, the first one can be derived from equation (7) as where (ℎ+1) 1/ (⋅) denotes the pdf of a two-parameters Weibull model with replaced by (ℎ + 1) 1/ . So, whenever possible, (9) can be used to derive mgf of the TIIGE-W model from those of a two-parameters Weibull distribution. Consider Ψ (⋅) , the complex parameter Wright generalized hypergeometric function with numerator and denominator parameters (Kilbas et al., 2006, Equation (1.9)) defined by the series where , ∈Bbb , , ≠ 0, = 1, , = 1, and the series converges for 1 + ∑ =1 − ∑ =1 > 0, compare with Mathai and Saxena (1978) and Srivastava et al. (1982). This function was originally introduced by Wright (1935). Let be a random variable having the pdf (4), we can write the mgf of the TIIGE-W model as Hypergeometric functions are included as in-built functions in most popular algebraic mathematical software packages, so the special function in (10) and hence (11) can be easily evaluated by the software packages Maple, Matlab and Mathematica using known procedures.

Incomplete moments
The main applications of the first incomplete moment refer to the mean deviations and the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The th incomplete moment, say ( ), of can be expressed using (6) as is easily calculated from (3) and 1 ( ) is the first incomplete moment given by (12) with = 1. Now, we provide two ways to determine 1 and 2 . First, a general equation for is the first incomplete moment of the exp-G distribution. A second general formula for 1 ( ) is given by 1 can be computed numerically. These equations for 1 ( ) can be applied to construct Bonferroni and Lorenz curves defined for a given probability by ( ) = 1 ( )/( 1 ′ ) and ( ) = 1 ( )/ 1 ′ , respectively, where 1 ′ = ( ) and = ( ) is the qf of at . For the TIIGE-W model we get where (. , . ) is the lower incomplete gamma function.

moment residual life and reversed residual life
The th moment of the residual life, say ( ) = [( − ) | > ], = 1,2, …, uniquely determines ( ). The th moment of the residual life of is given by Another interesting function is the mean residual life (MRL) function or the life expectation at age defined by 1 ( ) = [( − )| > ], which represents the expected additional life length for a unit which is alive at age . The MRL of can be obtained by setting = 1 in the last equation.
The th moment of the reversed residual life, say ( ) = [( − ) | ≤ ], for > 0 and = 1,2, …, uniquely determines ( ). We obtain Then, the th moment of the reversed residual life of becomes The mean inactivity time (MIT), also called the mean reversed residual life function, is given by 1 ( ) = [( − )| ≤ ], and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, ). The MIT of the TIIGE-G family can be obtained easily by setting = 1 in the above equation. For the TIIGE-W model we get

Order statistics
Suppose 1 , … , is a random sample from any TIIGE-G distribution. Let : denote the th order statistic. The pdf of : can be expressed as Following similar algebraic developments of Nadarajah et al. (2015), we can write the density function of : as is given in Section 1 and the quantities + −1, can be determined with + −1,0 = 0 + −1 and recursively for ≥ 1 Equation (13) is the main result of this section. It reveals that the pdf of the TIIGE-G order statistics is a linear combination of exp-G density functions. So, several mathematical quantities of the TIIGE-G order statistics such as ordinary, incomplete and factorial moments, mean deviations and several others can be determined from those quantities of the Exp-G distribution. For example, for the TIIGE-W model we get

Stress-strength model
The stress-strength model is the most widely approach used for reliability estimation. This model is used in many applications of physics and engineering such as strength failure and system collapse. In stress-strength modeling, = Pr( 2 < 1 ) is a measure of reliability of the system when it is subjected to random stress 2 and has strength 1 .
The system fails if and only if the applied stress is greater than its strength and the component will function satisfactorily whenever 1 > 2 . can be considered as a measure of system performance and naturally raised in electrical and electronic systems. Other interpretations can be that, the reliability, say , of the system is the probability that the system is strong enough to overcome the stress imposed on it. Let 1 and 2 be two independent random variables with TIIGE( 1 , 1 , ) and TIIGE( 2 , 2 , ) distributions. Then, the reliability is defined by

Characterizations
The problem of characterizing a distribution is an important problem, in its own right, which can help the investigator to see if their model is the correct one. This section deals with various characterizations of TIIGE distribution. These characterizations are presented in three directions: ( ) based on the ratio of two truncated moments; ( ) in terms of the hazard function and ( ) based on the conditional expectation of certain functions of the random variable. It should be noted that characterization ( ) can be employed also when the does not have a closed form. We present our characterizations ( ) − ( ) in three subsections.

Characterizations based on two truncated moments
This subsection deals with the characterizations of TIIGE distribution based on the ratio of two truncated moments. Our first characterization employs a theorem of Glänzel (1987), see Theorem 1 of Appendix A .The result, however, holds also when the interval is not closed since the condition of Theorem 1 is on the interior of . The general solution of the differential equation in Corollary 3.1 is where is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 3.1 with = 0.

Characterization in terms of the hazard function
It is known that the hazard function, ℎ , of a twice differentiable distribution function, , satisfies the first order differential equation For many univariate continuous distributions, this is the only characterization available in terms of the hazard function. The following characterization establishes a non-trivial characterization of TIIGE in terms of the hazard function which is not of the above trivial form. Proposition 3. 2. Let : Ω → ℝ be a continuous random variable. Then, has (4) if and only if its hazard function ℎ ( ) satisfies the differential equation which is the hazard function of (4).

Characterization based on the conditional expectation of certain functions of the random variable
In this subsection we employ a single function of and characterize the distribution of in terms of the truncated moment of ( ). The following proposition has already appeared in Hamedani's previous work (2013), so we will just state it as a proposition here, which can be used to characterize TIIGE distribution.

Maximum likelihood method
Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The MLEs enjoy desirable properties and can be used for constructing confidence intervals and also for test statistics. The normal approximation for these estimators in large samples can be easily handled either analytically or numerically. Here, we consider the estimation of the unknown parameters of the new family from complete samples only by maximum likelihood. Let 1 , … , be a random sample from the TIIGE-G model with a ( + 2) × 1 parameter vector =( , , ) ú , where is a × 1 baseline parameter vector. The log-likelihood function for is given by when → ∞, the distribution of ̂ can be approximated by a multivariate normal (0, (̂) −1 ) distribution to construct approximate confidence intervals for the parameters. Here, (̂) is the total observed information matrix evaluated at ̂. The method of the re-sampling bootstrap can be used for correcting the biases of the MLEs of the model parameters. We can compute the maximum values of the unrestricted and restricted log-likelihoods to obtain likelihood ratio (LR) statistics for testing some submodels of the TIIGE-G model. Hypothesis tests of the type 0 : = 0 versus 1 : ≠ 0 , where is a vector formed with some components of and 0 is a specified vector, can be performed using LR statistics. For example, the test of 0 : = 1 versus 1 : 0 is equivalent to comparing the TIIGE and G distributions and the LR statistic is given by = 2{ℓ(̂,̂,̂) − ℓ(1,1,̃)},where ̂,̂ and ̂ are the MLEs under and ̃ is the estimate under 0 .

Simulation study
In this section, we survey the performance of the MLEs of the Type II General Exponential Lomax (TIIGELO) distribution with respect to sample size . This performance is done based on the following simulation study: 1.Generate 1000 samples of size from TIIGELO distribution. The inversion method was used to generate samples.  Figure 2 shows how the four mean squared errors vary with respect to n. The mean squared errors for each parameter decrease to zero as → ∞. Table 1 shows the simulation study results for TIIGE-LO(15,2,20,1) and TIIGE-LO(2,2,4,6). In summary, the biases and MSEs for each parameter decreased to zero and appeared reasonably small at = 300. Clearly, the rate of convergence of MSE for and is less than and .  Table 1: Any information that is needed based on the output interpretation

Special T GE models
In this section, we provide two special models of the TIIGE family. These special models generalize some wellknown distributions reported in the literature. They correspond to the baseline Lomax (LO) and Lindley (L) distributions and illustrate the flexibility of the new family.

Data analysis
In this section, we use two real data sets to compare the fits of the TIIGE-G family with others commonly used lifetime family of distributions. In each case, the parameters of models are estimated by maximum likelihood (Section 4) using the optim function in R program. First, we describe the data sets and give the MLEs (the corresponding standard errors) of the model parameters and the values of the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC) and Anderson-Darling ( * ) statistics. The lower the values of these criteria shows the better fitted model to dataset. Its worth to mention that over-parametrization is penalized in these criteria. Finally, we provide the histograms of the data sets to have a visual comparison of the fitted density functions.     Table 5.  Table 6 presents that the estimates and standard errors of models parameters. Table 7 present Goodness of fit critera for fitted models. This table shows that the TIIGE-L model gives a better fit to this data than the other distributions.

Conclusions
A new class of distributions called the Type II General Exponential class is introduced and studied. We provide a comprehensive treatment of some of its mathematical properties including ordinary and incomplete moments, generating function, asymptotics, order statistics and the QS order, moment of residual life and reversed residual life. We estimate the model parameters by the maximum likelihood method. We assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors by means of a simulation study. The potentiality of the proposed models is illustrated by means of three real data sets.