Exponential modified discrete Lindley distribution

Transcription

1 Yilmaz et al. SpringerPlus 6 5:66 DOI.86/s RESEARCH Open Access Eponential modified discrete Lindley distribution Mehmet Yilmaz *, Monireh Hameldarbandi and Sibel Aci Kemaloglu *Correspondence: yilmazm@ science.anara.edu.tr Department of Statistics, Faculty of Science, Anara University, 6 Tandogan, Anara, Turey Full list of author information is available at the end of the article Abstract In this study, we have considered a series system composed of stochastically independent M-component where M is a random variable having the zero truncated modified discrete Lindley distribution. This distribution is newly introduced by transforming on original parameter. The properties of the distribution of the lifetime of above system have been eamined under the given circumstances and also parameters of this new lifetime distribution are estimated by using moments, maimum lielihood and EM-algorithm. Keywords: Modified discrete Lindley distribution, Eponential- modified, Discrete Lindley distribution, Method of moments, Maimum lielihood estimation, EM-algorithm Mathematics Subject Classification: 46N3, 6F, 6E5 Bacground Under the name of the new lifetime distribution, about 4 studies have been done in the recent 5 years. In particular, the compound distributions obtained by eponential distribution are applicable in the fields such as electronics, geology, medicine, biology and actuarial. Some of these wors can be summarized as follows: Adamidis and Louas 998 and Adamidis et al. 5 introduced a two-parameter lifetime distribution with decreasing failure rate by compounding eponential and geometric distribution. In the same way, eponential-poisson EP and eponential-logarithmic EL distributions were given by Kus 7 and Tahmasbi and Rezaei 8, respectively. Chahandi and Ganjali 9 introduced eponential-power series distributions EPS. Barreto-Souza and Baouch 3 introduced a new three-parameter distribution by compounding eponential and Poisson Lindley distributions, named the eponential Poisson Lindley EPL distribution. Eponential-Negative Binomial distribution is introduced by Hajebi et al. 3. Furthermore, Gui et al. 4 have considered the Lindley distribution which can be described as a miture of the eponential and gamma distributions. This idea has helped them to propose a new distribution named as Lindley Poisson by compounding the Lindley and Poisson distributions. Because most of those distributions have decreasing failure rate. They have important place in reliability theory. Lots of those lifetime data can be modelled by compound 6 The Authors. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a lin to the Creative Commons license, and indicate if changes were made.

2 Yilmaz et al. SpringerPlus 6 5:66 Page of distributions. Although these compound distributions are quite comple, new distributions can fit better than the nown distributions for modelling lifetime data. Probability mass function of the discrete Lindley distribution obtained by discretizing the continuous survival function of the Lindley distribution Gómez-Déniz and Calderín-Ojeda ; Eq. 3, Baouch et al. 4; Eq. 3. This discrete distribution provided by authors above, is quite a comple structure in terms of parameter. In order to overcome problems in estimation process of the parameter of Lindley distribution, we propose a modified discrete Lindley distribution. Thus, estimation process of the parameters using especially the EM algorithm was facilitated. Afterwards, we propose a new lifetime distribution with decreasing hazard rate by compounding eponential and modified-zerotruncated discrete Lindley distributions. This paper is organized as follows: In Construction of the model section, we propose the two-parameter eponential-modified discrete Lindley EMDL distribution, by miing eponential and zero truncated modified discrete Lindley distribution, which ehibits the decreasing failure rate DFR property. In Properties of EMDL distribution section, we obtain moment generating function, quantile, failure rate, survival and mean residual lifetime functions of the EMDL. In Inference section, the estimation of parameters is studied by some methods such as moments, maimum lielihood and EM algorithm. Furthermore, information matri and observed information matri are also discussed in this section. The end of this section includes a detailed simulation study to see the performance of Moments with lower and upper bound approimations, ML and EM estimates. Illustrative eamples based on three real data sets are provided in Applications section. Construction of the model In this section, we first give the definition of the discrete Lindley distribution introduced by Gómez-Déniz and Calderín-Ojeda and Baouch et al. 4. We have achieved a more simplified discrete distribution than discrete Lindley distribution by taing instead of e in subsequent definition. Thus, we introduce a new lifetime distribution by compounding Eponential and Modified Discrete Lindley distributions, named the Eponential-Modified Discrete Lindley EMDL distribution. Discrete Lindley distribution A discrete random variable M is said to have Lindley distribution with the parameter >, if its probability mass function p.m.f is given by PM = m = e m e + e + m, m =,,,... + The cumulative distribution function of M will be given by PM m = + + m e m+, m =,,,... + Modified discrete Lindley distribution If is limited to the range,, then we replace ep by using the first degree Taylor epansion of ep in. The new discrete distribution is specified by the following probability mass function:

3 Yilmaz et al. SpringerPlus 6 5:66 Page 3 of PM = m = + m m +, for << and m =,,,... We call this distribution Modified Discrete Lindley MDL. Theorem MDL distribution can be represented as a miture of geometric and negative binomial distributions with miing proportion is, and a common success rate. + Proof If p.m.f in is rewritten as the following form PM = m = [ m ] + [ m + m] = w f m+w f m, + + then f indicates p.m.f of a geometric random variable with success probability and f indicates p.m.f of a negative binomial random variable which denotes the number of trials until the second success, with common success probability. w = + and w = + denote component probabilities; in other words these are called the miture weights Fig.. Note that MDL distribution has an increasing hazard rate while a geometric distribution has a constant hazard rate. So, MDL distribution is more useful than geometric distribution for modelling the number of rare events. When the is closed to zero, then MDL can occure different shapes than the p.m.f of a Geometric distribution. This situation made the distribution thinner right tail than a distribution which is compounded with eponential distribution. Thus, this proposed compound distribution can be usefull for modelling lifetime data such as time interval between successive earthquaes, time period of bacteria spreading, recovery period of the certain disease. Eponential modified discrete Lindley distribution Suppose that M is a zero truncated MDL random variable with probablity mass function πm = PM = m M > = + m m + and X, X,..., X M are i.i.d. with probability density function h; β = βe β, >. Let X = minx, X,..., X M, then g m; β = mβe mβ and g, m = g mπm = β + mm + m e mβ. Thus, we can obtain the marginal probability density function of X as f ;, β = + βe β 3 e β, > e β 3 3 where, and β >. Henceforth, the distribution of the random variable X having the p.d.f in 3 is called shortly EMDL. By changing of variables r = e β in cumulative integration of 3, the distribution function can be found as follows: [ F;, β = + e β 3 e β e β ].

4 Yilmaz et al. SpringerPlus 6 5:66 Page 4 of Geo NegBin, MDL =. = m =.5 = m m Fig. P.m.f of geometric, negative binomial and modified discrete Lindley 5 5 m Following figure shows different shapes of p.d.f of EMDL random variable for various values of and β Fig.. Properties of EMDL distribution In this section the important characteristics and features in mathematical statistics and realibility which are moment generating function and moments, quantiles, survival, hazard rate and mean residual life functions of the EMDL distribution are introduced. We will also give a relationship with Loma and Eponential-Poisson distributions. Moment generating function and moments Moment generating function of X is given by Mt = E e t = + j j + j β βj t j= for t <β. Hence a closed form of.th raw moment of X is epressed by E X Ŵ + = β + j + j j, j= for =,,... Here for > raw moments can be calculated numerically for given values of since infinite series above can be represented by polylog functions.

5 Yilmaz et al. SpringerPlus 6 5:66 Page 5 of.5.4 =.3, β=. =.5, β=. =.7, β=..5 =.3, β=.4 =.5, β=.4 =.7, β=.4 f.3 f..5 f =.3, β=.7 3 =.5, β=.7 =.7, β= f =.3, β= 4 =.5, β= =.7, β= Fig. P.d.f of EMDL random variable for different parameter values.5.5 First and second raw moments are evaluated respectively as [ ] EX =, β + ln [ E X = β ln + + ] = 4 5 Quantile function Quantile function of X is obtained simply by inverting F;, β = q as follows [ ] q+3a 9A log log +4 qa q+4a q = β where < q < and A =.5 = log log +. In particular, the first quartile of X is [ 3 ] +3A β 3 +4A 9A +3A, the median of X is [ log log.5 = +3A β 9A +A +4A ],

6 Yilmaz et al. SpringerPlus 6 5:66 Page 6 of and the third quartile of X is.75 = log log [ +3A β +4A 9A +A ]. Survival, hazard rate and mean residual life functions The survival function of X is given by Fig. 3 S = + e β 3 e β. e β 6 From 3 and 6 it is easy to verify that the hazard rate function of X is h = f [ ] S = β 3 r r3 r = β r r + r3 r [ ] = β r 3 3 r 7 with h = β+ + β and lim h = β where r = e β. As it can be seen immediately from last two statements on the right side of 7, h is a monotonically decreasing function and bounded from below with β see Fig =., β=. =.5, β=. =.9, β= =., β=.4 =.5, β=.4 =.9, β=.4 S.5. S =., β=.7 =.5, β=.7 =.9, β=.7.4. =., β= =.5, β= =.9, β= S.6.4 S Fig. 3 Survival function of EMDL random variable for selected parameter values 4 6 8

7 Yilmaz et al. SpringerPlus 6 5:66 Page 7 of.5.4 =.3, β=. =.5, β=. =.7, β=..5 =.3, β=.4 =.5, β=.4 =.7, β=.4 h.3 h =.3, β=.7 =.5, β=.7 =.7, β= =.3, β= =.5, β= =.7, β= h h Fig. 4 Hazard rate function of EMDL random variable for selected parameter values The mean residual life function of X is given by mrl = EX X > = β r r r ln r 3r r where r = e β. Note that mrl β holds for >. We can see this result immediately below by letting ln r = r z dz. Then applying the mean value theorem, we have the upper bound for ln r as r r. If this upper bound is written above, then mrl β 3 r 3 r β. We have the following graphs of mrl for different values of parameter and β Fig. 5. Relationship of the other distribution Let consider the following transformation of X Y = eβx. Then the probability density function of Y can be obtained as 3 f Y y = y + y + 3

8 Yilmaz et al. SpringerPlus 6 5:66 Page 8 of.5 8 mrl 6 4 =., β=. =.5, β=. =.9, β= mrl.5 =., β=.4.5 =.5, β=.4 =.9, β= mrl.5 =., β=.7 =.5, β=.7 =.9, β= Fig. 5 Mrl function of EMDL random variable for different parameter values mrl.6.4 =., β=. =.5, β= =.9, β= It can be easily seen that distribution of Y is a miture of two Loma distributions with common scale paramater 3, and α = and α = respectively. Thus, + and + represent the weight probabilities of miture components. Inference In this section the estimation techniques of the parameters of the EMDL distribution are studied using the moments, maimum lielihood and EM algorithm. In particular, because first two moments of the distribution have a very comple structure, we have developed bounds to get a solution more easily. Fisher information matri and asymptotic confidence ellipsoid for the parameters and β are also obtained. A detailed simulation study based on four estimation mehods is located at the end of this section. Estimation by moments Let X, X,..., X n be a random sample from EMDL distribution and m and m represent the first two sample moments. Then from 4 and 5, we will have the following system of equations [ ] ln m =, 8 β + m = β [ ln + I]. + 9 where I = =.

9 Yilmaz et al. SpringerPlus 6 5:66 Page 9 of Moment estimates of and β can be obtained by solving equations above. However, Eqs. 8 and 9 have no eplicit analytical solutions for the parameters. Thus, the estimates can be obtained by means of numerical procedures such as Newton-Raphson method. Since we can only get the symbolic computation for I, the calculation process taes too long during simulations. Therefore, we will find the lower and upper bounds for I. Theorem For [, ], I lies between 3 ln + I ln ln and ln + 4 i.e. Proof lower bound Let write inequality + for all, then holds. We have the following lower bound for I when summation is made over + I = + = = = +. According to convergence test comparison test of infinite series, since = is a convergent geometric series, two infinite series in the right hand side of inequality above are both convergent. By using Fubini s theorem for these series respectively we have = z dz = = ln z dz and + = z dz = = ln z dz By subtracting first term from the second and adding, then we get the lower bound for I. upper bound Let write inequality for all =, 3,..., then we have the upper bound for as below: [ ] + Let s add and subtract the term in bounds above, then +. +

10 Yilmaz et al. SpringerPlus 6 5:66 Page of First term can be rewritten following form = + =3 =3 = + + Thus, can be epressed by +. + The latter is combined with the epressions and together then we have + + ln +. If this result is placed in position in the bracets in the epression, and adding the term, then the upper bound is obtained. Graph below shows that these bounds are eligible for I, so, this leads us to solve moment estimate by using these bounds Fig. 6. Now let s go bac to the moments estimation problem. From the Eq. 8 we get the equality for β and replace it in 9, then we have the following equation to get a solution for ln + I + ln m =. m Solution was obtained by putting lower and upper limits in place of I, and applying Newton Raphson s method..4. Lower Bound I Upper Bound Fig. 6 Lower and upper bounds for I

11 Yilmaz et al. SpringerPlus 6 5:66 Page of Estimation by maimum lielihood Let =,,..., n be an observation of size n from the EMDL distribution with parameters and β. The log lielihood l = l, β; for, β is n n l, β; = n ln β + n ln β i + ln 3 e β i + i= i= 3 n 3 ln e β i and subsequently differentiating 3 with respect to and β yields the lielihood equations for, β l The solution of two equations above does not have a closed form, therefore numerical techniques can be used to solve the above system of equations. We investigate below conditions for the solution of this system of equations for β and. Proposition If n < n i= e β i, then the equation l/ = has at least one root in,, where β is the true value of the parameter. Proof Let ω denote the function on the RHS of the epression l/, then it is clear that lim ω = + and lim ω = 4n 3 + ni= 3 e β i 3 n i= e β i. Therefore, the equation ω = has at least one root in,, if n n i= e β i <. Proposition If [ is the true value ] of the parameter, the root of the equation l/ β = lies in the interval. X i= =n n l β = n n β i + i= i= i= 3, X e β i 3 e β 3 i n i e β i 3 e β 3 i + + n e β i e β i = n i e β i e β i = i= i= Proof Let ωβ denote the function on the RHS of the epression l/ β, then ωβ n n β i + i= n i= i e β i 3 e β i Note that 3 e β i + and e β i. Hence, ωβ n n β i + i= + n i. i= Therefore, ωβ when β + +. On the other hand, ωβ n n β i 3 i= n i= i e β i e β i.

12 Yilmaz et al. SpringerPlus 6 5:66 Page of By noting e β i and e β i. Hence, ωβ n n β 3 + i. Therefore, ωβ when β 3. Thus, there is at least one root of ωβ = in the interval. Recently, EM algorithm has been used by several authors X 3, X + + i= to find the ML estimates of compound distributions parameters. EM algorithm which is used to mae maimizing the complete data loglielihood is useful when observed log lielihood equations are difficult to solve. However EM algorithm plays a crucial role for getting parameter estimates in such compound distribution as long as equations obtained in E-step are more simple and clear. Estimation by EM algorithm The hypothetical complete-data, m density function is given by f, m;, β = β + mm + m e βm for ǫr +, m =,,..., ǫ,, β>. Here, and β are the parameters of the eponential-zero truncated Lindley distribution. According to E-step of EM cycle, we need to compute the conditional epectation of M with given X =. Therefore, immediately let s write conditional probability mass function as below: PM = m ;, β = r3 mm + r m, 3 r 4 for m =,,..., where r = e β. By using equation 4, we can find the conditional epectation of M to complete E-step as [ ] δ;, β = EM ;, β = 6. 3 r r M-step of each iteration requires maimization of complete-data lielihood function defined over, β. Let s l c indicate complete-data log lielihood function, i.e. ln L, β;, m then l c = n ln β + n ln n ln ln n m i β i= n ln m i m i + Hence, the lielihood equations can be verified by evaluating l c below: i= n m i i. i= = and l c β = as ni= + + = m i n ni= β = m i i n 5

13 Yilmaz et al. SpringerPlus 6 5:66 Page 3 of The M-step is completed with the missing observations of M i replaced by δ i ; t, β t. Thus, iterative solution of the system of equations in 5 is given by t+ = β t+ = ni= i t n n ni= i i t + ni= t 4 i n ni= i t n ni= t + 6 i n where i t = δ i ; t, β t and r i t = t e βt i. The information matri We first calculate the elements of epected Hessian matri of l with respect to the distribution of X. According to that, let a ij s denote epected values of the second derivatives of l with respect to, β where i, j =,. Then we have a = E l = n 4n + + n + 3 r 3 r r 3 dr 3 Inverse of the Fisher-information matri of single observation, i.e., I, β indicates asymptotic variance-covariance matri of ML estimates of, β. Hence, joint distribution of maimum lielihood estimator for, β is asymptotically normal with mean, β and variance-covariance matri I r 3 r r 5 dr = n 4n + + n ln l a = E β = n β + n r r β + 3 r r 5 l n a = a = E = β β +, β. Namely, r ln dr We have the simulated data sets with sample size of n = 5 from the EMDL distribution with nown parameters as =.6 and β =.3. Based on the asymptotic normal r r 3 r r 5 ln r Thus, Fisher information matri, I n, β of sample size n for, β is as follows: I n, β = E l E l ] β a a E l β E = [ l a a β n [ ˆ ˆβ ] [ ] [ ] AN, I β., β dr

14 Yilmaz et al. SpringerPlus 6 5:66 Page 4 of distribution, confidence ellipsoid of ML estimates for, β can be drawn at the 95 % confidence level as follows. Firstly, we present the asymptotic distribution of the ML estimates, [ ] [ ] [ ] [ ] ˆ AN, ˆβ [ ].6 Now, let µ = indicate the center of ellipsoid, and observed information matri.3 [ ] is calculated as Î, β = note that Î P I. Then the confidence [ ] [ ] ˆ ˆ ellipsoid at the level 95 % is defined by 5 µ Î µ 5.99 where 5.99 ˆβ ˆβ is a critical value of the chi-squared distribution with two degrees of freedom with upper percentiles 95 % Fig. 7. Simulation study We conduct a simulation study generating samples, each of which has a sample size of n =,, 5,. We computed the moment using lower and upper bounds and ML Newton-Raphson and EM algorithm estimates of the parameters for every sample size level with different values of and β. From each generated sample of a given size n the root mean square errors RMSE of four estimates are also calculated. These results are tabulated in Table. It is observed from the tables that when β >, the ML estimates of and β are better than the others with respect to the RMSE. When >β, the moment estimates both bounds are as good as ML and EM estimates. Even for small sample size n, moment estimates are a little better. Applications We illustrate the applicability of EMDL distribution by considering three different data sets which have been eamined by a lot of other researchers. First data set is tried to be modeled by Transmuted Pareto and Lindley Distributions, second and third data sets.5 β Fig. 7 Confidence region for ˆ, ˆβ

15 Yilmaz et al. SpringerPlus 6 5:66 Page 5 of Table Simulation results for moment, ML, EM estimates for different parameter values Parameter ; β Sample size Moment estimates lower bound ˆ; ˆβ Moment estimates upper bound ˆ; ˆβ ML estimates ˆ; ˆβ EM algorithm ˆ; ˆβ.;..389; ; ; ; ; ; ; ; ; ; ;.63.55;.7.366; ; ; ; ; ; ; ;.3.97; ; ; ; ;.95.36;.37.5;.653.5; ;.5.48; ; ;.36.;..344; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;.76.44;.994.8; ; ; ; ;.3455; ; ; ; ; ; ; ; ; ; ; ; 8.8.3; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;.88.93; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;..377; ; ; ; ;.83.48; ; ;.75.37; ; ; ; ; ;.99.3;.49.33; ; ; ;.39.54;.34.35; ; ;..538;.3.57;.93.39;.93.56;.64.84;.4.674;.7.678; ;..45;.59

16 Yilmaz et al. SpringerPlus 6 5:66 Page 6 of Table continued Parameter ; β Sample size Moment estimates lower bound ˆ; ˆβ Moment estimates upper bound ˆ; ˆβ ML estimates ˆ; ˆβ EM algorithm ˆ; ˆβ.;..5543; ; ; ;.74.68; ; ; ; ; ; ; ; ; ; ; ; ; ; ;.933.;.334.8; ;.769.4; ; ;.4.46; ; ;.535.4; ; ;.5.343;.5.;.446; ; ; ; ; ; ; ; ; ; ; ; ;.48.4; ; ; ;.358.;.697.3;.5.553;.77.38; ; ; ; ;.66.39;.767.4; ;.36.85;..49; ; ;.6684.; ; ; ; ; ;.63.95; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;.55.99; ; ; ; ; ; ;..4846;.3.573;.63.74;.43.48;.8.7;.9.9; ;.8.34; ;.83.79; ;.9.4;.6.5; ; ;..889; ;.6.988;.33.68;.3.38; ;.5.79;.6.756;.3.998;..579;.95.75;.5.643;.99.5;..67;.99.89;.3.68;..777;.7.6;..489;.7.959; ;.87.63; ;.6.7; ; ;.6.658;..497; ;.4.94; ; ; ;.8.33; ;.8.67; ;.43.5; ;.6.668;.3.667;.84.53; ;.99.83;.8.698; ;.7.633; ;.9.688;.4.53;.6

17 Yilmaz et al. SpringerPlus 6 5:66 Page 7 of Table continued Parameter ; β Sample size Moment estimates lower bound ˆ; ˆβ Moment estimates upper bound ˆ; ˆβ ML estimates ˆ; ˆβ EM algorithm ˆ; ˆβ.6;.6795;.8.585; ;.67.47; ;.8.344;.55.65; ; ;.37.84; ;.5.4; ; ; ; ; ;.4.378; ;.5.36; ; ; ;.96.57; ;.6.679; ; ;.346.6;.6.39; ; ;.36.6; 3.566; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;.7.559; ; ; ; ; ;.8.698; ; ;.778.3; ; ; ; ; ;..643; ; ;.86.4;.36.89;.7.439; ;.9.33; ;.79.38;.3.773;.83.;.3.73; ;..938;.99.; ;.85.73;.4.853;.94.3;.3.795;.96.5;.6.986;.3.473;..7465;.9.98; ;.3.6;..85;.93.36;.4.968;..8;.9.9;..646; ; ;.98.63; ;..934; ;.6.45; ;.5.3; ;.995.8; ;.4.685;.46.87; ; ; ;..77;.887.7;.9.864;.4.35; ;.89.;.6.785; ; ;.3.87;.8.96;.93.88; ;.6.8;.99.9;.5889; ; ; ; ; ; ; ; ; ; ;.99.6; ; ; ;.83.65; ; ; ; ; ;.5.77;.7.875; ; ; ; ; ; ;.7.8; ; ;.79.9; 3.535; ; ; ; ;.8.976; ; ; ;.3.33; ; ; ; ;.77.8; ; ; 3.6.6; ;.87.8; ; ;.9.94; ; ; ; ; ; ;.967.; ;.977.7;.4539

18 Yilmaz et al. SpringerPlus 6 5:66 Page 8 of are tried to be modeled by the Eponential-Poisson EP and Eponential-Geometric EG distributions. In order to compare distributional models, we consider some criteria as K-S Kolmogorow-Smirnow, LL LogL, AIC Aaie information criterion and BIC Bayesian information criterion for the data sets. Data Set The data consist of the eceedances of flood peas in m 3 /s of the Wheaton River near Carcross in Yuon Territory, Canada. The data consist of 7 eceedances for the years , rounded to one decimal place. These data were analyzed by Choulaian and Stephens and are given in Table. Later on, Beta-Pareto distribution was applied to these data by Ainsete et al. 8. Merovcia and Puab 4 made a comparison between Pareto and transmuted Pareto distribution. They showed that better model is the transmuted Pareto distribution TP. Bourguignon et al. 3 proposed Kumaraswamy Kw Pareto distribution Kw-P. Tahir et al. 4 have proposed weibull-pareto distribution WP and made a comparison with Beta Eponentiated Pareto BEP distriubtion. Nasiru and Luguterah 5 have proposed different type of weibull-pareto distribution NWP. Mahmoudi concluded that the Beta- Generalized Pareto BGP distribution fits better to these data than the GP, BP, Weibull and Pareto models. We fit data to EMDL distribution and get parameter estimates as ˆ =.778, ˆβ =.695. According to the model selection criteria AIC, or BIC tabulated in Table 3, it is said that EMDL taes fifth place in amongst proposed models. Table Eceedances of Wheaton river flood data Table 3 Model selection criteria for river flood data Model K-S LL AIC BIC T. Pareto Pareto EP BP Kw-P WP NWP BEP BGP EMDL

19 Yilmaz et al. SpringerPlus 6 5:66 Page 9 of Data Set The data set given in Table 4, contains the time intervals in days between coal mine accidents caused death of or more men. Firstly, this data set was obtained by Maguire et al. 95. There were lots of models on this data set such as Adamidis and Louas 998 and Kus 7. They suggested to use Eponential-Geometric EG and Eponential-Poisson EP distributions respectively. On the other hand, Yilmaz et al. 6 have proposed two-component mied eponential distribution MED for modeling this data set. In addition to these three models, we try to fit this data set by using EMDL distribution and we get the parameter estimates as ˆ =.539 and ˆβ =.5. We have only K-S and p values which are tabulated in Table 5 to mae a comparison. According to Table 5, EMDL distribution fits better than EG distribution. Data Set3 The data set in Table 6 obtained by Kus 7 includes the time intervals in days of the successive earthquaes with magnitudes greater than or equal to 6 Mw. Kus 7 has used this data set to show the applicability of the EP distribution and he made a comparison between EG and EP distributions with K-S statistic. Parameter esitmates of EMDL distribution are ˆ =.354, ˆβ =.3. Calculated K-S statistic for EMDL can be seen in Table 7, according to this, EMDL distribution gives the best fit to earthquae data in three models. Table 4 The time intervals in days between coal mine accidents Table 5 K-S and p values for EP, EG, MED and EMDL Model K-S p value EP EG MED EMDL Table 6 Time intervals of the successive earthquaes in North Anatolia fault zone

20 Yilmaz et al. SpringerPlus 6 5:66 Page of Table 7 K-S and p values for EP, EG and EMDL Model K-S p value EP EG EMDL Conclusions In this paper we have proposed a new lifetime distribution, which is obtained by compounding the modified discrete Lindley distribution MDL and eponential distribution, referred to as the EMDL. Some statistical characteristics of the proposed distribution including eplicit formulas for the probability density, cumulative distribution, survival, hazard and mean residual life functions, moments and quantiles have been provided. We have proposed bounds to solve moment equations. We have derived the maimum lielihood estimates and EM estimates of the parameters and their asymptotic variance-covariance matri. Simulation studies have been performed for different parameter values and sample sizes to assess the finite sample behaviour of moments, ML and EM estimates. The usefulness of the new lifetime distribution has been demonstrated in three data sets. EMDL distribution fits better for the third data set consisting of the times between successive earthquaes in North Anatolia fault zone than the EP and EG. Authors contributions This wor was carried out in cooperation among all the authors MY, MH and SAK. All authors read and approved the final manuscript. Author details Department of Statistics, Faculty of Science, Anara University, 6 Tandogan, Anara, Turey. Graduate School of Natural and Applied Sciences, Anara University, 6 Disapi, Anara, Turey. Acnowledgements This research has not been funded by any entity. Competing interests The authors declare that they have no competing interests. Received: March 6 Accepted: September 6 References Adamidis K, Dimitraopoulou T, Louas S 5 On an etension of the eponential-geometric distribution. Stat Probab Lett 733:59 69 Adamidis K, Louas S 998 A lifetime distribution with decreasing failure rate. Stat Probab Lett 39:35 4 Ainsete A, Famoye F, Lee C 8 The beta-pareto distribution. Statistics 46: Baouch HS, Jazi MA, Nadarajah S 4 A new discrete distribution. Statistics 48: 4 Barreto-Souza W, Baouch HS 3 A new lifetime model with decreasing failure rate. Statistics 47: Bourguignon M, Silva RB, Zea LM, Cordeiro GM 3 The Kumaraswamy Pareto distribution. J Stat Theory Appl :9 44 Chahandi M, Ganjali M 9 On some lifetime distributions with decreasing failure rate. Comput Stat Data Anal 53: Choulaian V, Stephens MA Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 434: Gómez-Déniz E, Calderín-Ojeda E The discrete Lindley distribution: properties and applications. J Stat Comput Simul 8:45 46 Gui W, Zhang S, Lu X 4 The Lindley-Poisson distribution in lifetime analysis and its properties. Hacet J Math Stat 436:63 77 Hajebi M, Rezaei S, Nadarajah S 3 An eponential-negative binomial distribution. REVSTAT Stat J :9 Kus C 7 A new lifetime distribution. Comput Stat Data Anal 59:

21 Yilmaz et al. SpringerPlus 6 5:66 Page of Maguire BA, Pearson ES, Wynn AHA 95 The time intervals between industrial accidents. Biometria 39/:68 8 Mahmoudi E The beta generalized Pareto distribution with application to lifetime data. Math Comput Simul 8:44 43 Merovcia F, Puab L 4 Transmuted pareto distribution. ProbStat Forum 7: Nasiru S, Luguterah A 5 The new Weibull-Pareto distribution. Pa J Stat Oper Res :3 4 Tahir MH, Cordeiro GM, Alzaatreh A, Mansoor M, Zubair M 6 A new Weibull-Pareto distribution: properties and applications. Commun Stat Simul Comput 45: Tahmasbi R, Rezaei S 8 A two-parameter lifetime distribution with decreasing failure rate. Comput Stat Data Anal 58: Yilmaz M, Potas N, Buyum B 6 A classical approach to modeling of coal mine data. Chaos, compleity and leadership. Springer, New Yor, pp 65 73