# Exponential-modified discrete Lindley distribution

- Mehmet Yilmaz
^{1}Email authorView ORCID ID profile, - Monireh Hameldarbandi
^{2}and - Sibel Acik Kemaloglu
^{1}

**Received: **21 March 2016

**Accepted: **11 September 2016

**Published: **26 September 2016

## 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 examined under the given circumstances and also parameters of this new lifetime distribution are estimated by using moments, maximum likelihood and EM-algorithm.

## Keywords

## Mathematics Subject Classification

## Background

Under the name of the “*new lifetime distribution*”, about 400 studies have been done in the recent 5 years. In particular, the compound distributions obtained by exponential distribution are applicable in the fields such as electronics, geology, medicine, biology and actuarial. Some of these works can be summarized as follows: Adamidis and Loukas (1998) and Adamidis et al. (2005) introduced a two-parameter lifetime distribution with decreasing failure rate by compounding exponential and geometric distribution. In the same way, exponential-Poisson (EP) and exponential-logarithmic (EL) distributions were given by Kus (2007) and Tahmasbi and Rezaei (2008), respectively. Chahkandi and Ganjali (2009) introduced exponential-power series distributions (EPS). Barreto-Souza and Bakouch (2013) introduced a new three-parameter distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. Exponential-Negative Binomial distribution is introduced by Hajebi et al. (2013). Furthermore, Gui et al. (2014) have considered the Lindley distribution which can be described as a mixture of the exponential 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 distributions. Although these compound distributions are quite complex, new distributions can fit better than the known 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 2011; Eq. 3, Bakouch et al. 2014; Eq. 3). This discrete distribution provided by authors above, is quite a complex 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 exponential and modified-zero-truncated discrete Lindley distributions.

This paper is organized as follows: In “Construction of the model” section, we propose the two-parameter exponential-modified discrete Lindley (EMDL) distribution, by mixing exponential and zero truncated modified discrete Lindley distribution, which exhibits 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, maximum likelihood and EM algorithm. Furthermore, information matrix and observed information matrix 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 approximations), ML and EM estimates. Illustrative examples 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 (2011) and Bakouch et al. (2014). We have achieved a more simplified discrete distribution than discrete Lindley distribution by taking \(1-\theta\) instead of \(e^{-\theta }\) in subsequent definition. Thus, we introduce a new lifetime distribution by compounding Exponential and Modified Discrete Lindley distributions, named the Exponential-Modified Discrete Lindley (EMDL) distribution.

### Discrete Lindley distribution

*M*is said to have Lindley distribution with the parameter \(\theta >0\), if its probability mass function (p.m.f) is given by

*M*will be given by

### Modified discrete Lindley distribution

###
**Theorem 1**

*MDL distribution can be represented as a mixture of geometric and negative binomial distributions with mixing proportion is*
\(\frac{\theta }{1+\theta }\)
*, and a common success rate*
\(\theta\).

###
*Proof*

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 \(\theta\) 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 exponential distribution. Thus, this proposed compound distribution can be usefull for modelling lifetime data such as time interval between successive earthquakes, time period of bacteria spreading, recovery period of the certain disease.

### Exponential modified discrete Lindley distribution

Suppose that *M* is a zero truncated *MDL* random variable with probablity mass function \(\pi \left( m\right) =P(M=m\)
\(\vert {M>0})=\frac{{\theta }^2}{\left( 1+2\theta \right) }{\left( 1-\theta \right) }^{m-1}\left( m+2\right)\) and \(X_1,X_2,\ldots, X_M\) are i.i.d. with probability density function \(h\left( x;\beta \right) =\beta e^{-\beta x},\ x>0\). Let \(X=min\left( X_1,X_2,\ldots ,X_M\right)\), then \(g\left( x\vert {m};\beta \right) =m\beta e^{-m\beta x}\) and \(g\left( x,m\right) =g\left( x\vert {m}\right) \pi (m)=\frac{\beta {\theta }^2}{\left( 1+2\theta \right) }m\left( m+2\right) {\left( 1-\theta \right) }^{m-1}e^{-m{\beta}x }\).

*X*as

*θ*∈ (0, 1) and

*β*> 0. Henceforth, the distribution of the random variable

*X*having the p.d.f in (3) is called shortly EMDL. By changing of variables \(r=\left( 1-\theta \right) e^{-\beta x}\) in cumulative integration of (3), the distribution function can be found as follows:

*EMDL*random variable for various values of \(\theta\) and \(\beta\) (Fig. 2).

## 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 Lomax and Exponential-Poisson distributions.

### Moment generating function and moments

*X*is given by

*k*.th raw moment of

*X*is expressed by

*polylog*functions.

### Quantile function

*X*is obtained simply by inverting \(F(x;\theta ,\beta )=q\) as follows

*X*is

*X*is

*X*is

### Survival, hazard rate and mean residual life functions

*X*is given by (Fig. 3)

*X*is

*h*(

*x*) is a monotonically decreasing function and bounded from below with \(\beta\) (see Fig. 4).

*X*is given by

*mrl*(

*x*) for different values of parameter \(\theta\) and \(\beta\) (Fig. 5).

### Relationship of the other distribution

*X*

*Y*can be obtained as

*Y*is a mixture of two Lomax distributions with common scale paramater \(\frac{\theta }{1\,-\,\theta }\), and \(\alpha =1\) and \(\alpha =2\) respectively. Thus, \(\frac{3\theta }{1\,+\,2\theta }\) and \(\frac{1-\theta }{1+2\theta }\) represent the weight probabilities of mixture components.

## Inference

In this section the estimation techniques of the parameters of the *EMDL* distribution are studied using the moments, maximum likelihood and EM algorithm. In particular, because first two moments of the distribution have a very complex structure, we have developed bounds to get a solution more easily. Fisher information matrix and asymptotic confidence ellipsoid for the parameters \(\theta\) and \(\beta\) are also obtained. A detailed simulation study based on four estimation mehods is located at the end of this section.

### Estimation by moments

*EMDL*distribution and \(m_1\) and \(m_2\) represent the first two sample moments. Then from (4) and (5), we will have the following system of equations

Moment estimates of \(\theta\) and \(\beta\) can be obtained by solving equations above. However, Eqs. (8) and (9) have no explicit 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 \(I\left( \theta \right)\), the calculation process takes too long during simulations. Therefore, we will find the lower and upper bounds for \(I\left( \theta \right)\).

###
**Theorem 2**

*For*\(\theta \in \left[ 0,1\right]\), \(I\left( \theta \right)\)

*lies between*\(\frac{\theta }{1-\theta }\ln {\left( \theta \right) }+\frac{3-\theta }{2}\)

*and*\(\frac{\theta \left( 2-\theta \right) }{2\left( 1-\theta \right) }\ln {\left( \theta \right) }+\frac{7-5\theta }{4}\)

*i.e.*

###
*Proof*

*k*, then \(\frac{{\left( 1-\theta \right) }^k}{k^2}\ge \frac{{\left( 1-\theta \right) }^k}{k(k+1)}\) holds. We have the following lower bound for \(I\left( \theta \right)\) when summation is made over

*k*

### Estimation by maximum likelihood

*n*from the

*EMDL*distribution with parameters \(\theta\) and \(\beta\). The log likelihood \(\ell\) = \(\ell (\theta ,\beta ;\ x)\) for \((\theta ,\beta )\) is

We investigate below conditions for the solution of this system of equations for \(\beta\) and \(\theta\).

###
**Proposition 1**

*If*
\(\frac{n}{2}<\sum _{i=1}^{n}e^{-\beta x_{i}}\)
*, then the equation*
\(\partial \ell /\partial \theta =0\)
*has at least one root in*
\(\left( 0,1\right)\)
*, where*
\(\beta\)
*is the true value of the parameter.*

###
*Proof*

Let \(\omega \left( \theta \right)\) denote the function on the RHS of the expression \(\partial \ell /\partial \theta\), then it is clear that \(\lim \limits _{\theta \rightarrow 0}\omega \left( \theta \right) =+\infty\) and \(\lim \limits _{\theta \rightarrow 1}\omega \left( \theta \right) =\frac{4n }{3}+\frac{1}{3}\sum _{i=1}^{n}e^{-\beta x_{i}}-3\sum _{i=1}^{n}e^{-\beta x_{i}}\). Therefore, the equation \(\omega \left( \theta \right) =0\) has at least one root in \(\left( 0,1\right)\), if \(\frac{n}{2}-\sum _{i=1}^{n}e^{-\beta x_{i}}<0\). \(\square\)

###
**Proposition 2**

*If*
\(\theta\)
*is the true value of the parameter, the root of the equation*
\(\partial \ell /\partial \beta =0\)
*lies in the interval*
\(\left[ \frac{1}{ \overline{X}}\frac{\theta }{\left( 3-2\theta \right) },~\frac{1}{\overline{X} }\frac{2+\theta }{\left( 1+2\theta \right) }\right]\).

###
*Proof*

### Estimation by EM algorithm

*x*,

*m*) density function is given by

*M*with given \(X=x\). Therefore, immediately let’s write conditional probability mass function as below:

*M*to complete E-step as

### The information matrix

*X*. According to that, let \(a_{ij}\)’s denote expected values of the second derivatives of \(\ell\) with respect to \(\theta ,\beta\) where \((i,j=1,2)\). Then we have

*n*for \(\left( \theta ,\beta \ \right)\) is as follows:

### Simulation study

*n*the root mean square errors (RMSE) of four estimates are also calculated. These results are tabulated in Table 1.

Simulation results for moment, ML, EM estimates for different parameter values

Parameter \(\left( \theta ;\beta \right)\) | Sample size | Moment estimates (lower bound) \(rmse\left( \hat{\theta };\hat{\beta }\right)\) | Moment estimates (upper bound) \(rmse\left( \hat{\theta };\hat{\beta }\right)\) | ML estimates \(rmse\left( \hat{\theta };\hat{\beta }\right)\) | EM algorithm \(rmse\left( \hat{\theta };\hat{\beta }\right)\) |
---|---|---|---|---|---|

(0.01; 0.01) | 10 | (0.3819; 0.4290) (0.4305; 0.5161) | (0.5330; 0.7897) (0.6306; 0.9422) | (0.5560; 1.1057) (0.6350; 1.3591) | (0.4490; 0.7892) (0.5419; 1.0079) |

20 | (0.3473; 0.4560) (0.4555; 0.5957) | (0.4871; 0.6023) (0.5500; 0.7010) | (0.3662; 0.5454) (0.4537; 0.6899) | (0.5202; 0.7779) (0.5953; 0.9440) | |

50 | (0.2734; 0.4128) (0.2933; 0.4678) | (0.1971; 0.2388) (0.2319; 0.3022) | (0.2927; 0.3971) (0.4197; 0.5767) | (0.2559; 0.3208) (0.2837; 0.3513) | |

100 | (0.2086; 0.2952) (0.2361; 0.3722) | (0.2050; 0.2653) (0.2250; 0.3083) | (0.1975; 0.2521) (0.2428; 0.3155) | (0.1806; 0.2556) (0.2135; 0.3216) | |

(0.01; 0.1) | 10 | (0.3441; 4.7886) (0.3940; 6.1487) | (0.4563; 6.0390) (0.4968; 7.6046) | (0.1563; 1.2578) (0.2539; 2.0050) | (0.5476; 8.4679) (0.6203; 10.1551) |

20 | (0.4427; 6.1054) (0.5348; 8.0019) | (0.5069; 8.2839) (0.5310; 9.0880) | (0.1382; 1.4474) (0.3346; 3.2439) | (0.2243; 2.8182) (0.3571; 4.4648) | |

50 | (0.2810; 3.9128) (0.2982; 4.2335) | (0.3375; 4.8599) (0.3980; 6.1233) | (0.1457; 1.7248) (0.1635; 1.9789) | (0.3682; 5.9521) (0.4436; 7.3848) | |

100 | (0.2980; 4.2913) (0.3269; 4.9345) | (0.2281; 2.7206) (0.2440; 2.9904) | (0.2180; 3.1057) (0.2560; 3.6473) | (0.2187; 2.9054) (0.2819; 3.8716) | |

(0.01; 1) | 10 | (0.3455; 31.2778) (0.4174; 37.4113) | (0.3251; 31.8195) (0.3651; 40.4183) | (0.0320; 6.6267) (0.0714; 15.6975) | (0.1561; 21.8481) (0.1863; 24.7316) |

20 | (0.4703; 66.2730) (0.5181; 78.6781) | (0.4256; 62.2380) (0.4959; 81.2181) | (0.0232; 1.8407) (0.0515; 3.9342) | (0.2677; 44.3097) (0.3238; 56.3634) | |

50 | (0.3570; 57.4291) (0.4121; 75.3072) | (0.3372; 45.2799) (0.4119; 63.8644) | (0.0485; 6.3230) (0.0690; 9.3401) | (0.2985; 42.1803) (0.3549; 53.6298) | |

100 | (0.2481; 35.1331) (0.2583; 37.8049) | (0.3825; 50.6405) (0.4070; 54.8860) | (0.0412; 5.3581) (0.0563; 8.0360) | (0.2263; 32.0492) (0.2549; 37.9289) | |

(0.01; 3) | 10 | (0.3955; 155.0823) (0.4273; 168.6059) | (0.3385; 118.2894) (0.3807; 151.7710) | (0.0169; 12.4376) (0.0274; 32.4726) | (0.2980; 215.2239) (0.3521; 305.7791) |

20 | (0.4581; 185.6736) (0.5048; 211.9931) | (0.4414; 180.4513) (0.5424; 249.3666) | (0.0565; 22.2828) (0.0913; 38.1805) | (0.2077; 72.5677) (0.2928; 101.4907) | |

50 | (0.2804; 123.0569) (0.2960; 137.4734) | (0.3819; 129.9446) (0.4090; 145.2682) | (0.0301; 9.2323) (0.0470; 14.5592) | (0.1684; 68.5255) (0.2332; 96.9123) | |

100 | (0.2034; 78.9574) (0.2238; 90.6749) | (0.2601; 104.8366) (0.3032; 130.2829) | (0.0115; 3.6741) (0.0196; 6.4544) | (0.1746; 66.2565) (0.1975; 78.9489) | |

(0.1; 0.01) | 10 | (0.3717; 0.0426) (0.3629; 0.0535) | (0.5123; 0.0529) (0.4926; 0.0719 | (0.4609; 0.0831) (0.4811; 0.1192) | (0.6697; 0.1255) (0.6698; 0.1715) |

20 | (0.3217; 0.0443) (0.3398; 0.0530) | (0.4694; 0.0702) (0.4458; 0.0791 | (0.4358; 0.0750) (0.4491; 0.0991) | (0.3200; 0.0492) (0.3301; 0.0603) | |

50 | (0.3540; 0.0474) (0.3450; 0.0524) | (0.2736; 0.0319) (0.2540; 0.0341) | (0.3502; 0.0459) (0.3586; 0.0492) | (0.1795; 0.0220) (0.1538; 0.0213) | |

100 | (0.1570; 0.0193) (0.1319; 0.0193) | (0.2562; 0.0264) (0.1824; 0.0204) | (0.1674; 0.0172) (0.1678; 0.0176) | (0.1996; 0.0220) (0.1450; 0.0159) | |

(0.1; 0.1) | 10 | (0.5543; 0.5408) (0.5215; 0.5529) | (0.5115; 0.6531) (0.4591; 0.7114) | (0.6810; 0.8712) (0.6672; 0.9190) | (0.5306; 0.6258) (0.5184; 0.7046) |

20 | (0.4436; 0.5750) (0.4105; 0.5916) | (0.4167; 0.3775) (0.4206; 0.4234) | (0.4159; 0.4117) (0.4641; 0.4481) | (0.4431; 0.5409) (0.4599; 0.6300) | |

50 | (0.3695; 0.4175) (0.3499; 0.4394) | (0.2921; 0.2933) (0.2211; 0.2334) | (0.1822; 0.1846) (0.1566; 0.1769) | (0.2421; 0.3087) (0.1937; 0.2976) | |

100 | (0.1999; 0.2411) (0.1416; 0.2194) | (0.1993; 0.1916) (0.1396; 0.1535) | (0.2411; 0.2668) (0.2137; 0.2548) | (0.2038; 0.2215) (0.1343; 0.1502) | |

(0.1; 1) | 10 | (0.4461; 4.5711) (0.4257; 4.1085) | (0.5073; 5.6156) (0.4897; 7.4046) | (0.1549; 2.0957) (0.1488; 2.7522) | (0.6790; 7.3133) (0.6839; 7.5797) |

20 | (0.3167; 3.8858) (0.3364; 5.1648) | (0.4691; 6.4103) (0.4813; 8.6758) | (0.1318; 1.4181) (0.1402; 1.7552) | (0.2355; 2.6073) (0.1795; 2.2731) | |

50 | (0.2459; 2.3518) (0.2212; 2.1697) | (0.2131; 2.1511) (0.1553; 1.7007) | (0.2238; 2.5573) (0.2083; 2.4655) | (0.2574; 2.8351) (0.1860; 2.2699) | |

100 | (0.1941; 2.2066) (0.1392; 1.7672) | (0.2142; 2.3646) (0.1853; 2.2036) | (0.1851; 2.0202) (0.1492; 1.8207) | (0.2388; 2.8016) (0.1965; 2.6684) | |

(0.1; 3) | 10 | (0.3444; 13.6255) (0.3562; 20.5479) | (0.4058; 11.4606) (0.3983; 11.9361) | (0.0083; 0.1631) (0.0950; 2.8788) | (0.2777; 8.0010) (0.3672; 10.6504) |

20 | (0.3146; 7.9388) (0.3077; 7.7036) | (0.1761; 6.0556) (0.1258; 6.7467) | (0.0829; 2.6625) (0.0944; 2.9822) | (0.3651; 13.6145) (0.4316; 16.6203) | |

50 | (0.1978; 5.9274) (0.1392; 4.4230) | (0.2566; 8.6629) (0.2078; 8.3575) | (0.0943; 2.9838) (0.1105; 3.5267) | (0.2816; 9.8190) (0.2150; 8.2843) | |

100 | (0.1460; 4.0423) (0.0941; 2.5510) | (0.1929; 5.4612) (0.1277; 3.9084) | (0.1019; 3.3346) (0.0845; 3.2275) | (0.2194; 7.7674) (0.1791; 7.1549) | |

(0.6; 0.01) | 10 | (0.4846; 0.0113) (0.2573; 0.0063) | (0.7014; 0.0143) (0.2481; 0.0081) | (0.7000; 0.0109) (0.2921; 0.0052) | (0.7637; 0.0108) (0.3041; 0.0051) |

20 | (0.4787; 0.0083) (0.2079; 0.0036) | (0.6509; 0.0109) (0.2240; 0.0061) | (0.5200; 0.0089) (0.2843; 0.0048) | (0.5983; 0.0111) (0.2889; 0.0064) | |

50 | (0.6111; 0.0106) (0.1988; 0.0033) | (0.6812; 0.0103) (0.2380; 0.0037) | (0.6782; 0.0115) (0.1719; 0.0026) | (0.7056; 0.0103) (0.1998; 0.0021) | |

100 | (0.5729; 0.0095) (0.1725; 0.0025) | (0.6413; 0.0099) (0.1502; 0.0021) | (0.6071; 0.0099) (0.1892; 0.0032) | (0.6108; 0.0100) (0.1777; 0.0027) | |

(0.6; 0.1) | 10 | (0.4892; 0.1172) (0.1959; 0.1301) | (0.6964; 0.1187) (0.1631; 0.0498) | (0.6865; 0.1116) (0.2722; 0.0345) | (0.6737; 0.1339) (0.2742; 0.0610) |

20 | (0.6058; 0.1000) (0.2497; 0.0474) | (0.6651; 0.1040) (0.1942; 0.0334) | (0.5737; 0.0956) (0.3423; 0.0644) | (0.5784; 0.1081) (0.3023; 0.0697) | |

50 | (0.5760; 0.1028) (0.1671; 0.0279) | (0.6347; 0.1043) (0.1510; 0.0251) | (0.6598; 0.1162) (0.1668; 0.0321) | (0.6670; 0.1184) (0.2531; 0.0473) | |

100 | (0.6451; 0.0990) (0.1823; 0.0208) | (0.6198; 0.0950) (0.1734; 0.0270) | (0.6303; 0.1047) (0.1747; 0.0229) | (0.6188; 0.1014) (0.1253; 0.0206) | |

(0.6; 1) | 10 | (0.6795; 1.2128) (0.1585; 0.9660) | (0.6059; 1.0671) (0.1417; 0.5625) | (0.6832; 1.0181) (0.3404; 0.5152) | (0.6502; 1.5538) (0.3275; 1.1916) |

20 | (0.6957; 1.2371) (0.2184; 0.5714) | (0.6458; 1.0511) (0.2042; 0.3870) | (0.5259; 0.8140) (0.2643; 0.4453) | (0.6212; 1.0367) (0.2875; 0.5246) | |

50 | (0.6684; 1.0214) (0.1378; 0.1628) | (0.6477; 1.0105) (0.1361; 0.3014) | (0.6550; 0.9959) (0.2374; 0.3766) | (0.6311; 0.9262) (0.2507; 0.3527) | |

100 | (0.6354; 1.0162) (0.1679; 0.2285) | (0.6323; 0.9949) (0.1456; 0.2346) | (0.6120; 1.0161) (0.1392; 0.2164) | (0.5899; 0.9988) (0.1893; 0.3216) | |

(0.6; 3) | 10 | (0.5661; 2.9674) (0.2097; 1.3517) | (0.5572; 3.1071) (0.2856; 2.7416) | (0.6525; 4.0879) (0.3489; 3.0165) | (0.5420; 2.8224) (0.2985; 1.7626) |

20 | (0.6527; 3.6483) (0.2922; 1.9112) | (0.6443; 3.0669) (0.1563; 0.8436) | (0.6340; 3.2548) (0.2618; 1.4175) | (0.6871; 3.6108) (0.2115; 1.4077) | |

50 | (0.6667; 3.4335) (0.2030; 1.0722) | (0.5592; 2.7817) (0.2527; 1.4259) | (0.6313; 3.0814) (0.2326; 0.9826) | (0.6583; 3.1923) (0.1929; 1.1181) | |

100 | (0.6198; 3.0107) (0.1390; 0.6702) | (0.5831; 2.7708) (0.1301; 0.6725) | (0.6169; 3.0523) (0.1450; 0.6430) | (0.6374; 3.1514) (0.1758; 0.8689) | |

(0.9; 0.01) | 10 | (0.6143; 0.0071) (0.3933; 0.0048) | (0.7565; 0.0086) (0.2204; 0.0036) | (0.8192; 0.0107) (0.2439; 0.0033) | (0.7333; 0.0109) (0.3113; 0.0036) |

20 | (0.6665; 0.0079) (0.3182; 0.0032) | (0.7730; 0.0083) (0.2121; 0.0032) | (0.7321; 0.0086) (0.2844; 0.0021) | (0.9318; 0.0099) (0.1212; 0.0025) | |

50 | (0.6855; 0.0085) (0.2732; 0.0024) | (0.8523; 0.0094) (0.1123; 0.0013) | (0.7905; 0.0096) (0.2251; 0.0026) | (0.9186; 0.0103) (0.1473; 0.0021) | |

100 | (0.7465; 0.0090) (0.2198; 0.0016) | (0.8935; 0.0103) (0.0622; 0.0011) | (0.8215; 0.0093) (0.1361; 0.0014) | (0.9268; 0.0100) (0.0812; 0.0009) | |

(0.9; 0.1) | 10 | (0.6462; 0.0862) (0.3381; 0.0378) | (0.7013; 0.0908) (0.2623; 0.0309) | (0.8431; 0.1020) (0.1934; 0.0374) | (0.9185; 0.1006) (0.2451; 0.0267) |

20 | (0.8678; 0.1015) (0.1223; 0.0214) | (0.8388; 0.0995) (0.1228; 0.0269) | (0.7909; 0.1014) (0.2685; 0.0406) | (0.8270; 0.0988) (0.2316; 0.0419) | |

50 | (0.7725; 0.0956) (0.1698; 0.0121) | (0.7720; 0.0887) (0.2027; 0.0209) | (0.8624; 0.1042) (0.2035; 0.0226) | (0.7946; 0.0890) (0.2022; 0.0216) | |

100 | (0.7852; 0.0952) (0.1698; 0.0153) | (0.8895; 0.1031) (0.0870; 0.0180) | (0.9620; 0.1093) (0.1088; 0.0157) | (0.9496; 0.1016) (0.1028; 0.0099) | |

(0.9; 1) | 10 | (0.5889; 0.7639) (0.3829; 0.4564) | (0.6355; 0.9238) (0.3500; 0.4046) | (0.8911; 1.2604) (0.1467; 0.4797) | (0.8847; 1.0496) (0.1907; 0.3918) |

20 | (0.6091; 0.7758) (0.3424; 0.4521) | (0.6895; 0.9191) (0.2611; 0.2828) | (0.7729; 0.8300) (0.2886; 0.3351) | (0.9399; 1.1832) (0.1065; 0.3144) | |

50 | (0.6475; 0.7636) (0.3288; 0.3296) | (0.8115; 0.9372) (0.2038; 0.1594) | (0.8992; 1.0015) (0.1707; 0.2070) | (0.8752; 0.9595) (0.1696; 0.1752) | |

100 | (0.7548; 0.8831) (0.2393; 0.2185) | (0.7883; 0.8887) (0.1605; 0.1674) | (0.9435; 1.1117) (0.1080; 0.1885) | (0.8962; 1.0487) (0.1629; 0.1792) | |

(0.9; 3) | 10 | (0.5325; 2.3452) (0.4575; 1.3839) | (0.8471; 3.0594) (0.1205; 0.6364) | (0.9126; 2.8102) (0.1976; 0.6489) | (0.8290; 2.9299) (0.2637; 1.3058) |

20 | (0.6708; 2.3101) (0.3300; 0.9030) | (0.8174; 3.1643) (0.1580; 0.7843) | (0.8827; 3.0024) (0.1854; 0.7117) | (0.8202; 2.8625) (0.2537; 0.7691) | |

50 | (0.8508; 3.2611) (0.1162; 0.6055) | (0.8560; 2.8071) (0.1108; 0.4488) | (0.8717; 3.1389) (0.1911; 0.9120) | (0.9140; 2.9894) (0.1193; 0.3799) | |

100 | (0.8442; 2.7397) (0.1149; 0.4302) | (0.8867; 2.9404) (0.0769; 0.3865) | (0.8969; 2.9607) (0.1110; 0.3967) | (0.9193; 2.9727) (0.1070; 0.4539) |

It is observed from the tables that when \(\beta >\theta\), the ML estimates of \(\theta\) and \(\beta\) are better than the others with respect to the RMSE. When \(\theta >\beta\), 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 examined 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 are tried to be modeled by the Exponential-Poisson (EP) and Exponential-Geometric (EG) distributions. In order to compare distributional models, we consider some criteria as K-S (Kolmogorow-Smirnow), \(-2LL\)(−2LogL), AIC (Akaike information criterion) and BIC (Bayesian information criterion) for the data sets.

**Data Set1** The data consist of the exceedances of flood peaks (in m^{3}/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 Choulakian and Stephens (2001) and are given in Table 2. Later on, Beta-Pareto distribution was applied to these data by Akinsete et al. (2008). Merovcia and Pukab (2014) made a comparison between Pareto and transmuted Pareto distribution. They showed that better model is the transmuted Pareto distribution (TP). Bourguignon et al. (2013) proposed Kumaraswamy (Kw) Pareto distribution (Kw-P). Tahir et al. (2014) have proposed weibull-Pareto distribution (WP) and made a comparison with Beta Exponentiated Pareto (BEP) distriubtion. Nasiru and Luguterah (2015) have proposed different type of weibull-pareto distribution (NWP). Mahmoudi (2011) concluded that the Beta-Generalized Pareto (BGP) distribution fits better to these data than the GP, BP, Weibull and Pareto models.

Exceedances of Wheaton river flood data

1.7 | 2.2 | 14.4 | 1.1 | 0.4 | 20.6 | 5.3 | 0.7 |

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 |

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 | 1.9 | 2.8 |

Model selection criteria for river flood data

Model | K-S | −2LL | AIC | BIC |
---|---|---|---|---|

T. Pareto | 0.389 | 572.401 | 578.4 | 580.9 |

Pareto | 0.332 | 606.200 | 608.2 | 610.4 |

EP | 0.199 | 574.600 | 578.6 | 583.2 |

BP | 0.175 | 567.400 | 573.4 | 580.3 |

Kw-P | 0.170 | 542.400 | 548.4 | 555.3 |

WP | – | 498.793 | 502.8 | 507.3 |

NWP | – | 158.326 | 162.3 | 166.9 |

BEP | – | 496.111 | 504.1 | 513.2 |

BGP | 0.071 | 486.200 | 496.2 | 507.6 |

EMDL | 0.116 | 503.574 | 507.6 | 512.1 |

**Data Set2** The data set given in Table 4, contains the time intervals (in days) between coal mine accidents caused death of 10 or more men. Firstly, this data set was obtained by Maguire et al. (1952). There were lots of models on this data set such as Adamidis and Loukas (1998) and Kus (2007). They suggested to use Exponential-Geometric (EG) and Exponential-Poisson (EP) distributions respectively. On the other hand, Yilmaz et al. (2016) have proposed two-component mixed exponential distribution (2MED) 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 \(\hat{\theta }=\ 0.5239\) and \(\hat{\beta }=\ 0.0025\). We have only K-S and p values which are tabulated in Table 5 to make a comparison.

The time intervals (in days) between coal mine accidents

378 | 96 | 59 | 108 | 54 | 275 | 498 | 228 | 217 | 19 | 156 |

36 | 124 | 61 | 188 | 217 | 78 | 49 | 271 | 120 | 329 | 47 |

15 | 50 | 1 | 233 | 113 | 17 | 131 | 208 | 275 | 330 | 129 |

31 | 120 | 13 | 28 | 32 | 1205 | 182 | 517 | 20 | 312 | 1630 |

215 | 203 | 189 | 22 | 23 | 644 | 255 | 1613 | 66 | 171 | 29 |

11 | 176 | 345 | 61 | 151 | 467 | 195 | 54 | 291 | 145 | 217 |

137 | 55 | 20 | 78 | 361 | 871 | 224 | 326 | 4 | 75 | 7 |

4 | 93 | 81 | 99 | 312 | 48 | 566 | 1312 | 369 | 364 | 18 |

15 | 59 | 286 | 326 | 354 | 123 | 390 | 348 | 338 | 37 | 1357 |

72 | 315 | 114 | 275 | 58 | 457 | 72 | 745 | 336 | 19 |

K-S and p values for EP, EG, 2MED and EMDL

Model | K-S | p value |
---|---|---|

EP | 0.0625 | 0.7876 |

EG | 0.0761 | 0.5524 |

2MED | 0.0578 | 0.8386 |

EMDL | 0.0752 | 0.5436 |

**Data Set3**The data set in Table 6 obtained by Kus (2007) includes the time intervals (in days) of the successive earthquakes with magnitudes greater than or equal to 6 Mw. Kus (2007) 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 \(\hat{\theta }=0.3540\), \(\hat{\beta }=0.0003\). Calculated K-S statistic for EMDL can be seen in Table 7, according to this, EMDL distribution gives the best fit to earthquake data in three models.

Time intervals of the successive earthquakes in North Anatolia fault zone

1163 | 3258 | 323 | 159 | 756 | 409 |

501 | 616 | 398 | 67 | 896 | 8592 |

2039 | 217 | 9 | 633 | 461 | 1821 |

4863 | 143 | 182 | 2117 | 3709 | 979 |

K-S and p values for EP, EG and EMDL

Model | K-S | p value |
---|---|---|

EP | 0.0972 | 0.9772 |

EG | 0.1839 | 0.3914 |

EMDL | 0.0706 | 0.9991 |

## Conclusions

In this paper we have proposed a new lifetime distribution, which is obtained by compounding the modified discrete Lindley distribution (MDL) and exponential distribution, referred to as the EMDL. Some statistical characteristics of the proposed distribution including explicit 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 maximum likelihood estimates and EM estimates of the parameters and their asymptotic variance-covariance matrix. 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 earthquakes in North Anatolia fault zone than the EP and EG.

## Declarations

### Authors' contributions

This work was carried out in cooperation among all the authors (MY, MH and SAK). All authors read and approved the final manuscript.

### Acknowledgements

This research has not been funded by any entity.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Adamidis K, Dimitrakopoulou T, Loukas S (2005) On an extension of the exponential-geometric distribution. Stat Probab Lett 73(3):259–269MathSciNetView ArticleMATHGoogle Scholar
- Adamidis K, Loukas S (1998) A lifetime distribution with decreasing failure rate. Stat Probab Lett 39(1):35–42MathSciNetView ArticleMATHGoogle Scholar
- Akinsete A, Famoye F, Lee C (2008) The beta-Pareto distribution. Statistics 42(6):547–563MathSciNetView ArticleMATHGoogle Scholar
- Bakouch HS, Jazi MA, Nadarajah S (2014) A new discrete distribution. Statistics 48(1):200–240MathSciNetView ArticleMATHGoogle Scholar
- Barreto-Souza W, Bakouch HS (2013) A new lifetime model with decreasing failure rate. Statistics 47(2):465–476MathSciNetView ArticleMATHGoogle Scholar
- Bourguignon M, Silva RB, Zea LM, Cordeiro GM (2013) The Kumaraswamy Pareto distribution. J Stat Theory Appl 12(2):129–144MathSciNetView ArticleGoogle Scholar
- Chahkandi M, Ganjali M (2009) On some lifetime distributions with decreasing failure rate. Comput Stat Data Anal 53(12):4433–4440MathSciNetView ArticleMATHGoogle Scholar
- Choulakian V, Stephens MA (2001) Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43(4):478–484MathSciNetView ArticleGoogle Scholar
- Gómez-Déniz E, Calderín-Ojeda E (2011) The discrete Lindley distribution: properties and applications. J Stat Comput Simul 81(11):1405–1416MathSciNetView ArticleMATHGoogle Scholar
- Gui W, Zhang S, Lu X (2014) The Lindley-Poisson distribution in lifetime analysis and its properties. Hacet J Math Stat 43(6):1063–1077MathSciNetMATHGoogle Scholar
- Hajebi M, Rezaei S, Nadarajah S (2013) An exponential-negative binomial distribution. REVSTAT Stat J 11(2):191–210MathSciNetMATHGoogle Scholar
- Kus C (2007) A new lifetime distribution. Comput Stat Data Anal 51(9):4497–4509MathSciNetView ArticleMATHGoogle Scholar
- Maguire BA, Pearson ES, Wynn AHA (1952) The time intervals between industrial accidents. Biometrika 39(1/2):168–180View ArticleMATHGoogle Scholar
- Mahmoudi E (2011) The beta generalized Pareto distribution with application to lifetime data. Math Comput Simul 81(11):2414–2430MathSciNetView ArticleMATHGoogle Scholar
- Merovcia F, Pukab L (2014) Transmuted pareto distribution. ProbStat Forum 07:1–11MathSciNetGoogle Scholar
- Nasiru S, Luguterah A (2015) The new Weibull-Pareto distribution. Pak J Stat Oper Res 11(1):103–114MathSciNetView ArticleMATHGoogle Scholar
- Tahir MH, Cordeiro GM, Alzaatreh A, Mansoor M, Zubair M (2016) A new Weibull-Pareto distribution: properties and applications. Commun Stat Simul Comput 45(10):3548–3567MathSciNetView ArticleGoogle Scholar
- Tahmasbi R, Rezaei S (2008) A two-parameter lifetime distribution with decreasing failure rate. Comput Stat Data Anal 52(8):3889–3901MathSciNetView ArticleMATHGoogle Scholar
- Yilmaz M, Potas N, Buyum B (2016) A classical approach to modeling of coal mine data. Chaos, complexity and leadership. Springer, New York, pp 65–73Google Scholar