EXTENDED EXPONENTIATED POWER LINDLEY DISTRIBUTION

Ranjbar, V.; Alizadeh, M.; Hademani, G. G. Article EXTENDED EXPONENTIATED POWER LINDLEY DISTRIBUTION Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Ranjbar, V.; Alizadeh, M.; Hademani, G. G. (2018) : EXTENDED EXPONENTIATED POWER LINDLEY DISTRIBUTION, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, NY, Vol. 19, Iss. 4, pp. 621-643, https://doi.org/10.21307/stattrans-2018-033 This Version is available at: https://hdl.handle.net/10419/207916 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ STATISTICS IN TRANSITION new series, December 2018 621 STATISTICS IN TRANSITION new series, December 2018 Vol. 19, No. 4, pp. 621–643, DOI 10.21307/stattrans-2018-033 EXTENDED EXPONENTIATED POWER LINDLEY DISTRIBUTION V. Ranjbar1,M. Alizadeh2,G. G. Hamedani3 ABSTRACT In this study, we introduce a new model called the Extended Exponentiated Power Lindley distribution which extends the Lindley distribution and has increasing, bathtub and upside down shapes for the hazard rate function. It also includes the power Lindley distribution as a special case. Several statistical properties of the distribution are explored, such as the density, hazard rate, survival, quantile functions, and moments. Estimation using the maximum likelihood method and inference on a random sample from this distribution are investigated. A simulation study is performed to compare the performance of the different parameter estimates in terms of bias and mean square error. We apply a real data set to illustrate the applicability of the new model. Empirical findings show that proposed model provides better fits than other well-known extensions of Lindley distributions. Key words:Power Lindley distribution, Structural properties, Failure-time, Maximum likelihood estimation. 1. Introduction The statistical analysis and modeling of lifetime data are essential in almost all applied sciences such as, biomedical science, engineering, nance, and insurance, amongst others. A number of one-parameter continuous distributions for modelling lifetime data has been introduced in statistical literature including exponential, Lindley, gamma, lognormal, and Weibull. The exponential, Lindley and Weibull distributions are more popular than the gamma and lognormal distributions because the survival functions of the gamma and the lognormal distributions cannot be expressed in closed forms and both require numerical integration. The Lindley distribution is a very well-known distribution that has been extensively used over the past decades for modeling data in reliability, biology, insurance, and lifetime analysis. It was introduced by Lindley (1985) to analyze failure time data, especially in applications of modeling stress-strength reliability. The motivation for introducing the Lindley distribution arises from its ability to model failure time data with increasing, decreasing, unimodal and bathtub shaped hazard rates. It may also be mentioned that the Lindley distribution belongs to an exponential family and it can be written as a mixture of an exponential and a gamma distributions. This distribution represents a good alternative to the exponential failure time distributions that suffer from not exhibiting unimodal and bathtub shaped failure rates (Bakouch et al. (2012)). The properties and inferential procedure for the Lindley distribution were studied by 1Golestan University, Gorgan, Iran. E-mail: vahidr[email protected] 2Persian Gulf University, Bushehr, Iran. E-mail: [email protected] 3Marquette University, Milwaukee, USA. E-mail: [email protected] 622 V. Ranjbar, et. al: Extended exponentiated power... Ghitany et al. (2008, 2013). They show via a numerical example that the Lindley distribution gives better modeling than the one based on the exponential distribution when hazard rate is unimodal or bathtub shaped. Furthermore, Mazucheli and Achcar (2011) showed that many of the mathematical properties are more exible than those of the exponential distribution and proposed the Lindley distribution as a possible alternative to exponential or Weibull distributions. The need for extended forms of the Lindley distribution arises in many applied areas. The emergence of such distributions in the statistics literature is quite recent. For some extended forms of the Lindley distribution and their applications, the interested reader is referred to Kumaraswamy Lindley (Cakmakyapan and Ozel, (2014)), beta odd log-logistic Lindley (Cordeiro et al., (2015)), generalized Lindley (Nadarajah et al., (2011)), quasi Lindley distributions (Shanker and Mishra, (2013) ). The probability density function (pdf) and cumulative distribution function (cdf ) of the power Lindley distribution are given respectively by f(x) = λ2 1+λβxβ−1e−λxβ, F(x) = 1−(1+λxβ 1+λ)e−λxβ.(1) It can be seen that this distribution is a mixture of Exponential and gamma distributions. Having only one parameter, the Lindley distribution does not provide enough exibility for analyzing different types of lifetime data. To increase the exibility for modeling purposes it will be useful to consider further alternatives to this distribution. Our purpose here is to provide a generalization that may be useful for more complex situations. Once the proposed distribution is quite exible in terms of pdf and hazard rate function (hrf), it may provide an interesting alternative for describing income distributions and can also be applied in actuarial science, nance, bioscience, telecommunications and modeling lifetime data. Therefore, goal is to introduce a new distribution using the Lindley distribution. Alizadeh et.al (2017), introduced a new class of exponentiated distributions which called Extended Exponentiated distribution (EE-G). The cdf and pdf of this family are given by F(x;α,γ,ξ) = ZG(x;ξ)α 1−G(x;ξ)γ 0 dt (1+t)2dt =G(x;ξ)α G(x;ξ)α+1−G(x;ξ)γ(2) f(x;α,γ,ξ) = g(x;ξ)G(x;ξ)α−1[α+(γ−α)G(x;ξ)γ] [G(x;ξ)α+1−G(x;ξ)γ]2,(3) where α,γ>0are two shape parameters and ξis the vector of parameters for baseline cdf G. For α=γ, it contains exp-G family of distributions. Taking G(x;ξ) as power Lindley distribution with parameters λ,β, we introduce a new extension of STATISTICS IN TRANSITION new series, December 2018 623 Exponentiated power Lindley distribution. The article is outlined as follows: In Section 2, we introduce the EE-PL distribution and provide plots of the density and hazard rate functions. Shapes, quantile function, moments, and moment generating function are also obtained. Moreover, mean deviation, Lorenz and Bonferroni curves, order statistics and finally a simulation study are presented in this section. In section 3, the asymptotic properties and extreme values are obtained. Estimation by the method of maximum likelihood and an explicit expression for the observed information matrix are presented in Section 4. The characterizations of EE-PL distribution are presented in Section 5. The Applications to real data sets are considered in Section 6. Finally, Section 7 offers some concluding remarks. 2. Main properties 2.1. Probability Density and Cumulative Distribution Functions Inserting (1) in (2), the cdf of the EE-PL with four parameters (α,β,γ,λ>0)is defined by F(x;α,β,γ,λ) = h1−(1+λxβ 1+λ)e−λxβiα h1−(1+λxβ 1+λ)e−λxβiα+1−h1−(1+λxβ 1+λ)e−λxβiγ,x≥0(4) The corresponding pdf for x>0is given by f(x;α,β,γ,λ) = λ2βxβ−1(1+xβ)e−λxβ"1−(1+λxβ 1+λ)e−λxβ#α−1 ×nα+(γ−α)h1−(1+λxβ 1+λ)e−λxβiγo (1+λ)nh1−(1+λxβ 1+λ)e−λxβiα+1−h1−(1+λxβ 1+λ)e−λxβiγo2,(5) where λis a scale parameter α,βand γare the shape parameters. Here, αand βgovern the skewness of (5). A random variable Xwith the pdf (5) is denoted by X∼EE −PL(α,β,γ,λ). It is easy to see that: •For β=1, we obtain Extended Generalized Lindley by Ranjbar et al. (2018). •For α=γ, we obtain Exponentiated power Lindley. •For α=γand β=1, we obtain Generalized Lindley. •For α=γ=1, we obtain Power Lindley. •For α=γ=β=1, we obtain Lindley. Some of the possible shapes of the density function (5) for the selected parameter values are illustrated in Figure 1. As seen in Figure 1, the density function can take various forms depending on the parameter values. It is evident that the EE-LP distribution is much more flexible than the power Lindley distribution, i.e. the additional shape parameter allows 624 V. Ranjbar, et. al: Extended exponentiated power... 02468 0.0 0.1 0.2 0.3 0.4 0.5 x density λ=1.0, α=1.3, γ=2.0, β=1.0 λ=1.0, α=1.3, γ=2.0, β=1.0 λ=1.0, α=1.3, γ=2.0, β=1.0 λ=1.0, α=1.3, γ=2.0, β=1.0 λ=1.0, α=1.3, γ=2.0, β=1.0 02468 0.0 0.2 0.4 0.6 x density λ=1.0, α=2.0, γ=1.5, β=1.3 λ=0.7, α=2.0, γ=1.5, β=1.3 λ=0.5, α=2.0, γ=1.5, β=1.3 λ=0.3, α=2.0, γ=1.5, β=1.3 λ=0.1, α=2.0, γ=1.5, β=1.3 02468 0.0 0.2 0.4 0.6 x density λ=1.0, α=2.0, γ=1.5, β=1.3 λ=1.0, α=1.7, γ=1.5, β=1.3 λ=1.0, α=1.5, γ=1.5, β=1.3 λ=1.0, α=1.3, γ=1.5, β=1.3 λ=1.0, α=1.0, γ=1.5, β=1.3 02468 0.0 0.2 0.4 0.6 x density λ=1.0, α=0.3, γ=2.0, β=1.3 λ=1.0, α=0.3, γ=1.5, β=1.3 λ=1.0, α=0.3, γ=1.0, β=1.3 λ=1.0, α=0.3, γ=0.7, β=1.3 λ=1.0, α=0.3, γ=0.5, β=1.3 Figure 1: Plots of Pdf of the EE-PL model for selected λ,α,γand β. for a high degree of flexibility of the EE-PL distribution. Both unimodal and monotonically decreasing and increasing shapes appear to be possible. 2.2. Survival and Hazard Rate Functions Central role is played in the reliability theory by the quotient of the pdf and survival function. We obtain the survival function corresponding to (4) as S(x;λ,α,γ,β) = 1−h1−(1+λxβ 1+λ)e−λxβiγ h1−(1+λxβ 1+λ)e−λxβiα+1−h1−(1+λxβ 1+λ)e−λxβiγ.(6) In reliability studies, the hrf is an important characteristic and fundamental to the design of safe systems in a wide variety of applications. Therefore, we discuss these properties of STATISTICS IN TRANSITION new series, December 2018 625 01234 0.0 1.0 2.0 3.0 x hazard λ=1, α=1.3, γ=2, β=1 λ=1, α=1.3, γ=2, β=1 λ=1, α=1.3, γ=2, β=1 λ=1, α=1.3, γ=2, β=1 λ=1, α=1.3, γ=2, β=1 02468 01234567 x hazard λ=1.0, α=2, γ=1.5, β=1.3 λ=0.7, α=2, γ=1.5, β=1.3 λ=0.5, α=2, γ=1.5, β=1.3 λ=0.3, α=2, γ=1.5, β=1.3 λ=0.1, α=2, γ=1.5, β=1.3 01234 0.0 1.0 2.0 3.0 x hazard λ=1, α=2.0, γ=1.5, β=1.3 λ=1, α=1.7, γ=1.5, β=1.3 λ=1, α=1.5, γ=1.5, β=1.3 λ=1, α=1.3, γ=1.5, β=1.3 λ=1, α=1.0, γ=1.5, β=1.3 012345 0.0 1.0 2.0 3.0 x hazard λ=1, α=0.3, γ=2.0, β=1.3 λ=1, α=0.3, γ=1.5, β=1.3 λ=1, α=0.3, γ=1.0, β=1.3 λ=1, α=0.3, γ=0.7, β=1.3 λ=1, α=0.3, γ=0.5, β=1.3 Figure 2: Hazard rate functions of the EE-PL model for selected λ,α,γand β. the EE-LP distribution. The hrf of Xtakes the form h(x;λ,α,γ,β) = λ2βxβ−1(1+xβ)e−λxβ"1−(1+λxβ 1+λ)e−λxβ#α−1 ×(α+(γ−α)"1−(1+λxβ 1+λ)e−λxβ#γ)/ "( 1−(1+λxβ 1+λ)e−λxβ!α +1− 1−(1+λxβ 1+λ)e−λxβ!γ) ×(1−"1−(1+λxβ 1+λ)e−λxβ#γ)#,x>0.(7) Plots of the hrf of the EE-PL distribution for several parameter values are displayed in Figure 2. Figure 2 shows that the hrf of the EE-PL distribution can have very flexible shapes, such as increasing, decreasing, bathtub followed by upside down bathtub, and bathtub shapes for the selected values of the model parameters. This attractive flexibility makes the hrf of the EE-PL distribution useful and suitable for non-monotone empirical hazard behaviors which are more likely to be encountered or observed in real life situations. 2.3. Mixture representations for the pdf and cdf In this subsection, we provide alternative mixture representations for the pdf and cdf of X. Some useful expansions for (4) can be derived by using the concept of power series. We 626 V. Ranjbar, et. al: Extended exponentiated power... have [1−(1+λ 1+λxβ)e−λxβ]α= ∞ ∑ i=1 (−1)iα i[1+λ 1+λxβ)e−λxβ]i = ∞ ∑ i=1 i ∑ k=0 (−1)i+kα ii k[1−(1+λ 1+λxβ)e−λxβ]k = ∞ ∑ k=0 ∞ ∑ i=k (−1)i+kα ii k[1−(1+λ 1+λxβ)e−λxβ]k = ∞ ∑ k=0 ak[1−(1+λ 1+λxβ)e−λxβ]k, where ak=ak(α) = ∑∞ i=k(−1)i+kα ii k. Also [1−(1+λ 1+λxβ)e−λxβ]α+1−[1−(1+λ 1+λxβ)e−λxβ]γ= ∞ ∑ k=0 bk[1−(1+λ 1+λxβ)e−λxβ]k, where b0=a0(α) + 1−a0(γ)and bk=ak(α)−ak(γ)for k≥1. Then using the ratio of two power series, we can write F(x) = ∑∞ k=0ak[1−(1+λ 1+λxβ)e−λxβ]k ∑∞ k=0bk[1−(1+λ 1+λxβ)e−λxβ]k = ∞ ∑ k=0 ck[1−(1+λ 1+λxβ)e−λxβ]k,(8) where c0=a0 b0and for k≥1, ck=1 b0 [ak−1 b0 k ∑ r=1 brck−r].(9) Equation (8)shows that we can write the cdf of EE-PL as a Linear combination of generalized lindly distribution. Then we can write f(x) = ∞ ∑ k=0 ck+1 (k+1)λ2βxβ−1(1+xβ) 1+λe−λxβ[1−(1+λ 1+λxβ)e−λxβ]k. 2.4. Moments and Moment Generating Function Some of the most important features and characteristics of a distribution can be studied through moments (e.g. tendency, dispersion, skewness and kurtosis). Now we obtain ordinary moments and the moment generating function (mgf) of the EE-PL distribution. We define and compute A(a1,a2,a3,a4;λ,β) = Z∞ 0 xa1(1+xβ)a2e−a3xβ1−(1+λ 1+λxβ)e−λxβa4 dx.(10) STATISTICS IN TRANSITION new series, December 2018 627 Using generalized binomial expansion, one can obtain A(a1,a2,a3,a4;λ,β) = ∞ ∑ l,r=0 l ∑ k=0 (−1)la4 ll ka2 r(λ 1+λ)l× Γ(a1+1 β+k+r) β(λl+a3) a1+1 β+k+r. (11) Next, the nth moment of the EE-PL distribution is given by E[Xn] = λ2β 1+λ ∞ ∑ k=0 kckA(n+β−1,1,λ,k;λ,β).(12) For integer values of k, let µ0 k=E(Xk)and µ=µ0 1=E(X), then one can also find the kth central moment of the EE-PL distribution through the following well-known equation µk=E(X−µ)k= k ∑ r=0k rµ0 r(−µ)k−r.(13) The moment generating function of a random variable provides the basis of an alternative route to analytical results compared with working directly with its pdf and cdf. Using (12) and (13), we obtain MX(t) = EhetXi=λ2 1+λ ∞ ∑ k=0 (k+1)ck+1A(k+1,λ,0,λ−t). Using (13), the variance, skewness and kurtosis measures can be obtained. Skewness measures the degree of the long tail and kurtosis is a measure of the degree of tail heaviness. For the EE-PL distribution, The skewness can be computed as S=µ3 µ3/2 2 =µ0 3−3µ0 2µ0 1+2µ03 1 (µ0 2−µ02 1)3/2 and the kurtosis is based on octiles as K=µ4 µ2 2 =µ0 4−4µ0 1µ0 3+6µ02 1µ0 2−3µ04 1 µ0 2−µ02 1 . When the distribution is symmetric S=0, and when the distribution is right (or left) skewed S>0(or S <0). As Kincreases, the tail of the distribution becomes heavier. These measures are less sensitive to outliers and they exist even for distributions without moments. We present first four ordinary moments, skewness and kurtosis of the EE-PL distribution for various values of the parameters in Table 1. Plots for skewness and kurtosis, when λ=2, are presented in Figure 3. Next, we define and compute B(a1,a2,a3,a4;y,λ,β) = Zy 0 xa1(1+xβ)a2e−a3xβ1−(1+λ 1+λxβ)e−λxβa4 dx.(14) 628 V. Ranjbar, et. al: Extended exponentiated power... Table 1: Moments, skewness, and kurtosis of the EE-PL dist. for the some parameter values. λ α β γ µ0 1µ0 2µ0 3µ0 4Skewness Kurtosis 2.0 0.5 0.5 0.5 0.457 1.788 16.97 289.197 7.4082 164.75 2.0 0.5 0.5 1.0 0.413 0.457 0.783 1.7888 2.3382 3.052 2.0 0.5 0.5 3.0 0.608 0.468 0.413 0.4004 0.2704 0.243 2.0 0.5 1.0 0.5 0.752 3.384 33.41 575.819 5.6285 172.40 2.0 0.5 1.0 1.0 0.571 0.752 1.413 3.3844 1.7824 3.070 2.0 0.5 1.0 3.0 0.688 0.592 0.571 0.5947 0.0234 0.261 2.0 0.5 2.0 0.5 1.176 6.223 65.03 1142.245 4.3521 182.28 2.0 0.5 2.0 1.0 0.750 1.176 2.445 6.2232 1.3397 3.107 2.0 0.5 2.0 3.0 0.757 0.716 0.750 0.8367 -0.1566 0.288 2.0 2.0 0.5 0.5 0.655 2.028 17.57 291.944 7.0008 156.70 2.0 2.0 0.5 1.0 0.646 0.655 0.981 2.0282 2.1655 2.572 2.0 2.0 0.5 3.0 0.818 0.708 0.646 0.619 0.4272 0.134 2.0 2.0 1.0 0.5 0.985 3.744 34.46 580.961 5.4788 167.33 2.0 2.0 1.0 1.0 0.806 0.985 1.678 3.7447 1.7687 2.701 2.0 2.0 1.0 3.0 0.882 0.822 0.806 0.8272 0.2475 0.140 2.0 2.0 2.0 0.5 1.437 6.729 66.76 1151.513 4.3391 179.76 2.0 2.0 2.0 1.0 0.986 1.437 2.781 6.7291 1.4076 2.814 2.0 2.0 2.0 3.0 0.943 0.941 0.986 1.0779 0.0508 0.150 From the generalized binomial expansion, we have B(a1,a2,a3,a4;a,λ,β) = ∞ ∑ l,r=0 l ∑ k=0 (−1)la4 ll ka2 r(λ 1+λ)l×γ(a1+1 β+k+r,y 1 β λl+a3) β(λl+a3) a1+1 β+k+r, (15) where γ(λ,z) = Rz 0tλ−1e−tdt denotes the incomplete gamma function. Now, the nth incomplete moment of the EE-PL distribution is found to be mn(y) = E[Xn|X<y] = λ2β 1+λ ∞ ∑ k=0 (k+1)ck+1B(n+β−1,1,λ,k,y;λ,β).(16) 2.5. Mean Deviations, Lorenz and Bonferroni Curves Mean deviation about the mean and mean deviation about the median as well as Lorenz and Bonferroni curves for the EE-PL distribution are presented in this section. Bonferroni and Lorenz curves are widely used tool for analyzing and visualizing income inequality. Lorenz curve, L(p) can be regarded as the proportion of total income volume accumulated by those units with income lower than or equal to the volume y, and Bonferroni curve, B(p) is the scaled conditional mean curve, that is, ratio of group mean income of the population. STATISTICS IN TRANSITION new series, December 2018 635 η(x)q1(x)−q2(x) = q1(x) 2(1−"1− 1+λxβ 1+λ!e−λxβ#α)>0f or x >0 Conversely, if ηis given as above, then s0(x) = η0(x)q1(x) η(x)q1(x)−q2(x)= αβλ2xβ−11+xβe−λxβh1−1+λxβ 1+λe−λxβiα−1 1−h1−1+λxβ 1+λe−λxβiα,x>0, and hence s(x) = −λlog(1−"1− 1+λxβ 1+λ!e−λxβ#α),x>0. Now, according to Theorem 1, Xhas density (2.2). Let X:Ω→(0,∞)be a continuous random variable and let q1be as in Proposition (5.1). Then, Xhas pd f (2.2)if and only if there exist functions q2and ηdefined in Theorem 1 satisfying the differential equation η0(x)q1(x) η(x)q1(x)−q(x)= αβλ2xβ−11+xβe−λxβh1−1+λxβ 1+λe−λxβiα−1 1−h1−1+λxβ 1+λe−λxβiα,x>0. The general solution of the differential equation in Corollary (5.1) is η(x) = (1−"1− 1+λxβ 1+λ!e−λxβ#α)−1 ×  −Zαβλ2xβ−11+xβe−λxβ"1− 1+λxβ 1+λ!e−λxβ#α−1 (q1(x))−1q2(x)dx +D , where Dis a constant. Note that a set of functions satisfying the above differential equation is given in Proposition (5.1) with D=1 2. For α=γ=1,q1(x)≡1and q2(x) = e−λxβ,we have η(x) = 1 2e−λxβ,x>0,s0(x) = λβxβ−1, x>0and η(x) = eλxβ−Zλβxβ−11+xβe−λxβq2(x)dx +D. 636 V. Ranjbar, et. al: Extended exponentiated power... 5.2. Characterization in terms of the hazard function It is known that the hazard function, hF, of a twice differentiable distribution function, F, satisfies the first order differential equation f0(x) f(x)=h0 F(x) hF(x)−hF(x). 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 EE-PL,for α=γ=1, in terms of the hazard function which is not of the above trivial form. Let X:Ω→(0,∞)be a continuous random variable. Then, Xhas pdf (5), for α=γ=1, if and only if its hazard function hF(x)satisfies the differential equation h0 F(x)−(β−1)x−1hF(x) = λ2β2x2(β−1)1+λ+λxβ−2,x>0, with the initial condition hF(0) = 0for β>1. Proof. If Xhas pd f (2.2), for α=γ=1, then clearly the above differential equation holds. Now, if the differential equation holds, then d dx nx−(β−1)hF(x)o=λ2β2(1+xβ 1+λ+λxβ) or hF(x) = λ2βxβ−11+xβ 1+λ+λxβx>0, which is the hazard function of (2.2). 5.3. Characterization in terms of the reverse hazard function The reverse hazard function, rF, of a twice differentiable distribution function, F, is defined as rF(x) = f(x) F(x),x∈support o f F. In this subsection we present characterization of EE-PL distribution in terms of the reverse hazard function. Let X:Ω→(0,∞)be a continuous random variable. Then, Xhas pd f (2.2)if and only if its reverse hazard function rF(x)satisfies the differential equation STATISTICS IN TRANSITION new series, December 2018 637 r0 F(x)+λβxβ−1rF(x) = λ2βe−λxβ 1+λ d dx      xβ−11+xβnα+(γ−α)h1−1+λxβ 1+λe−λxβiγo h1−1+λxβ 1+λe−λxβin1+h1−1+λxβ 1+λe−λxβiα−h1−1+λxβ 1+λe−λxβiγo    , x>0. 5.4. Characterization based on the conditional expectation of certain function of the random variable In this subsection we employ a single function ψof Xand characterize the distribution of X,for α=γ=1, in terms of the conditional expectation of ψ.The following proposition has already appeared in Hamedani’s previous work (2013), so we will just state it here which can be used to characterize EE-PL distribution. Let X:Ω→(a,b)be a continuous random variable with cd f F . Let ψ(x)be a differentiable function on (a,b)with limx→a+ψ(x) = 1. Then for δ6=1, E[ψ(X)|X>x] = δψ (x),x∈(a,b), if and only if ψ(x) = (1−F(x)) 1 δ−1,x∈(a,b). For α=γ=1,(a,b) = (0,∞),ψ(x) = 1+λxβ 1+λe−λxβand δ=λ 1+λ, Proposition 5.4 provides a characterization of the EE-PL distribution. 6. Application In this section, we illustrate the fitting performance of the EE-PL distribution using a real data set. For the purpose of comparison, we fitted the following models to show the fitting performance of EE-PL distribution by means of real data set: i) Lindley Distribution, L(λ). ii) Power Lindley distribution, PL(β,λ). iii) Generalized Lindley, GL(α,λ), (Nadarajah et al. (2011)), with distribution function given by F(x) = 1−(1+λx 1+λ)e−λxα . iv) Beta Lindley, BL(α,β,λ), with distribution function given by F(x) = ZL(x,λ) 0 tα−1(1−t)β−1dt. 638 V. Ranjbar, et. al: Extended exponentiated power... v) Exponentiated power Lindley distribution, EPL(α,β,λ), with distribution function given by F(x) = 1−(1+λxβ 1+λ)e−λxβ!α . vi) Odd log-logistic power Lindley distribution OLL−PL(α,β,λ), (Alizadeh et al. (2017)), with distribution function given by F(x) = PL(x,β,λ)α PL(x,β,λ)α+(1−PL(x,β,λ))α. vii) Kumaraswamy Power Lindley, KPL(α,β,γ,λ)(Broderick et al. (2012)) F(x) = 1−[1−PL(x,β,λ)α]γ. viii) Odd Burr-Power Lindley, OBu −PL(α,β,γ,λ)(Altun et al.(2017a)) F(x) = 1−1−PL(x,β,λ)α PL(x,β,λ)α+(1−PL(x,β,λ))α. ix) Extended Exponential Lindley, EE −L(α,γ,λ), Ranjbar, et al. (accepted (2018)), F(x) = L(x,λ)α L(x,λ)α+1−(1−L(x,λ))γ. Estimates of the parameters of EE-PL distribution, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Cramer Von Mises and Anderson-Darling statistics (W∗and A∗) are presented for each dataset. We have also considered the Kolmogorov- Smirnov (K-S) statistic and its corresponding p-value and the minimum value of the minus log-likelihood function (-Log(L)) for the sake of comparison. Generally speaking, the smaller values of AIC,BIC,W∗and A∗, the better fit to a data set. All the computations were carried out using the software R. Note that initial values of model parameters are quite important to obtain the correct MLEs of parameters. To avoid local minima problem, we first obtain the parameter estimate of the Lindley distribution. Then, the estimated parameter of the Lindley distribution is used as the initial value of the parameter of the PL and GL distributions. Then, the estimated parameters of PL distribution, λand β, is used as the initial values of the EE-PL distribution. This approach is quite useful to obtain correct parameter estimates of extended models. The data are the exceedances of flood peaks (in m3/s) of the Wheaton River near Carcross in Yukon Territory, Canada. The data consist of 72 exceedances for the years 1958- 1984 rounded to one decimal place. These data were analyzed by Akinsete et al. (2008). The ML estimates of the parameters and the goodness-of-fit test statistics for the real data set is presented in Table 3 and 4 respectively. As we can see, the smallest values of AIC,BIC,A∗,W∗and −lstatistics and the largest p-values belong to the EE-PL distribution. Therefore the EE-PL distribution outperforms the other competitive considered distribution in the sense of this criteria. The OLL-L distribution provides the second best fit and this data set. STATISTICS IN TRANSITION new series, December 2018 639 Table 2: The data set. 1.7 2.2 14.4 1.1 0.4 20.6 5.3 0.7 1.9 13.0 12.0 9.3 1.4 18.7 8.5 25.5 11.6 14.1 22.1 1.1 2.5 14.4 1.7 37.6 0.6 2.2 39.0 0.3 15.0 11.0 7.3 22.9 1.7 0.1 1.1 0.6 9.0 1.7 7.0 20.1 0.4 2.8 14.1 9.9 10.4 10.7 30.0 3.6 5.6 30.8 13.3 4.2 25.5 3.4 11.9 21.5 27.6 36.4 2.7 64.0 1.5 2.5 27.4 1.0 27.1 20.2 16.8 5.3 9.7 27.5 2.5 27.0 Table 3: Parameter ML estimates and their standard errors (in parentheses) for the data set. Model α β γ λ Lindley(λ)– – – 0.153 (0.013) GL(α,λ)0.508(0.076) – – 0.104 (0.0149) PL(β,λ)– 0.700 (0.057) – 0.338 (0.055) BL(α,β,λ)0.555(0.098) 0.274 (0.239) – 0.333 (0.272) EPL(α,β,λ)0.730(0.235) 0.915 (0.595) – 0.300 (0.279) OLLPL(α,β,λ)0.557(0.178) 1.073 (0.244) – 0.154 (0.091) KPL(α,β,γ,λ)1.675(2.433) 0.453 (0.432) 7.563 (11.736) 0.279 (0.522) OBu(α,β,γ,λ)24.91(25.654) 0.024 (0.032) 41.25 (22.520) 0.984 (0.149) EEL(α,γ,λ)0.618(0.101) – 2.770 (1.704) 0.169 (0.028) EEPL(α,β,γ,λ)4.521(3.067) 0.472 (0.094) 55.07 (58.193) 1.551 (0.643) Table 4: Goodness-of-fit test statistics for the data set. Model AIC BIC p −value W∗A∗−l Lindley(λ)530.423 532.700 0.001 0.139 0.852 264.211 GL(α,λ)509.349 513.902 0.276 0.132 0.822 252.674 PL(β,λ)508.443 512.996 0.405 0.123 0.766 252.103 BL(α,β,λ)510.206 517.036 0.297 0.150 0.866 252.221 EPL(α,β,λ)510.425 517.255 0.395 0.147 0.854 252.212 OLLPL(α,β,λ)507.937 514.767 0.471 0.093 0.592 250.968 KPL(α,β,γ,λ)512.221 521.328 0.371 0.152 0.866 252.110 OBu(α,β,γ,λ)511.212 520.319 0.401 0.140 0.799 251.606 EEL(α,γ,λ)508.931 515.761 0.174 0.101 0.662 251.465 EEPL(α,β,γ,λ)500.594 509.701 0.994 0.026 0.180 246.297 In addition, the profile log-likelihood functions of the EE-PL distribution are plotted in Figure 4. These plots reveal that the likelihood equations of the EE-PL distribution have solutions that are maximizers. Here, we also applied likelihood ratio (LR) tests. The LR tests can be used for comparing the EE-PL distribution with its sub-models. For example, the test of H0:β=1against H1: β=1is equivalent to comparing the EE-PL and EE-L distributions with each other. For this 640 V. Ranjbar, et. al: Extended exponentiated power... 01234 −700 −600 −500 −400 −300 lambda Profile log−likelihood 3.0 4.0 5.0 6.0 −249.5 −249.0 −248.5 −248.0 −247.5 −247.0 −246.5 alpha Profile log−likelihood 0.0 0.2 0.4 0.6 0.8 1.0 −700 −600 −500 −400 −300 beta Profile log−likelihood 48 50 52 54 56 58 −246.40 −246.38 −246.36 −246.34 −246.32 gamma Profile log−likelihood Figure 4: The profile log-likelihood functions of the EE-PL distribution. test, the LR statistic can be calculated by the following relation LR =hl(b α,b β,b γ,bλ)−l(c α∗,1,b γ∗,c λ∗)i, where c α∗,b γ∗and c λ∗are the ML estimators of α,γand λ, respectively, obtained under H0. Under the regularity conditions and if H0is assumed to be true, the LR test statistic converges in distribution to a chi square with rdegrees of freedom, where requals the difference between the number of parameters estimated under H0and the number of parameters estimated in general, (for H0:β=1, we have r=1). Table 5 gives the LR statistics and the corresponding p-values. Table 5: The LR test results. Hypotheses LR p-value EE-PL versus Lindley H0:α=β=γ=135.828 <0.0001 EE-PL versus PL H0:α=γ=111.612 0.0030 EE-PL versus GL H0:α=γ,β=112.754 0.0017 EE-PL versus EPL H0:α=γ11.830 0.0005 EE-PL versus EE-L H0:β=110.336 0.0013 From Table 5, we observe that the computed p-values are too small so we reject all the null hypotheses and conclude that the EE-PL fits the first data better than the considered sub-models according to the LR criterion. We also plotted the fitted pdfs and cdfs of the considered models for the sake of visual comparison, in Figure 4. Figure 4 suggests that the EE-PL fits the skewed data very well. 7. Conclusion In this paper, a new distribution called Extended Exponentiated Power Lindley (EE-PL) distribution is introduced. The statistical properties of the EE-PL distribution including the hazard STATISTICS IN TRANSITION new series, December 2018 641 density 0 10203040506070 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Lindley PL GL EPL EE−P L density 0 10203040506070 0.00 0.01 0.02 0.03 0.04 0.05 0.06 BL KPL OBu OLL−P L EE−PL 0204060 0.0 0.2 0.4 0.6 0.8 1.0 x Fn(x) Lindley PL GL EPL EE−P L 0204060 0.0 0.2 0.4 0.6 0.8 1.0 x Fn(x) BL KPL OBu OLL−P L EE−PL Figure 5: Fitted densities and distribution functions for the data set. and reverse hazard functions, quantile function, moments, incomplete moments, generating functions, mean deviations, Bonferroni and Lorenz curves, order statistics and maximum likelihood estimation for the model parameters are given. 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