1. Introduction

Recently, several families of continuous uni-variate distributions have been constructed by extending common families of continuous models. These generalized distributions give more flexibility by adding one or more parameters to the baseline model. Many classes can be cited such as the Marshall-Olkin-G family by Marshall and Olkin [25], transmuted exponentiated generalized-G family by Yousof et al. [34], Burr X-G by Yousof et al. [35], type I half-logistic family by Cordeiro et al. [12], Zografos-Balakrishnan odd log-logistic family of distributions by Cordeiro et al. [13], a new generalized two-sided family of distributions by Korkmaz and Genç [22], generalized odd log-logistic family by Cordeiro et al. [10], odd-Burr generalized family by Alizadeh et al. [4], beta Weibull G by Yousof et al. [36], exponentiated generalized-G Poisson family by Aryal and Yousof [8], type I general exponential class by Hamedani et al. [20] and beta transmuted-H by Afify et al. [2] among others.

Recently, Gleaton and Lynch [17] introduced a class of distributions called the odd log-logistic family with one extra shape parameter α > 0 with cumulative distribution function (cdf)

where G(x; ξ) is the baseline cdf depending on the vector parameter ξ and G¯(x;ξ)=1-G(x;ξ) and ξ the parameter vector of associated parameters from baseline distribution. Alzaatreh et al. [5] defined the T − X family with cdf where r(t) is the probability density function (pdf) of the random variable of T ∈ [a, b] for −∞ < a < b < ∞, G(x; ξ) is cdf of the baseline random variable X and W [G(x)] is a function of the baseline cdf which satisfies the following conditions: i)W [G(x)] ∈ [a, b], ii)W [G(x)] is differentiable and monotonically non-decreasing, iii)lim_x→−∞W [G(x)] = a and lim_x→∞W [G(x)] = b. The pdf of the T − X family is given by On the other hand, an extension of the Lindley distribution, denoted by exponential Lindley (EL), was introduced by Gomez et al. [18]. The cdf and pdf of this extension are given, respectively, by and where x > 0, β > 0 and θ > 0. Clearly, distribution of this extension is exponential distribution for β = 0 and is Lindley distribution for β = 1. Also, this distribution is more flexible than exponential and Lindley distributions.

The goal of this paper is to introduce a new flexible and wider family of the distributions based on T–X family using the EL as the generator. The paper is organized as follows. The new family is introduced in Section 2. Some members and sub-families of the new family are introduced in Section 3. In Section 4, the series expansions for cdf and pdf of the new family are presented. In Section 5, some of its mathematical properties are derived. Section 6 deals with some characterizations of the new family. In Section 7, the maximum likelihood method is presented. In Section 8, a simulation study is performed to evaluate the efficiency of the Maximum Likelihood method. In Section 9, we illustrate the importance of the new family by means of three applications to real data sets. The paper is concluded in Section 10.

2. The new family

We define the new family by taking W [G(x; ξ)] = −log[1 − H(x; ξ)] and r(t) = g_EL (t; β, θ). From (1.1), (1.2), (1.3) and (1.4), we introduce cdf and pdf of the new family as

and respectively, where G(x) = G(x; ξ) is the baseline cdf, g(x) = g(x; ξ)is the baseline pdf, α > 0, β > 0 and θ > 0 are three extra parameters. We call it exponential Lindley odd log-logistic family and denote it by ELOLL−G(α, β, θ, ξ). Further, the ELOLL-G family is in fact a mixture of the exponential-odd log-logistic-G family (EOLL), whose cdf is 1-(G¯(x)αG¯(x)α+G(x)α)θ and gamma(2, θ)-odd log-logistic-G family (GaOLL), whose cdf is 1-[1-log(G¯(x)αG¯(x)α+G(x)α)θ](G¯(x)αG¯(x)α+G(x)α)θ, with mixing rate θ/(θ + β)and β/(θ + β) respectively. So, we express the ELOLL-G family as As a result, we can say that the ELOLL-G family is a mixture family. The hazard rate function (hrf) of this family is given by, The quantile function (qf) of ELOLL-G is given as follows: if U has a uniform U(0, 1) the solution of the non-linear equation x=Q(U)=QG(TU1/α[TU1/α+(1-TU)1/α]-1) has random number on the ELOLL−G(α, β, θ, ξ), where Q_G(·) = G⁻¹(·) is the qf of the baseline distribution, Q_EL(u) is the solution of (1.3) that is the qf of the EL distribution and T_U = (1 − e^−Q_EL(U)). Hence, if U is a uniform random variable on U(0,1), then X = Q(U) has the ELOLL-G distribution. Also, the mixture representation in (2.3) can be used to generate random number from the ELOLL-G distribution.

3. Some special cases of the ELOLL-G family

The ELOLL-G family includes important sub-families. We mention these sub-families in Table (1). Also, the (2.1) and (2.2) will be most tractable when g(x) and G(x) have simple analytic forms. Here, we provide three special models of the ELOLL-G family. These special models generalize some well-known distributions reported in the literature. We note that since the cdf, pdf and hrf of any ELOLL-G distribution will be easily determined by (2.1), (2.2) and (2.4), we dont give them in following sub-sections.

4. Expansions for cdf and pdf

In this Section, we will obtain cdf and pdf expansions of the ELOLL-G family using Taylor series of log(1-z)=-∑k=0∞zk+1(k+1),|z|<1 and others series expansions. Then equation (2.1) can be written as

where the quantities A, B, C and D are defined in Appendix A . The last equation can be written as where b_k and h_k are defined in the Appendix A. Then where ϒ0=d0b0 and Ω0=p0h0 for k ≥ 1 and then where Ψk=∑i=0∞θβ(θ+β)(i+1)Ωk, υ₀ = 1 − ζ_k = 1 − (ϒ_k +Ψ_k) and for k ≥ 1 we have υ₀ = −ζ_k and Π_k(x) is the cdf of the exponentiated-G (exp-G) family with power parameter (γ). Equation (4.1) reveals that the density of ELOLL-G family is a linear combination of exp-G densities. So, some mathematical properties of this family can be determined from those of the exp-G distribution. By differentiating (4.1), we obtain the same mixture representation where π_γ(x) = γg(x)G(x)^γ−1 represents the exponentiated-G family density with power parameter γ > 0. The equations (4.1) and (4.2) are the main results of this section.

5. Properties

The r^th moment of X, say µ′_r, follows from (4.2) as

Henceforth, Y_γ denotes the exp-G distribution with power parameter (γ). For γ > 0, we have E(Yγr)=γ ∫-∞∞xr g(x;ξ) G(x;ξ)γ-1 dx, which can be computed numerically in terms of the baseline quantile function (qf) Q_G (u; ξ) = G⁻¹ (u; ξ) as E(Yγr)=γ∫01QG(u;ξ)ruγ-1du. The n^th central moment of X, say M_n, is given by Here, we provide two formulae for the mgf M_X (t) = E(e^tX) of X. Clearly, the first one can be derived from equation (5) as MX(t)=∑k=0∞vk+1Mk+1(t), where M_k+1 (t) is the mgf of Y_k+1. Hence, M_X (t) can be determined from the exp-G generating function. A second formula for M_X (t) follows from (4.2) as MX(t)=∑k=0∞υk+1ς(t,k), where ς(t,k)=∫01exp[tQG(u)]ukdu and Q_G(u) is the qf corresponding to G(x; φ), i.e., Q_G(u) = G⁻¹(u; φ). For the ELOLL-W model we get

Using the series expansion (1-z)a=∑i=0∞(ai)(-z)i for |z| < 1, one can expand (k+1)βαβxβ-1exp[-(αx)β]{1-exp[-(αx)β]}k as

where f_(h+1)^1/γλ(·) denotes the pdf of a two-parameter Weibull distribution with λ replaced by (h + 1)^1/γ λ. So, whenever possible, (5.1) can be used to derive moment generating function of the ELOLL-W model from those of a two-parameter Weibull distribution. Consider _pΨ_q(·) as the complex parameter Wright generalized hypergeometric function with p numerator and q denominator parameters (Kilbas et al. [21], Equation (1.9)) defined by the series where α_j, β_k ∈ B_k, A_j, B_k ≠ 0, j=1,p¯, k=1,q¯ and the series converges for 1+∑j=1qBj-∑j=1pAj>0, compare with Mathai and Saxena [26] and Srivastava et al. [31]. This function was originally introduced by Wright [33]. Let X be a random variable having the pdf (3.2). Following similar algebraic developments of Nadarajah et al. [28], we can write the moment generating function (mgf) of the ELOLL-W model Hypergeometric functions are included as in-built functions in most popular algebraic mathematical software packages, so the special function in (5.2) and hence (5.3) can be easily evaluated by the software packages Maple, Matlab and Mathematica using known procedures.

Suppose X₁,...,X_n is a random sample from an ELOLL-G distribution. Let X_i:n denote the i^th order statistic. The pdf of X_i:n can be expressed as

Following similar algebraic developments of Nadarajah et al. [29], we can write the density function of X_i:n as where br,k=n!(r+1)(i-1)!υr+1(r+k+1)∑j=0n-i(-1)jfj+i-1,k(n-i-j)!j!, υ_r+1 is given in Section 3 and the quantities f_j+i−1,k can be determined with fj+i-1,0=υ0j+i-1 and recursively for k ≥ 1 Equation (5.4) reveals that the pdf of the ELOLL-G order statistics is a linear combination of exp-G density functions.

6. Characterizations

This section deals with various characterizations of ELOLL distribution. These characterizations are presented in two directions: (i) based on truncated moments and (ii) in terms of the hazard function. It should be noted that characterization (i) can be employed also when the cdf does not have a closed form. We present our characterizations (i) and (ii) in two subsections.

7. Maximum Likelihood Estimation (MLE) and Inference

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 x₁,...,x_n be a random sample from ELOLL-G model with a (q+3)×1 parameter vector Ξ =(α, β, θ, ξ) ^⊤ , where ξ is a q×1 baseline parameter vector. The log-likelihood function for Ξ can be expressed as

where ti=G(xi,ξ)α+G¯(xi,ξ)α. The log-likelihood function can be maximized by solving the following nonlinear normal equations which are available if needed. To solve these equations, it is more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ(Ξ).

The likelihood ratio (LR) statistic can be used for comparing the some sub-models of ELOLLG model. For example, the LR statistic can be used to discriminate between the ELLOL-Lomax and L-Lomax since they are nested models, which is equivalently to test H₀ : α = β = 1. The LR statistic reduces to w=2[ℓ(α^,β^,σ^,β^)-ℓ(1,1,σ˜,β˜)], where (α^,β^,σ^,β^) are the unrestricted MLEs and (1,1,σ˜,β˜) are the restricted estimates under H₀. The statistic w is asymptotically (as n → ∞) distributed as χk2, where k is difference of two parameter vectors of nested models. For example, k = 2 for above hypothesis test.

8. A simulation study

In this subsection, we perform the simulation study using the exponential Lindley odd log-logistic exponential distribution (ELOLL-E) which is the generalization of the exponential distribution with cdf F (x; λ) = 1 − exp(−λx), x > 0, λ > 0. To see the performance of MLE’s of this distributions, we generate 1,000 samples of sizes 20, 50 and 100 from ELOLL-E using inverse of the its cdf for θ = 2. The results of the simulation are reported in Table (2). From this Table, we observe that the estimates approach true values as the sample size increases whereas the standard devations of the estimates decrease.

9. Data analysis

In this section, we introduce three applications using well-known data sets to show the flexibility and applicability of the proposed models over other models. First, we describe the data sets and we fit some distributions to these data sets using MLE method. Then we compare proposed distributions with several member of distribution families. The model selection is applied using the estimated log-likelihood (ℓ^), Kolmogorov-Smirnov (K-S) statistics, Akaike information criterion (AIC), Consistent Akaike information criteria (CAIC), Bayesian information criterion (BIC), and Hannan-Quinn information criterion (HQIC). The AIC, CAIC, BIC and HQIC are by given by AIC=-2ℓ^+2p, CAIC=-2ℓ^+2pn(n-k-1)-1, BIC=-2ℓ^+plogn and HQIC=-2ℓ^+plog(logn), where p is the number of the estimated model parameters and n is sample size. When searching the best fit among others to data, the distribution with the smallest AIC, CAIC, BIC, HQIC and K-S values and the biggest log-likelihood and p values of the K-S statistics is chosen. All calculations are obtained by maxLik routine in R programme.

First data set studied by Murthy et al. [27], which represent failure times for a particular windshield device. This data set has been analyzed by Cordeiro et al. [14]. Using this data set, we fit the ELOLL-N, Lindley-normal (L-N) (Çakmakyapan and Ozel, [9]), exponential-normal (E-N) or Lehmann type II exponentiated-normal (Alzaatreh et al., [5]; Cordeiro et al. [15]), McDonaldnormal (McN) (Alexander et al. [3]), normal-normal{exponential} (NNE) (Alzaatreh et al. [6]), normal-Cauchy{log-logistic} (NCLL) (Alzaatreh, et al. [7]), logistic-normal (LN) (Tahir et al. [32]), generelized Kumaraswamy-normal (GKw-N) (Cordeiro et al. [11]) and generalized odd log-logistic normal (GOLLN) (Cordeiro, et al. [10]) distributions models. The results of this application are listed in Table 3.

As second data analysis, we analyze the data set studied by Abouammoh et al. [1], which represent the lifetime in days of 40 patients suffering from leukemia from one of the Ministry of Health Hospitals in Saudi Arabia. By using this data set we compare the ELOLL-W distribution with logistic-Weibull (LW) (Tahir et al. [32]), odd-Burr-Weibull (OBW) (Alizadeh et al. [4]), Lindley-Weibull (L-W) (Çakmakyapan and Ozel [9]) and exponential-Weibull (E-W) or ordinary Weibull distributions. The results of this application are listed in Table 4.

The third real data consists of the number of successive failure for the air conditioning system reported of each member in a fleet of 13 Boeing 720 jet airplanes. The pooled data with 213 observations was considered by Proschan [30] and Kuş [23]. For this data, we use the ELOLL-Lx distribution and compare it Zografos-Balakrishan odd-loglogistic-Lomax (ZBOLL-Lx) (Cordeiro et al. [13]), Kumaraswamy Lomax (Kw-Lx) (Lemonte and Cordeiro [24]), beta-Lomax (B-Lx) (Lemonte and Cordeiro [24]) and its sub-models. The results of the application are in Table 5.

From Tables 3–5, we see that the ELOLL-N, ELOLL-W and ELOLL-Lx models have the lowest AIC, CAIC, BIC, HQIC and K-S values and has the biggest estimated log-likelihood and p-value of the K-S statistics among all the fitted models. So they could be chosen as the best models under these criteria.

The histograms of three data sets and the estimated pdfs and cdfs of the application models are displayed in Figures 4. From the Figures 4, we show that the ELOLL-G models provide the good fit to these data sets as compared to other models.

Fig. 4:

The fitted pdfs and the fitted cdfs for three data sets

A comparison of the proposed distributions with some of their sub-models using LR statistics is performed in Table 6. From Table 6, we can conclude that the ELOLL-N and ELOLL-W models yield a better fit to these data than the other two and three sub-models respectively. Also, there is no difference among the fits to the current data using the ELLOL-Lx, L-Lx and E-Lx models. In addition, these models provide a better representation of the data than the other sub-models based on the LR test at the 5% significance level.

10. Conclusions

A new family of distributions called the exponential Lindley odd log-logistic G family is introduced and studied. The new family generalizes the Lindley-G family [9], Lehmann Type II-G family [19] and odd Burr-G family [4] as well as it introduces new distribution families such as exponential Lindley-G family and Lindley odd log-logistic-G family. We provide some mathematical properties of the new family including ordinary, generating function and order statistics. Characterizations based on truncated moments as well as in terms of the hazard function are presented. The maximum likelihood is used for estimating the model parameters. We give a simulation study for the maximum likelihood estimators by using the special member of new family. Finally, the usefulness of the family is illustrated by means of three real data sets. The new models provide consistently better fits than other competitive models for these data sets.