Odd Generalized N-H Generated Family of Distributions with Application to Exponential Model

A new family of distributions called the odd generalized N-H is introduced and studied. Four new special models are presented. Some mathematical properties of the odd generalized N-H family are studied. Explicit expressions for the moments, probability weighted, quantile function, mean deviation, order statistics and Rényi entropy are investigated. Characterizations based on the truncated moments, hazard function and conditional expectations are presented for the generated family. Parameter estimates of the family are obtained based on maximum likelihood procedure. Two real data sets are employed to show the usefulness of the new family.


Introduction
In recent years, several classes have been defined by adding one or more parameters to generate new distributions. These distributions extend well-known distributions as well as provide great flexibility to model specific real data. Some of the well-known generators are the beta-G by Eugene et al. (2002), Kumaraswamy-G by Cordeiro  In this paper, we introduce a new generated family of distributions using the NH distribution as a generator. The NH distribution is introduced by (Nadarajah and Haghighi, 2011) which has the following cumulative distribution function (cdf) where C = e. The associated probability density function (pdf ) corresponding to (1) is as follows where λ is the scale parameter and α is the shape parameter. We are interested in modeling the random variable X of this odds using the NH model (with scale parameter λ = 1) given by (1). The cdf of OGNH-G family can be expressed as follows , ∈ ℝ, , > 0, where, α, β are two shape parameters, and ( ; ) is a baseline cdf, which depends on a parameter vector . The distribution function (3) provides a broadly odd half logistic generated distributions. Therefore, the pdf of the OGNH-G family is as follows . (4) Hereafter, a random variable has pdf (4) will be denoted by ∼ − . The survival function, hazard rate and reversed hazard rate functions are, respectively, given by .
This paper can be sorted as follows. In the next section, Characterizations based on the truncated moments, hazard function and conditional expectations are presented for the OGNH-G family. Section 3 provides some general mathematical properties of the family. Section 4 gives the estimation of the parameters of the family using the maximum likelihood method. In Section 5, some new special models of the generated family are considered. Some statistical properties, estimation and simulation study for odd generalized NH exponential model are derived in Section 6. The application of the odd generalized NH exponential distribution to a real data set is presented in Section 7. At the end, concluding remarks are outlined in Section 8.

Characterizations of OGNH-G Distribution
This section is devoted to the characterizations of the OGNH-G distribution in the following directions: (i) based on the ratio of two truncated moments and (ii) in terms of the hazard. Note that (i) can be employed also when the cdf does not have a closed form. We also like to point out that the characterization (i) is stable in the sense of weak convergence. We present our characterizations (i)-(ii) in two subsections.

Characterizations based on two truncated moments
This subsection is devoted to the characterizations of OGNH-G distribution based on the ratio of two truncated moments. Our first characterization employs a theorem due to Glänzel (1987), see Theorem 1 of Appendix A. The result, however, holds also when the interval H is not closed, since the condition of the Theorem is on the interior of H. ( ; ) ) ) }and 2 ( ) = 1 ( ) ( ; ) for ∈ ℝ.
The random variable X has pdf (4) if and only if the function defined in Theorem 1 is of the form Proof. Suppose the random variable X has pdf (4), then Further, Conversely, if is of the above form, then  where D is a constant. We like to point out that one set of functions satisfying the above differential equation is given in Proposition A.1 with D=1/2. Clearly, there are other triplets ( 1 , 2 , )which satisfy conditions of Theorem 1.

Characterization in terms of hazard function
The hazard function, ℎ , of a twice differentiable distribution function, F, satisfies the following first order differential equation It should be mentioned that for many univariate continuous distributions, the above equation is the only differential equation available in terms of the hazard function. In this subsection we present a non-trivial characterization of OGNH-G distribution in terms of the hazard function.

Proposition 2.2.1Let
: Ω → be a continuous random variable. The random variable X has pdf (4) if and only if its hazard functionℎ , satisfies the following differential equation Proof. If X has pdf (4), then clearly the above differential equation holds. If the differential equation holds, then from which we arrive at the hazard function corresponding to the pdf (4).

Some Statistical Properties
This section provides some statistical properties of OGNH-G family of distributions.

Quantile function
Let denotes a random variable has the pdf (4), the quantile function, say ( ) of is given by where, u is a uniform distribution on the interval (0,1) and G -1 (.) is the inverse function of G(.).

3.2.
Useful representation In this subsection, a useful expansion of the pdf and cdf for OGNH-G is provided. Since the exponential series is Inserting (6) in (4) then, By using binomial theory for| | < 1, and is a positive real non integer. Then, by applying the binomial theorem (7)in the previous density function of OGNH-G family becomes Since, the binomial expansion By inserting (9) in (8) then, the pdf (8) can be written as follows .

3.3.
The probability weighted moments For a random variable , the probability-weighted moments (PWMs), denoted by , , can be calculated through the following relation The PWMs of OGNH-G is obtained by substituting (10) and (12) into (13), and replacing h with s, as follows Then,

Moments
Since the moments are necessary and important in any statistical analysis, especially in applications. Therefore, we derive the ℎ moment for the OGNH-G family. If has the pdf (10), then ℎmoment is obtained as follows For a random variable , it is known that, the moment generating function is defined as , ( +1)+ −1 .

3.5.
The mean deviation For random variable with pdf ( ), cdf ( ), the mean deviation about the mean and mean deviation about the median, are defined by which is the first incomplete moment.

Rényi entropy
The entropy of a random variable is a measure of variation of uncertainty and has been used in many fields such as physics, engineering and economics. As mentioned by (Rényi 1961), the Rényi entropy is defined by By applying the binomial and exponential theory in the pdf (4), then the pdf ( ) can be expressed as follows Therefore, the Rényi entropy of OGNH generated family of distributions is given by . These equations cannot be solved analytically and statistical software can be used to solve them numerically using iterative methods.

Special Models of the OGNH family
In this section, we discuss four special models of the OGNH generated family, namely, OGNH-uniform (OGNHU), OGNH-Lomax (OGNHL), OGNH-Rayleigh (OGNHR) and OGNH-exponential (OGNHE) distributions, respectively. The plots of the pdf and hrf of each special model are also sketched for different parametric values.

OGNH-Uniform Distribution
Consider the pdf and cdf of the uniformly distributed random variable given by ( ; ) = 1 , 0 < < , and ( ; ) = , respectively. Then the cdf of the OGNHU is given as The pdf corresponding (3.1) takes the following form For different values of parameters, the plots of pdf and hrf for the OGNHU distribution are displayed in Figure 1 below: Figure 1: Pdf and hrf of OGNHU distribution for varying values of parameters.

OGNH-Lomax Distribution
Let the Lomax distribution be the parent distribution with pdf and cdf given by For selected values of the model parameters, the graphs of pdf and hrf for the OGNHL distribution are illustrated in Figure 2 as:

OGNH-Rayleigh Distribution
The pdf and cdf of the Rayleigh random variable has the following form ( ) = 2 − 2 , , > 0, and ( ) = 1 − − 2 , respectively. Then, the cdf of OGNHR distribution is given by The graphical illustration of the pdf and hrf for the OGNHR are sketched below in Figure 3 as:

OGNH-Exponential Distribution
Considering the exponential distribution as a parent distribution with pdf and cdf given by ( ) = − , , > 0, and ( ) = 1 − − , respectively. Then, the cdf of OGNHE distribution is given by The density function corresponding to (3.7) becomes The graphical sketch of the pdf and hrf for the OGNHE are showed in Figure 4 as:

Properties and Estimation of OGNHE Distribution
In this section, we derived fundamental properties, quantile function, moments, of OGNHE distribution.

.1. Mathematical and statistical properties 5
The quntile function of the OGNHE distribution is given by Median of the OGNHE random variable is given by ]. .

Mean deviations
The probability-weighted moments of the OGNHE distribution is given by The r th moment of the OGNHE distribution is given by The moment generating function of the OGNHE distribution is given by .

MLEs and their Performances
The log-likelihood function for parameter vector = ( , , ) is obtained as follows The components of score vector are: The above equations cannot be solved analytically; rather analytical software is required to solve them numerically. Further, a numerical investigation is carried out to evaluate the performance of ML estimators for ONHE model. Performance of estimators is evaluated through their biases, and mean square errors (MSEs) for different sample sizes. A numerical study is performed using Mathematica (9) software. Different sample sizes are considered through the experiments at size n = 50, 100, 150and 200. In addition, the different values of parameters ( , , ) are considered. The experiment is repeated 10000 times. In each experiment, the estimates of the parameters are obtained by ML method of estimation. The MSEs and biases for the different estimates are reported from these experiments in Table  1.  Table 1 Continued of Table 1 7.

Real Life Applications
In this section, we have provided two applications using real data sets to illustrate efficiency of the new proposal. The first data set taken from Murthy et al. (2004) representing the failure times, while the second data set taken from the website: http://www.ceramics.nist.gov/srd/summary/ftmain.htm representing the fracture toughness of Alumina (Al2O3). For the interest of readers, we have provided the data in Table 2, and summarized in Table 3 Table 4, 5, 6 and 7 shows that the suggested method provides best fit than the other fitted distributions.

Conclusion
In this paper, we introduce the odd generalized NH-G family. Some of its properties are derived and some members of the family are defined. A member of the odd generalized NH -G family, namely, the odd generalized NH exponential distribution is defined and studied. Various properties of the odd generalized NH exponential distribution including, probability weighted moments, moments, Rényi entropy and order statistics are derived. Estimation and simulation issues for the OGNHE model are performed to study the behavior of the estimated parameters. The OGNHE is applied to fit a real data set. This application shows that the OGNHE can be preferred over some other well-known distributions.
Let : Ω → be a continuous random variable with the distribution function F and let 1 and 2 be two real functions defined on H such that is defined with some real function . Assume that 1 , 2 , 1 ( ), ∈ 2 ( )and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation 1 = 2 has no real solution in the interior of H. Then F is uniquely determined by the functions 1 , 2 and , particularly.
Where the function S is a solution of the differential equation / = / 1 1− 2 and C is the normalization constant, such that ∫ = 1.
We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see, Glänzel [2]), in particular, let us assume that there is a sequence { }of random variables with distribution functions { }such that the functions 1 , 2 and , ( ∈ ℕ) satisfy the conditions of Theorem 1 and let 1 → 1 , 2 → 2 for some continuously differentiable real functions 1 and 2 . Let, finally, X be a random variable with distribution F. Under the condition that 1 and 2 are uniformly integrable and the family{ }is relatively compact, the sequence converges to in distribution if and only if converges to , where This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions 1 , 2 and , respectively. It guarantees, for instance, the 'convergence' of characterization of the Wald distribution to that of the Lévy-Smirnov distribution if → ∞.
A further consequence of the stability property of Theorem 1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions. For such purpose, the functions 1 , 2 and, specially, should be as simple as possible. Since the function triplet is not uniquely determined it is often possible to choose as a linear function. Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics.