The odd generalized exponential family of distributions with applications
- Muhammad H Tahir†^{1}Email author,
- Gauss M Cordeiro†^{2},
- Morad Alizadeh†^{3},
- Muhammad Mansoor†^{4},
- Muhammad Zubair†^{4} and
- Gholamhossein G Hamedani†^{5}
https://doi.org/10.1186/s40488-014-0024-2
© Tahir et al.; licensee Springer. 2015
Received: 1 September 2014
Accepted: 15 December 2014
Published: 4 February 2015
Abstract
We propose a new family of continuous distributions called the odd generalized exponential family, whose hazard rate could be increasing, decreasing, J, reversed-J, bathtub and upside-down bathtub. It includes as a special case the widely known exponentiated-Weibull distribution. We present and discuss three special models in the family. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Rényi entropies and order statistics. For the first time, we obtain the generating function of the Fréchet distribution. Two useful characterizations of the family are also proposed. The parameters of the new family are estimated by the method of maximum likelihood. Its usefulness is illustrated by means of two real lifetime data sets. AMS Subject Classification Primary 60E05; secondary 62N05; 62F10
Keywords
Generalized exponential Hazard function Maximum likelihood Moments Order statistic Rényi entropyIntroduction
The art of proposing generalized classes of distributions has attracted theoretical and applied statisticians due to their flexible properties. Most of the generalizations are developed for one or more of the following reasons: a physical or statistical theoretical argument to explain the mechanism of the generated data, an appropriate model that has previously been used successfully, and a model whose empirical fit is good to the data. The last decade is full on new classes of distributions that become precious for applied statisticians. There are two main approaches for adding new shape parameter(s) to a baseline distribution.
1.1 Approach 1: Adding one shape parameter
The first approach of generalization was suggested by Marshall and Olkin (1997) by adding one parameter to the survival function \(\overline {G}(x)=1-G(x)\), where G(x) is the cumulative distribution function (cdf) of the baseline distribution. Gupta et al. (1998) added one parameter to the cdf, G(x), of the baseline distribution to define the exponentiated-G (“exp-G” for short) class of distributions based on Lehmann-type alternatives (see Lehmann 1953). Following Gupta et al. ’s class, Gupta and Kundu (1999) studied the two-parameter generalized-exponential (GE) distribution as an extension of the exponential distribution based on Lehmann type I alternative. The GE distribution is also known as the exponentiated exponential (EE) distribution. Since it is the most attractive generalization of the exponential distribution, the GE model has received increased attention and many authors have studied its various properties and also proposed comparisons with other distributions. Some significant references are: Gupta and Kundu (2001a; 2001b; 2002; 2003; 2004; 2006; 2007; 2008; 2011), Kundu et al. (2005), Nadarajah and Kotz (2006), Dey and Kundu (2009), Pakyari (2010) and Nadarajah (2011). In fact, the GE model has been proven to be a good alternative to the gamma, Weibull and log-normal distributions, all of them with two-parameters. The GE distribution can be used quite effectively for analyzing lifetime data which have monotonic hazard rate function (hrf) but unfortunately it cannot be used if the hrf is upside-down, J or reversed-J shapes. Differently, the new family can have increasing, decreasing, J, reversed-J, bathtub and upside-down bathtub and, therefore, it can be used effectively for analyzing lifetime data of various types. The generalization of the exp-G class to other distributions is beyond the scope of the paper.
1.2 Approach 2: Adding two or more shape parameters
A second approach of generalization was pioneered by Eugene et al. (2002) and Jones (2004) by defining the beta-generated (beta-G) class from the logit of the beta distribution. Further works on generalized distributions were the Kumaraswamy-G (Kw-G) by Cordeiro and de Castro (2011), McDonald-G (Mc-G) by Alexander et al. (2012), gamma-G type 1 by Zografos and Balakrishanan (2009) and Amini et al. (2014), gamma-G type 2 by Ristić and Balakrishanan (2012) and Amini et al. (2014), odd-gamma-G type 3 by Torabi and Montazari (2012), logistic-G by Torabi and Montazari (2014), odd exponentiated generalized (odd exp-G) by Cordeiro et al. (2013), transformed-transformer (T-X) (Weibull-X and gamma-X) by Alzaatreh et al. (2013), exponentiated T-X by Alzaghal et al. (2013), odd Weibull-G by Bourguignon et al. (2014), exponentiated half-logistic by Cordeiro et al. (2014a), T-X{Y}-quantile based approach by Aljarrah et al. (2014) and T-R{Y} by Alzaatreh et al. (2014), Lomax-G by Cordeiro et al. (2014b), logistic-X by Tahir et al. (2015a), a new Weibull-G by Tahir et al. (2015b) and Kumaraswamy odd log-logistic-G by Alizadeh et al. (2015).
We propose a new class of distributions called the odd generalized exponential (“OGE” for short) family, which is flexible because of the hazard rate shapes: increasing, decreasing, J, reversed-J, bathtub and upsidedown bathtub. In Section 2, we define the OGE family of distributions. Two special cases of this family are presented in Section 3. The density and hazard rate functions are described analytically in Section 4. A useful mixture representation for the pdf of the new family is derived in Section 5. In Section 6, we obtain explicit expressions for the moments, generating function, mean deviations and entropies. In Section 7, we determine a power series for the quantile function (qf). In Section 8, we investigate the order statistics. Section 9 refers to some characterizations of the OGE family. In Section 10, the parameters of the new family are estimated by the method of maximum likelihood. In Section 11, we illustrate its performance by means of two applications to real life data sets. The paper is concluded in Section 12.
The new family
Consider a parent distribution depending on a parameter vector ξ with cdf G(x;ξ), survival function \(\overline {G}(x;\boldsymbol {\xi })= 1-G(x;\boldsymbol {\xi })\) and probability density function (pdf) g(x;ξ). The cdf of the GE model with positive parameters λ and α is given by Π(x)=(1−e^{−λ x })^{ α } (for x>0). For α=1, it reduces to the exponential distribution with mean λ ^{−1}. In fact, the density and hazard functions of the GE and gamma distributions are quite similar. The GE density function is always right-skewed and can be used quite effectively to analyze skewed data. Its hrf can be increasing, decreasing and constant depending on the shape parameter in a similar manner of the gamma distribution. Its applications have been wide-spread as a model to power system equipment, rainfall data, software reliability and analysis of animal behavior, among others.
where α>0 and λ>0 are two additional parameters.
Some OGE models
SN | α | λ | G ( x ;ξ) | Reduced distribution |
---|---|---|---|---|
1 | 1 | - | G(x;ξ) | Odd exponential-G family (Bourguignon et al. 2014) |
2 | - | - | \(\frac {x}{1+x}\) | Generalized exponential distribution (Gupta and Kundu 1999) |
3 | 1 | - | \(\frac {x^{c}}{1+x^{c}}\) | Exponentiated Weibull distribution (Mudholkar and Srivastava 1993) |
4 | - | - | 1−e ^{−a x } | Generalized Gompertz distribution (El-Gohary et al. 2013) |
Suppose the system fails if all α components fail and let X denote the lifetime of the entire system. Then, the cdf of X is F(x;α,λ,ξ)=H(x;λ,ξ)^{ α }, which is identical to (1).
In Table 1, we provide some members of the OGE family.
Theorem 1 provides some relations of the OGE family with other distributions.
Theorem 1.
Let X∼O G E(α,λ,ξ).(a) If Y=G(X;ξ), then \(F_{Y}(y)=\left (1-\rm {e}^{-\lambda \frac {y}{1-y}}\right)^{\alpha },\,\,\,\, 0<y<1\), and (b) If \(\displaystyle Y=\frac {G(X;\boldsymbol {\xi })}{\overline {G}(X;\boldsymbol {\xi })}\), then Y∼G E(α,λ).
has the density function (2), where Q _{ G }(·)=G ^{−1}(·) is the baseline qf.
Special OGE distributions
Lifetime distributions play a fundamental role in survival analysis, biomedical science, engineering and social sciences. Typically, lifetime refers to human life length, the life span of a device before it fails or the survival time of a patient with serious disease from the diagnosis to the death. Here, we present three special OGE distributions that can be useful in applied survival analysis.
3.1 The OGE-Weibull (OGE-W) distribution
The OGE-W distribution is defined from (2) by taking \(\phantom {\dot {i}\!}G(x;\boldsymbol {\xi })=1-\mathrm {e}^ {-\theta \,x^{\beta }}\) and \(\phantom {\dot {i}\!}g(x;\boldsymbol {\xi })=\beta \,\theta \,x^{\beta -1}\,\rm {e}^{-\theta \,x^{\beta }}\) to be the cdf and pdf of the Weibull distribution with positive parameters β and θ, respectively, and ξ=(β,θ).
respectively, where α>0 and β>0 are shape parameters and λ>0 and θ>0 are scale parameters.
The applications of the OGE-W and EW distributions can be directed to model extreme value observations in floods, software reliability, insurance data, tree diameters, carbon fibrous composites, firmware system failure, reliability prediction and fracture toughness, among others.
3.2 The OGE-Fréchet (OGE-Fr) distribution
The OGE-Fr distribution is defined from (2) by taking \(G(x;\boldsymbol {\xi })=\mathrm {e}^{-(b/x)^{a}}\) and \(g(x;\boldsymbol {\xi })=a\,b^{a}\,x^{-(a+1)}\,\,\mathrm {e}^{-(b/x)^{a}}\phantom {\dot {i}\!}\) to be the cdf and pdf of the Fréchet distribution with parameters a and b, respectively, and ξ=(a,b).
respectively, where α>0 and a>0 are shape parameters and λ>0 and b>0 are scale parameters.
The OGE-Fr distribution has applications ranging from accelerated life testing, rainfall and floods, queues in supermarkets, sea currents, wind speeds, and track race records, among others.
3.3 The OGE-Normal (OGE-N) distribution
The OGE-N distribution is defined from (2) by taking G(x;ξ)=Φ[(x−μ)/σ] and g(x;ξ)=σ ^{−1} ϕ[(x−μ)/σ] to be the cdf and pdf of the normal distribution with parameters μ and σ, respectively, and ξ=(μ,σ).
respectively, where α>0 is a shape parameter, μ∈ℜ is a location parameter, and λ>0 and σ>0 are scale parameters.
Asymptotics and shapes
Proposition 1.
Proposition 2.
If x=x _{0} is a root of (8) then it corresponds to a local maximum if λ(x)>0 for all x<x _{0} and λ(x)<0 for all x>x _{0}. It corresponds to a local minimum if λ(x)<0 for all x<x _{0} and λ(x)>0 for all x>x _{0}. It refers to a point of inflexion if either λ(x)>0 for all x≠x _{0} or λ(x)<0 for all x≠x _{0}.
If x=x _{0} is a root of (9) then it refers to a local maximum if τ(x)>0 for all x<x _{0} and τ(x)<0 for all x>x _{0}. It corresponds to a local minimum if τ(x)<0 for all x<x _{0} and τ(x)>0 for all x>x _{0}. It gives an inflexion point if either τ(x)>0 for all x≠x _{0} or τ(x)<0 for all x≠x _{0}.
Useful expansions
where, for k≥0, b _{ k }=a _{ k+1}, and, for k≥1, \(a_{k}=\sum _{(i,j)\in I_{k}}\frac {(-1)^{i+j+1}\lambda ^{i}}{i!}\,\dbinom {-i}{j}, \,\,I_{k}=\{(i,j)|i+j=k;i=1,2,\ldots,j=0,1,2,\ldots \}. \)
where \(f_{m}=f_{m}(\alpha)=\sum _{l=m}^{\infty }\frac {(-1)^{l-m}}{l!}\,\binom {l}{m}\,(\alpha)_{l}\). Here and from now on, we use the notation (α)_{ l }=α(α−1)…(α−l+1) for the descending factorial with (α)_{0}=1.
where the coefficients c _{ n,i } (for n=1,2,…) can be determined from the recurrence equation \(c_{n,i}=(i\,b_{0})^{-1}\,\sum _{m=1}^{i}\,[\!m(n+1)-i]\,b_{m}\,c_{n,i-m}\) (for i≥1), and \(c_{n,0}={b_{0}^{n}}\).
where c _{ m,k } can be obtained from the quantities b _{0},…,b _{ k } as in equation (12).
where \(d_{k}=\sum _{m=0}^{\infty }f_{m}\,c_{m,k}\) (for k≥0), H _{ α+k }(x)=G(x)^{ α+k } denotes the cdf of the exp-G distribution with power parameter α+k.
where h _{ α+k }(x)=(α+k) G(x;ξ)^{ α+k−1} g(x;ξ) is the exp-G density function with power parameter α+k. Equation (15) reveals that the OGE density function is a linear combination of exp-G densities. Thus, some mathematical properties of the new model such as the ordinary and incomplete moments, and moment generating function (mgf) can be derived from those properties of the exp-G distribution. Some exp-G structural properties are studied by Mudholkar and Srivastava (1993), Mudholkar et al. (1995), Mudholkar and Hutson (1996), Gupta et al. (1998), Gupta and Kundu (2001a; 2001b), Nadarajah and Kotz (2006) and Nadarajah (2011).
Mathematical properties
6.1 Moments
The need for necessity and the importance of moments in Statistics especially in applications is obvious. Some of the most important features and characteristics of a distribution can be studied through moments. Let Y _{ k } be a random variable having the exp-G pdf h _{ α+k }(x) with power parameter α+k.
Expressions for moments of several exp-G distributions are given by Nadarajah and Kotz (2006), which can be used to obtain \(\mu _{n}^{\prime }\).
The double infinite series on the right-hand side converges for all parameter values.
where (for a>0) \(\tau (n,a)={\int _{0}^{1}} Q_{G}(u)^{n}\,u^{a} d u\). Cordeiro and Nadarajah (2011) obtained τ(n,a) for some well-known distribution such as the normal, beta, gamma and Weibull distributions, which can be applied to obtain raw moments of the corresponding OGE distributions.
For empirical purposes, the shapes of many distributions can be usefully described by what we call the first incomplete moment which plays an important role for measuring inequality, for example, income quantiles and Lorenz and Bonferroni curves.
The last integral can be computed for most G distributions.
6.2 Generating function
where M _{ k }(t) is the mgf of Y _{ k }. Hence, M(t) can be determined from the exp-G generating function.
for \(z\in \mathbb {C}\), where α _{ j }, \(\beta _{k} \in \mathbb {C}\), A _{ j }, B _{ k }≠0, j=1,…,p, k=1,…,q, which converges for \(1+ \sum _{j=1}^{q}B_{j}-\sum _{j=1}^{p}A_{j}>0\).
Combining (20) and the last equation gives the mgf of the OGE-W distribution.
The calculations of the integral in (21) involve the generalized hypergeometric function defined in equation (2.3.1.14) (Prudnikov et al. 1986, p. 322).
Numerical routines for computing the generalized hypergeometric function are available in most mathematical packages, e.g., Maple, Mathematica and Matlab. Nadarajah and Kotz (2005), and Nadarajah (2007) used also this result to obtain the properties of the distribution of the difference between two independent Gumbel variates, and to Iacbellis and Fiorentino (2000), and Fiorentino et al. (2006)’s model for peak streamflow.
where (for a>0) \(\rho (t,a)=\int _{-\infty }^{\infty } \rm {e}^{t\,x}\,G(x)^{a}\,g(x) dx= {\int _{0}^{1}}\,\exp [\!t\,Q_{G}(u)]\,u^{a}\,d u\).
We can obtain the mgf’s of several OGE distributions directly from equation (24). Equations (20) and (24) are the main results of this section.
6.3 Mean deviations
respectively, where \(\mu ^{\prime }_{1}=E(X)\), M=M e d i a n(X)=Q(0.5) is the median which comes from (4), \(F\left (\mu ^{\prime }_{1}\right)\) is easily calculated from the cdf (1) and m _{1}(z) is the first incomplete moment determined from (19) with n=1.
where \(J_{k}(z)=\int _{-\infty }^{z} x\,h_{\alpha +k}(x)dx\) is the first incomplete moment of Y _{ k }. Hence, the mean deviations in (25) depend only on quantity J _{ k }(z).
where \(\gamma (a,z)={\int _{0}^{z}}\,w^{a-1}\,\rm {e}^{-w} dw\) is the incomplete gamma function.
where \(T_{k}(z)=\int _{0}^{G(z)} Q_{G}(u)\,u^{\alpha +k-1} du\) depends on the baseline qf.
Applications of equations (26) and (27) can be directed to the Bonferroni and Lorenz curves defined for a given probability π by \(B(\pi) =m_{1}(q)/(\pi \mu ^{\prime }_{1})\) and \(L(\pi)=m_{1}(q)/\mu ^{\prime }_{1}\), respectively, where \(\mu ^{\prime }_{1}=E(X)\) and q=Q(π) is the qf of X at π.
6.4 Entropies
So, we can write:
Proposition 3.
Quantile power series
where the coefficients a _{ i }’s are suitably chosen real numbers which depend on the parameters of the G distribution. For several important distributions, such as the normal, Student t, gamma and beta distributions, Q _{ G }(u) does not have explicit expressions but it can be expanded as in equation (29). As a simple example, for the normal N(0,1) distribution, a _{ i }=0 for i=0,2,4,… and a _{1}=1, a _{3}=1/6, a _{5}=7/120 and a _{7}=127/7560,…
where \(e_{m}=\sum _{i=0}^{\infty } a_{i}\,d_{i,m}\) and, for i≥0, \(d_{i,0}={\delta _{0}^{i}}\) and, for m>1, \(d_{i,m}=(m\,\delta _{0})^{-1}\,\sum _{n=1}^{m}[n(i+1)-m]\,\delta _{n}\,d_{i,m-n}\).
Equations (31) and (32) are the main results of this section since from them we can obtain various OGE mathematical quantities. In fact, various of these properties can follow by using the second integral for special W(·) functions, which are usually more simple than if they are based on the first integral. Established algebraic expansions to determine mathematical quantities of the OGE distributions based on these equations can be more efficient then using numerical integration of the pdf (2), which can be prone to rounding off errors among others. For the great majority of these quantities, we can adopt twenty terms in the power series (31).
Order statistics
Equation (33) is the main result of this section. It reveals that the pdf of the OGE order statistics is a triple linear combination of exp-G distributions. So, several mathematical quantities of these order statistics like ordinary and incomplete moments, factorial moments, mgf, mean deviations and several others come from these quantities of the OGE distributions.
Characterization of the OGE family
Characterizations of distributions are important to many researchers in the applied fields. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution. Various characterizations of distributions have been established in many different directions. In this section, two main characterizations of the OGE family are presented. These characterizations are based on: (i) a simple relationship between two truncated moments; (ii) a single function of the random variable.
9.1 Characterizations based on truncated moments
In this subsection we present characterizations of the OGE family in terms of a simple relationship between two truncated moments. Our characterization results presented here will employ an interesting result due to Glänzel (1987) (Theorem 2, below). The advantage of these characterizations is that, the cdf F(x) does not require to have a closed-form and are given in terms of an integral whose integrand depends on the solution of a first order differential equation, which can serve as a bridge between probability and differential equation.
Theorem 2.
Clearly, Theorem 2 can be stated in terms of two functions q _{1} and η by taking q _{2}(x)≡1, which will reduce the condition given in Theorem 2 to E[q _{1}(X)|X≥x]=η(x). However, adding an extra function will give a lot more flexibility, as far as its application is concerned.
Proposition 4.
Proof.
Corollary 1.
Remark 1.
(b) Clearly there are other triplets of functions (q _{2},q _{1},η) satisfying the conditions of Theorem 2. We presented one such triplet in Proposition 4.
9.2 Characterizations based on single function of the random variable
In this subsection we employ a single function ψ of X and state characterization results in terms of ψ(X).
Proposition 5.
Proof.
Proof is straightforward. □
Remark 2. For \(\psi \left (x\right)=\left [1-e^{-\lambda \left (\frac {G\left (x;\mathbf {\xi }\right)}{\overline {G}\left (x;\mathbf {\xi }\right)}\right)}\right ]^{\frac {\alpha \left (1-\delta \right)}{\delta }}, x\in \left (0,\infty \right)\), Proposition 5 will give a cdf F(x) given by (1).
Estimation
Inference can be carried out in three different ways: point estimation, interval estimation and hypothesis testing. Several approaches for parameter point estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimates (MLEs) enjoy desirable properties and can be used when constructing confidence intervals and also in test-statistics. Large sample theory for these estimates delivers simple approximations that work well in finite samples. Statisticians often seek to approximate quantities such as the density of a test-statistic that depend on the sample size in order to obtain better approximate distributions. The resulting approximation for the MLEs in distribution theory is easily handled either analytically or numerically.
where \(V(x_{i};\boldsymbol {\xi })=G(x_{i};\boldsymbol {\xi })/\overline {G}(x_{i};\boldsymbol {\xi })\).
where v ^{(ξ)}(·) means the derivative of the function v with respect to ξ.
Setting these equations to zero and solving them simultaneously yields the MLEs \(\widehat {\Theta }= (\widehat {\alpha }, \widehat {\lambda }, \widehat {\boldsymbol {\xi }}\,)^{\top }\) of Θ=(α,λ,ξ)^{⊤}. These equations cannot be solved analytically, and analytical softwares are required to solve them numerically.
For interval estimation of the parameters, we obtain the 3×3 observed information matrix J(Θ)={U _{ rs }} (for r,s=α,λ,ξ _{ k }), whose elements are listed in Appendix A. Under standard regularity conditions, the multivariate normal \(N_{3}(0, J(\widehat \Theta)^{-1})\) distribution is used to construct approximate confidence intervals for the parameters. Here, \(J(\widehat \Theta)\) is the total observed information matrix evaluated at \(\widehat {\Theta }\). Then, the 100(1−γ)% confidence intervals for α, λ and ξ _{ k } are given by \(\hat {\alpha }\pm z_{\gamma ^{*}/2}\times \sqrt {var (\hat {\alpha })}\), \(\hat {\lambda }\pm z_{\gamma ^{*}/2}\times \sqrt {var (\hat {\lambda })}\) and \(\hat {\boldsymbol {\xi _{k}}}\pm z_{\gamma ^{*}/2}\times \sqrt {var (\hat {\boldsymbol {\xi _{k}}})}\), respectively, where the v a r(·)’s denote the diagonal elements of \(\phantom {\dot {i}\!}J(\widehat {\Theta })^{-1}\) corresponding to the model parameters, and \(z_{\gamma ^{*}/2}\phantom {\dot {i}\!}\) is the quantile (1−γ ^{∗}/2) of the standard normal distribution.
10.1 Simulation study
MLEs and standard deviations for various parameter values
Sample size | Actual values | Estimated values | Standard deviations | ||||||
---|---|---|---|---|---|---|---|---|---|
n | α | β | λ | \(\boldsymbol {\tilde {\alpha }}\) | \(\boldsymbol {\tilde {\beta }}\) | \(\boldsymbol {\tilde {\lambda }}\) | \(\boldsymbol {\tilde {\alpha }}\) | \(\boldsymbol {\tilde {\beta }}\) | \(\boldsymbol {\tilde {\lambda }}\) |
100 | 0.5 | 0.5 | 1 | 0.5264 | 0.5493 | 1.0198 | 0.0205 | 0.0188 | 0.0284 |
0.5 | 1 | 2 | 0.4894 | 1.3254 | 2.1484 | 0.0254 | 0.0709 | 0.0461 | |
0.5 | 1.5 | 1 | 0.5214 | 1.7122 | 1.0097 | 0.0249 | 0.0602 | 0.0284 | |
0.5 | 2 | 2 | 0.4897 | 2.6385 | 2.2290 | 0.0252 | 0.1421 | 0.0619 | |
1 | 0.5 | 1 | 1.0695 | 0.5210 | 1.0321 | 0.0429 | 0.0124 | 0.0244 | |
1 | 1 | 2 | 1.0063 | 1.1701 | 2.0607 | 0.0451 | 0.0470 | 0.0327 | |
1 | 1.5 | 1 | 1.0838 | 1.5895 | 1.0074 | 0.0476 | 0.0403 | 0.0267 | |
1 | 2 | 2 | 1.0753 | 2.2144 | 2.0797 | 0.0538 | 0.0765 | 0.0346 | |
1.5 | 0.5 | 1 | 1.6511 | 0.5250 | 1.0207 | 0.0853 | 0.0111 | 0.0264 | |
1.5 | 1 | 2 | 1.7591 | 1.0921 | 2.0900 | 0.1136 | 0.0355 | 0.0323 | |
1.5 | 1.5 | 1 | 1.6315 | 1.5700 | 1.0269 | 0.0670 | 0.0325 | 0.0247 | |
1.5 | 2 | 2 | 1.7981 | 2.2429 | 2.1356 | 0.1037 | 0.1227 | 0.0437 | |
300 | 0.5 | 0.5 | 1 | 0.4980 | 0.5261 | 0.9972 | 0.0064 | 0.0052 | 0.0086 |
0.5 | 1 | 2 | 0.4996 | 1.0557 | 2.0265 | 0.0070 | 0.0125 | 0.0111 | |
0.5 | 1.5 | 1 | 0.5014 | 1.5597 | 0.9960 | 0.0066 | 0.0154 | 0.0089 | |
0.5 | 2 | 2 | 0.4844 | 2.2030 | 2.0532 | 0.0079 | 0.0299 | 0.0123 | |
1 | 0.5 | 1 | 1.0319 | 0.5014 | 1.0132 | 0.0130 | 0.0036 | 0.0083 | |
1 | 1 | 2 | 0.9856 | 1.0539 | 2.0084 | 0.01450 | 0.01112 | 0.0097 | |
1 | 1.5 | 1 | 1.0277 | 1.5324 | 1.0063 | 0.0127 | 0.0116 | 0.0079 | |
1 | 2 | 2 | 1.0053 | 2.0929 | 2.0217 | 0.01536 | 0.0216 | 0.0097 | |
1.5 | 0.5 | 1 | 1.5457 | 0.5078 | 1.0068 | 0.0182 | 0.0038 | 0.0077 | |
1.5 | 1 | 2 | 1.5414 | 1.0310 | 2.0300 | 0.0250 | 0.0096 | 0.0094 | |
1.5 | 1.5 | 1 | 1.5356 | 1.5182 | 1.0085 | 0.0185 | 0.0098 | 0.0077 | |
1.5 | 2 | 2 | 1.5377 | 2.0670 | 2.0177 | 0.02377 | 0.02022 | 0.0085 |
Applications
In this section, we provide two applications to real data to illustrate the importance of the OGE family by means of the OGE-W, OGE-Fr and OGE-N models presented in Section 3. We consider θ=1, b=1 and μ=0 for the OGE-W, OGE-Fr and OGE-N models, respectively, due to the fact that one scale parameter is enough for fitting these univariate models. The MLEs of the parameters for the these models are calculated and four goodness-of-fit statistics are used to compare the new family with its sub-models.
The first real data set represents the survival times of 121 patients with breast cancer obtained from a large hospital in a period from 1929 to 1938 (Lee 1992). The data examined by Ramos et al. (2013) are: 0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0, 11.8, 12.2, 12.3, 13.5, 14.4, 14.4, 14.8, 15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6, 24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0, 40.0, 40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0, 51.0, 51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0, 69.0, 78.0, 80.0,83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0, 109.0, 111.0, 115.0, 117.0, 125.0, 126.0, 127.0, 129.0, 129.0, 139.0, 154.0.
The second data set consists of 63 observations of the strengths of 1.5 cm glass fibres, originally obtained by workers at the UK National Physical Laboratory. Unfortunately, the units of measurement are not given in the paper. The data are: 0.55, 0.74, 0.77, 0.81, 0.84, 0.93, 1.04, 1.11, 1.13, 1.24, 1.25, 1.27, 1.28, 1.29, 1.30, 1.36, 1.39, 1.42, 1.48, 1.48, 1.49, 1.49, 1.50, 1.50, 1.51, 1.52, 1.53, 1.54, 1.55, 1.55, 1.58, 1.59, 1.60, 1.61, 1.61, 1.61, 1.61, 1.62, 1.62, 1.63, 1.64, 1.66, 1.66, 1.66, 1.67, 1.68, 1.68, 1.69, 1.70, 1.70, 1.73, 1.76, 1.76, 1.77, 1.78, 1.81, 1.82, 1.84, 1.84, 1.89, 2.00, 2.01, 2.24. These data have also been analyzed by Smith and Naylor (1987).
The MLEs are computed using the Limited-Memory Quasi-Newton Code for Bound-Constrained Optimization (L-BFGS-B) and the log-likelihood function evaluated at the MLEs (\(\hat \ell \)). The measures of goodness of fit including the Akaike information criterion (AIC), Anderson-Darling (A ^{∗}), Cramér–von Mises (W ^{∗}) and Kolmogrov-Smirnov (K-S) statistics are computed to compare the fitted models. The statistics A ^{∗} and W ^{∗} are described in details in Chen and Balakrishnan (1995). In general, the smaller the values of these statistics, the better the fit to the data. The required computations are carried out in the R-language.
MLEs and their standard errors (in parentheses) for the first data set
Distribution | λ | α | β | a |
---|---|---|---|---|
OGE-W | 0.7173 | 8.1570 | 0.2208 | - |
(0.2019) | (3.6827) | (0.0276) | - | |
OE-W | 0.1114 | - | 0.3550 | - |
(0.0155) | - | (0.0123) | - | |
OGE-Fr | 0.4920 | 1.9766 | - | 0.6051 |
(0.2704) | (0.7609) | -(0.1314) | ||
OE-Fr | 0.1647 | - | - | 0.8627 |
(0.0303) | - | - | (0.0586) |
MLEs and their standard errors (in parentheses) for the second data set
Distribution | λ | α | β | σ |
---|---|---|---|---|
OGE-W | 0.7173 | 8.1570 | 0.2208 | - |
(0.2019) | (3.6827) | (0.0276) | - | |
OE-W | 0.0721 | - | 1.9603 | - |
(0.0162) | - | (0.0940) | - | |
OGE-N | 0.1010 | 3.1230 | - | 0.9884 |
(0.0798) | (1.5660) | - | (0.1484) | |
OE-N | 0.0121 | - | - | 0.7385 |
(0.0043) | - | - | (0.0364) |
The statistics \(\boldsymbol {\hat \ell }\) , AIC, A ^{ ∗ } , W ^{ ∗ } K-S and K-S p-value for the first data set
Distribution | \(\hat \ell \) | AIC | A ^{ ∗ } | W ^{ ∗ } | K-S | p-value (K-S) |
---|---|---|---|---|---|---|
OGE-W | -410.8347 | 827.6693 | 0.3111 | 0.0472 | 0.0460 | 0.9498 |
OE-W | -431.1625 | 866.3251 | 2.6578 | 0.4510 | 0.1430 | 0.0107 |
OGE-Fr | -414.2511 | 834.5023 | 0.6838 | 0.1011 | 0.0651 | 0.6492 |
OE-Fr | -416.3539 | 836.7078 | 0.8991 | 0.1367 | 0.0939 | 0.2095 |
The statistics \(\boldsymbol {\hat \ell }\) , AIC, A ^{ ∗ } , W ^{ ∗ } , K-S and K-S p-value for the second data set
Distribution | \(\hat \ell \) | AIC | A ^{ ∗ } | W ^{ ∗ } | K-S | p-value (K-S) |
---|---|---|---|---|---|---|
OGE-N | -14.1653 | 34.3306 | 0.8755 | 0.1548 | 0.1286 | 0.2483 |
OE-N | -17.5979 | 39.1958 | 0.9684 | 0.1557 | 0.1232 | 0.2945 |
OGE-W | -14.2733 | 34.5465 | 0.9645 | 0.1717 | 0.1334 | 0.2119 |
OE-W | -16.4613 | 36.9227 | 0.9619 | 0.1614 | 0.1377 | 0.1832 |
Concluding remarks
The generalized continuous distributions have been widely studied in the literature. We propose a new class of distributions called the odd generalized exponential family. We study some structural properties of the new family including an expansion for its density function. We obtain explicit expressions for the moments, generating function, mean deviations, quantile function and order statistics. The maximum likelihood method is employed to estimate the family parameters. We fit three special models of the proposed family to two real data sets to demonstrate the usefulness of the family. We use four goodness-of-fit statistics in order to verify which distribution provides better fit to these data. We conclude that these three special models provide consistently better fits than other competing models. We hope that the proposed family and its generated models will attract wider applications in several areas such as reliability engineering, insurance, hydrology, economics and survival analysis.
Appendix A
Declarations
Acknowledgements
The authors would like to thank the Editor and the two referees for careful reading and for their comments which greatly improved the paper.
Authors’ Affiliations
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