Research Article Volume 10 Issue 4
^{1}Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt
^{2}Department of Statistics, Mathematics and Insurance Benha University, Egypt
Correspondence: Rania M. A. Osman, Department of Statistics, Mathematics and Insurance, Benha University, Benha 13518, Egypt
Received: September 10, 2021  Published: November 25, 2021
Citation: Ahmed AHN, Nofal ZM, Osman RMA. The sevenparameter lindley distribution. Biom Biostat Int J. 2021;10(4):166174. DOI: 10.15406/bbij.2021.10.00344
In this paper, a seven parameter Lindley Distribution (SPL) distribution is proposed as a new generalization of the basic Lindley distribution. The structural properties of the new distribution are investigated. These include the compounding representation of the distribution, reliability analysis and statistical measures. Expressions for Lorenz and Bonferroni curves and Renyi entropy as a measure for uncertainty reduction are derived. Maximum likelihood estimation is used to evaluate the parameters. The new model contains twelve lifetime distributions as special cases such as the Lindley, Quasi Lindley, gamma, and exponential distributions, among others. This model has the advantage of being capable of modeling various shapes of aging and failure criteria. Finally, the usefulness of the new model for modeling reliability data is illustrated using a real data set.
Keywords: lindley distribution, mixture, reliability analysis, moment generating function, order statistics, maximum likelihood estimation
In many applied sciences such as medicine, engineering, and finance, amongst others, modeling and analyzing lifetime data are crucial. Several lifetime distributions have been used to model such kinds of data. The quality of the procedures used in a statistical analysis depends heavily on the assumed probability model or distribution along with relevant statistical methodologies. However, there remain many important problems where the real data does not follow any of the classical or standard probability models.
The one parameter family of distributions with density function
$\text{f}\left(x;\theta \right)=\frac{{\theta}^{2}}{\theta +1}\left(1+x\right){e}^{\theta x};x>0,\theta >0,\left(1\right)$
is used by Lindley^{1} to illustrate the difference between fiducial distribution and posterior distribution. Sankaran^{2} introduced the discrete PoissonLindley distribution by combining the Poisson and Lindley distributions. Ghitany et al.^{3} have discussed various properties of (4.1). Another discrete version of this distribution has been suggested by Deniz and Ojeda^{4} with applications in count data related to insurance. Ghitany et al.^{5} obtained sizebiased and zerotruncated version of PoissonLindley distribution with various properties and applications.
The aim of this chapter is to introduce a new generalization of Lindley^{1 }distribution. This generalization is flexible enough to model different types of lifetime data having different forms of failure rate. The new distribution can accommodate both decreasing and increasing failure rates as its antecessors, as well as unimodal and bathtub shaped failure rates. The Lindley distribution is generalized by mixing. Several authors have considered versions from usual density functions by following this idea. For instance, Zakerzadeh and Dolati^{6} considered a generalized Lindley distribution as a generalization of the usual Lindley distribution, Rama and Mishra^{7} studied quasiLindley distribution, Rama et al.^{8} introduced a new distribution for generalizing the Lindley model which called Janardan distribution, and Elbatal et al.^{9} have suggested a new generalized Lindley distribution.
The introduced model will be named Seven Parameter Lindley Distribution (SPL) distribution. The basic idea behind this generalization is to use a unified approach that accommodates all the proceeding generalizations of Lindley distribution abovementioned, and to add new model that could offer a better fit to lifetime data. The proposed distribution includes nine models as special cases plus three new models (all special cases are shown in section 2). The procedure used here is based on certain mixtures of two gamma distributions with various weights. The research examines various properties of the new distribution.
The rest of paper is organized as follows: Section 2 introduces the definition of the probability density function (pdf) of the SPL distribution including its cumulative distribution function (cdf) and the submodels of the new suggested model. The reliability analysis including the survival function, the hazard (or failure) rate function, the reversed hazard rate function, the cumulative hazard rate function, and the mean residual lifetime is explored in Section 3. The statistical properties of the new distribution such as the moments, the moment generating function, and the distribution of order statistics are investigated in Section 4, with a proposed algorithm for generating random data from the new distribution in this section. Section 5 introduces Lorenz and Bonferroni curves and Renyi entropy as measures of inequality and uncertainty, respectively. Section 6 discusses the estimation of parameters by using maximum likelihood estimation. Finally, Section 7 provides an application for modeling real data sets to illustrate the performance of the new distribution.
In this section, we introduce the pdf and the cdf of the five parameter Lindley distribution and then the special cases of the SPL distribution are mentioned.
Generalization
Let
${\text{f}}_{1}\left(x;\text{\alpha},\theta \right)=\frac{{\theta}^{\alpha}}{\Gamma \left(\text{\alpha}\right)}{x}^{\text{\alpha}1}{e}^{\theta x};x>0$
(2)
and
${\text{f}}_{2}\left(x;\text{\beta},\theta \right)=\frac{{\theta}^{\beta}}{\Gamma \left(\text{\beta}\right)}{x}^{\text{\beta}1}{e}^{\theta x};x>0,\theta ,\text{\beta}>0$ (3)
be gamma $\left(\alpha ,\theta \right)$ and gamma $\left(\beta ,\theta \right)$ densities, respectively.
We define a new seven parameter Lindley distribution as a mixture of (2) and (3) with Probabilities $p=\frac{{\theta}^{\varphi}k}{{\eta}^{\sigma}+{\theta}^{\varphi}k}$ and (1p), respectively, as follows:
$\text{f}\left(x;\theta ,\alpha ,\text{\beta},\text{k},\text{\eta},\varphi ,\text{\sigma}\right)={\text{\u0440f}}_{1}\left(x;\text{\alpha},\theta \right)+(1\text{\u0440}){\text{f}}_{2}\left(x;\text{\beta},\theta \right);\u0440\in \left(0,1\right).$Therefore, the pdf of the seven Parameter Lindley Distribution (SPL) distribution, is defined as
$\text{f}\left(x;\theta ,\alpha ,\text{\beta},\text{k},\text{\eta},\varphi ,\text{\sigma}\right)=\frac{{\theta}^{\varphi}k}{{\eta}^{\sigma}+{\theta}^{\varphi}k}[\frac{k{\theta}^{\varphi 1}{(\theta x)}^{\alpha 1}}{\Gamma \left(\alpha \right)}+\frac{{\eta}^{\sigma}{(\theta x)}^{\beta 1}}{\theta \Gamma \left(\beta \right)}{e}^{\theta x},x>0$We note that $\text{f}\left(x;\theta ,\alpha ,\text{\beta},\text{k},\text{\eta},\varphi ,\text{\sigma}\right)$ incorporates seven parameters namely $\theta >0,\text{\alpha}>0,\text{\beta}>0,\text{k}\ge 0,\varphi >0,\text{\sigma}>0$ and subject to $k$ and $\eta $ are not allowed to be simultaneously zeros. The corresponding cumulative distribution function (cdf) of the SPLD is
$\u03dc\left(x;\theta ,\alpha ,\text{\beta},\text{k},\text{\eta},\varphi ,\text{\sigma}\right)=\frac{\left[{\theta}^{\varphi}k{\gamma}_{\alpha}\left(\theta x\right)+{\eta}^{\sigma}{\gamma}_{\text{\beta}}\left(\theta x\right)\right]}{{\eta}^{\sigma}+{\theta}^{\varphi}k};x>0,\theta ,\alpha ,\text{\beta}>0,\text{k},\text{\eta}\ge 0$where
${Y}_{\alpha}\left(b\right)=\frac{Y\left(a,b\right)}{\Gamma \left(\alpha \right)}\underset{0}{\overset{b}{{\displaystyle \int}}}{t}^{a1}{e}^{t}dt$is known as the lower incomplete gamma function ratio. Also, the upper incomplete gamma function ratio is given by
${\Gamma}_{a}\left(b\right)=\frac{\Gamma \left(a,b\right)}{\Gamma \left(a\right)}=\frac{1}{\Gamma \left(a\right)}\underset{b}{\overset{\infty}{{\displaystyle \int}}}{t}^{a1}{e}^{t}dt$Figures 1&2 illustrate some of the possible shapes of the pdf and the cdf, respectively, of the SPLD for different values of the parameters $\theta ,\alpha ,\text{\beta},\text{k},\varphi ,\text{\sigma}$ and $\eta $ chosen from the ranges specified in Equation (4).
Submodels of the SPLD
It is clear that the seven parameter Lindley distribution is very flexible. Assigning particular numerical values of some subsets of the parameters yields several special generalizations of Lindley distribution. The special cases include nine distributions namely; the new generalized Lindley distribution (NGLD) introduced by Elbatal et al.,^{9} generalized Lindley distribution (GLD) introduced by Zakerzadeh and Dolati,^{6} quasi Lindley distribution (QLD) introduced by Rama and Mishra,^{7} Lindley distribution (LD) by Lindley,^{1} Erlang distribution, Janardan distribution introduced by Rama et al.,^{8} gamma distribution, the exponential distribution (ED), and Chisquare distribution. In addition to yield all the previous distributions, our generalization model allowed us to create new three distributions namely, the 4parameter Lindley type I (4p L type I) distribution, the 4parameter Lindley type II (4p L type II) distribution and the 2parameter Lindley (2p L) distribution.
In this section, we present the survival function, the hazard rate function, the reversed hazard rate function, the cumulative hazard rate function and the mean residual lifetime for the seven parameter Lindley distribution.
The survival function
The survival function $R\left(x\right)$ which is the probability of an item not failing prior to some time is defined by $R\left(x\right)=1F\left(x\right)$ . Therefore, the survival function of the SPL distribution is given
$R\left(x\right)=1\frac{1}{{\eta}^{\sigma}+{\theta}^{\varphi}k}\left[{\theta}^{\varphi}k{\gamma}_{\alpha}\left(\theta x\right)+{\eta}^{\sigma}{\gamma}_{\beta}\left(\theta x\right)\right];x>0$ (6)
The hazard rate function
The other characteristic of interest of a random variable is the hazard rate function, $h\left(x\right)$ the hazard rate function of the SPLD is given by
$h\left(x\right)=\frac{{\theta}^{2}\left[\frac{k{\theta}^{\varphi 1}{(\theta x)}^{\alpha 1}}{\Gamma \left(\alpha \right)}+\frac{{\eta}^{\sigma}{(\theta x)}^{\beta 1}}{\theta \Gamma \left(\beta \right)}\right]{e}^{\theta x}}{{\eta}^{\sigma}+{\theta}^{\varphi}k\left[{\theta}^{\varphi}k{\gamma}_{\alpha}\left(\theta x\right)+{\eta}^{\sigma}{\gamma}_{\beta}\left(\theta x\right)\right]}$ (7)
We note that $h\left(x\right)$ might be constant, increasing, decreasing, or bathtub shaped depending on the values of the parameters involved. For example, if $\eta =0$ and $\alpha =1$ then $h\left(x\right)=\theta $ , a constant, while for $\alpha \ge 1$ and $\beta \ge 2$ it will be increasing,$\beta \ge 2\alpha \le 1,\beta \le 2,\eta =0$ and it is going to be decreasing if , and the bathtubtype curve appears for $\alpha <1,\beta <2,$ and $\eta >0$ .
The next result describes some particular cases for the hazard rate function arising from the five parameter Lindley distribution by assigning relevant values of the parameters.
Theorem 1:
The hazard rate function of the particular cases from the five parameter Lindley distribution are given by
Proof:
(i)If $\alpha =1,\beta =2$ , and $\eta =\theta $ the failure rate is same as the $QLD\left(k,\theta \right).$ $h\left(x\right)=\frac{{\theta}^{2}\left(1+x\right)}{\theta +\theta x+1}.$
(ii) If $\alpha =1,\beta =2$ , and $\eta =\theta $ the failure rate is same as the $QLD\left(k,\theta \right).$ $h\left(x\right)=\frac{\theta \left(k+\theta x\right)}{k+\theta x+1}.$
(iii) If $\alpha =1,\beta =2$ , and $\eta =\theta $ the failure rate is same as the $ED(\theta )$ /$h\left(x\right)=\theta $ .
(iv) If $\alpha =k=1,\theta =\left(\theta /\eta \right)$ , and $\beta =2$ the failure rate is same as the $JD\left(\theta ,\eta \right)$ .
$h\left(x\right)=\frac{{\theta}^{2}\left(1+\eta x\right)}{\eta \left(\theta +{\eta}^{2}\right)+\theta {\eta}^{2}x}.$(v) if $k=\eta =1$ the failure rate is same as the $NGLD\left(\theta ,\alpha ,\beta \right)$ .
$h\left(x\right)=\frac{{\theta}^{2}\left[\frac{{(\theta x)}^{\alpha 1}}{\Gamma \left(\alpha \right)}+\frac{{(\theta x)}^{\beta 1}}{\theta \Gamma \left(\beta \right)}\right]{e}^{\theta x}}{1+\theta \left[\theta {Y}_{\alpha}\left(\theta x\right)+{Y}_{\text{\beta}}\left(\theta x\right)\right]}.$The reversed hazard rate function
The reversed hazard rate function $r\left(x\right)$ , of a random variable distributed according to the Spl after some simplifications is given by
(8)
The cumulative hazard rate function
Many generalized models have been proposed in reliability literature through the relationship between the reliability function and its cumulative hazard rate function given by .The cumulative hazard rate function of the SPL distribution is given by
(9)
where is the total number of failure or deaths over an interval of time, and is a nondecreasing function of satisfying.
The mean residual lifetime
The additional lifetime given that the component has survived up to time is called the residual life function of the component, the n^{th} expectation of the random variable that represent the remaining lifetime is called the mean residual lifetime (MRL) and is given by
or equivalently
While the hazard rate function provides information about a small interval after time (just after ), the MRL considers information about the whole interval after (all after ). The MRL as well as the hazard rate function or the reliability function is very important as each of them can be used to characterize a unique corresponding lifetime distribution.
The MRL function for SPL random variable is given by
(10)
The MRL function given in Equation (4.10) satisfies the following properties.
where is the first noncentral moment of the SPL Distribution (the mean of the distribution).
This section investigates the statistical properties of the SPL Distribution as the moments (non central and central), the moment generating function and an algorithm for random number generating.
The moment generating function
The following theorem gives the moment generating function (mgf) of SPL Distribution
( ).
Theorem 2:
If has the SPL Distribution ( ), then the mgf of say is given as follows
(11)
Proof:
using the expansion , one has
This completes the proof.
Depending on the previous theorem, we can conclude the basic statistical properties as follows:
(i) The noncentral moments are the coefficients of . In Equation (11), for . Therefore, the mean and the variance of the SPL random variable are, respectively, given by
(12)
and
(13)
Where is the second noncentral moment which is given by
(14)
The central moments can be obtained easily from the moments through the relation
Where
Then the central moments of the SPL distribution are given by
(15)
(iii) Finally, the coefficient of variation , the coefficient of skewness , and the coefficient of kurtosis of SPLdistribution are, respectively, obtained according the following relations
Distribution of order statistics
Let denote independent random variables from a distribution function with pdf , and then the pdf of (the order sample arrangement) is given by
(19)
Using Equations (4) and (5) into Equation (19), then the pdf of according to the SPL distribution is given by
(20)
Hence, the pdf of the largest order statistic and the smallest order statistic are, respectively, given by
(21)
and
(22)
Random variates generation
The probability density function of the SPL distribution can be expressed in terms of the gamma density function as follows
To generate random variates , for from SPL , we can use the following algorithm:
Set
In this section Lorenz and Bonferroni curves are introduced as measures of inequality. Also, Renyi entropy will be mentioned as an important measure of uncertainty.
Lorenz and bonferroni curves
Lorenz and Bonferroni curves are the most widely used inequality measures in income and wealth distribution.^{10}
In fact, Lorenz and Bonferroni curves are depending on the lengthbiased distribution with pdf defined by
(23)
Where is the pdf of the base distribution with mean
Accordingly, Lorenz and Bonferroni curves denoted by and respectively, defined by
(24)
where is the cdf of the lengthbiased distribution. Now, we shall derive the expressions of and based on and for SPLD.
It is easily shown that the pdf of the lengthbiased distribution can be obtained as Follows
(25)
With cdf defined by
(26)
It follows from (12), (24), and (26) that and are
(27)
and
(28)
Renyi entropy
If is a random variable having an absolutely continuous cdf and pdf , then the basic uncertainty measure for distribution (called the entropy of ) is defined as . Statistical entropy is a probabilistic measure of uncertainty or ignorance about the outcome of a random experiment and is a measure of a reduction in that uncertainty. Abundant entropy and information indices, among them the Renyi entropy, have been developed and used in various disciplines and contexts. Information theoretic principles and methods have become integral parts of probability and statistics and have been applied in various branches of statistics and related fields.
Renyi entropy is an extension of Shannon entropy. Renyi entropy of the SPLD is defined to be
(29)
Where and Renyi entropy tends to Shannon entropy as . Now,
(30)
Using then one has
(31)
Using the expansion: one can have
(32)
(33)
Using the gamma function to evaluate the integral in (33) and collecting the entire above evaluations then substitute into (29), the Renyi entropy of the SPLD can be written as
Where is a constant as
Estimation of the parameters
In this section, we use the method of likelihood to estimate the parameters involved and use them to create confidence intervals for the unknown parameters.
Let be a sample size from SPL distribution. Then the likelihood function is given by
Then,
(35)
Hence, the loglikelihood function becomes
(36)
Therefore, the maximum likelihood estimators (MLEs) of and are derived from the derivatives of .
They should satisfy the following equations
(37)
(38)
(39)
(40)
(41)
(42)
(43)
Where is the diagamma function, and it is defined as
To solve the equations (37) through (43), it is usually more convenient to use nonlinear optimization algorithms such as quasiNewton algorithm to numerically maximize the loglikelihood function. In order to compute the standard errors and asymptotic confidence intervals we use the usual large sample approximation, in which the MLEs can be treated as being approximately trivariate normal.
Hence as , the asymptotic distribution of the MLE is given by, see Zaindin et al.^{6}
Where ( ), and
is the approximate variancecovariance matrix with its elements obtained from
By solving this inverse dispersion matrix, these solutions will yield the asymptotic variances and co variances of these MLEs for and .
Approximate confidence intervals for and can be determinedas
, , ,
Where is the upperpercentile of the standard normal distribution.
In this section, we use a real data set to compare the fits of the SPL distribution with three submodels. In each case, the parameters are estimated by maximum likelihood as described in Section 4.6, using the R software.
The data set consist of uncensored data set from Nichols and Padgett on the breaking stress of carbon fibers (in Gba). The data are given below:
3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 3.56,
4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 1.57,
2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85,
1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03,
1.89, 2.88, 2.82, 2.05, 3.65, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96,
2.35, 2.55, 2.59, 2.03, 1.61, 2.12, 3.15, 1.08, 2.56, 1.80, 2.53.
The summary of the above data is given by
Units Minimum Ist Qu. Median Mean 3rd Qu. Maximum
66 0,390 2,178 2,835 2,760 3.278 4.900
In order to compare the two distribution models, we consider criteria like KS (Kolmogorov Smirnov), , AIC (Akaike information criterion), AICC (corrected Akaike information criterion), and BIC (Bayesian information criterion) for the data set. The better distribution corresponds to smaller KS, , AIC and AICC values:
and
Where denotes the loglikelihood function evaluated at the maximum likelihood estimates, is the number of parameters, and is the sample size.
Also, for calculating the values of KS we use the sample estimates of and b. Table 1&2 shows the parameter estimation based on the maximum likelihood and least square estimation, and gives the values of the criteria AIC, AICC, BIC, and KS test.
Distribution 
Parameters 
Author 

$\theta $ 
$\alpha $ 
$\beta $ 
$k$ 
$\eta $ 
$\sigma $ 
$\varnothing $ 

Gamma 


1 
1 
0 
1 
1 
Brown & Flood11 
ED 

1 
1 
1 
0 
1 
1 
Steffensen12 
LD 

1 
2 
1 
1 
1 
1 
Lindley1 
Erlang 

$v,v\ufffd\mathbb{N}$ 
1 
1 
0 
1 
1 
A. K. Erlang13 
QLD 

1 
2 

$\theta $ 
1 
1 
Rama & Mishra7 
GLD 


$\alpha +1$ 
1 

1 
1 
Zakerzadeh&Dolati6 
Janardan 
$\theta /\eta $ 
1 
2 
1 

1 
1 
Rama et al.8 
NGLD 



1 
1 
1 
1 
Elbatal et al.9 
Chisquare 
1/2 
$v/2,v\ufffd\mathbb{N}$ 
1 
1 
0 
1 
1 
Fisher14 
4p L type I 



1 

1 
1 
New 
4p L type II 




1 
1 
1 
New 
2p L 

1 
2 

1 
1 
1 
New 
5p L 
$\theta $ 
$\alpha $ 
$\beta $ 
$k$ 
1 
1 
New 
Table 1 The special cases of the SPL distribution

SPL 
FPLD 
Lindley 
Gamma 
exponential 


coef 
std.e 
coef 
std.e 
coef 
std.e 
coef 
std.e 
coef 
std.e 
alpha 
4.918145682 
3.1418 
4.918019 
2.273544 
4.9181 
2.2736 
7.48803 
1.27552 
 
 
beta 
13.32661771 
2.273564 
13.32648 
3.141746 
13.3266 
3.1418 
2.7135 
0.47806 
 
 
theta 
4.608695814 
1.033154 
4.608644 
1.033136 
4.6087 
1.0332 
 
 
0.362379 
0.044606 
phai 
0.033886236 
0.452362 
 
 
 
 
 
 
 
 
k 
1.471738523 
0.056224 
0.103372 
0.080925 
 
 
 
 
 
 
eta 
3.406750687 
0.187052 
6.104362 
0.001398 
59.0538 
46.2436 
 
 
 
 
sigma 
2.438262765 
0.236172 
 
 
 
 
 
 
 
 
AIC 
177.3931 
181.3931 
179.3931 
186.3351 
267.9887 

BIC 
182.0655 
192.3413 
188.1517 
190.7144 
270.1784 

AICC 
175.462 
182.3931 
180.0488 
186.5256 
268.0512 

HQIC 
171.3364 
185.7192 
182.854 
188.0656 
268.8539 

KS 
0.07 
0.070003 
0.0713806 
0.13285 
0.35811 

Pvalue 
0.9031 
0.9028 
0.8569 
0.1945 
8.89E08 
The values in Table 2 indicate that the SPL distribution leads to a better fit over all the other models.
A density plot compares the fitted densities of the models with the empirical histogram of the observed data (Figures 35). The fitted density for the SPL model is closer to the empirical histogram than the fits of the other models.^{1522}
Figure 3 Increasing, decreasing, constant, bathtub and upsidedown shapes for the hazad rate function of the SPLD.
There has been a great interest among statisticians and applied researchers in constructing flexible lifetime models to facilitate better modeling of survival data. Consequently, a significant progress has been made towards the generalization of some wellknown lifetime models and their successful application to problems in several areas. In this paper, we introduce a new fiveparameter distribution obtained using the idea of mixture of distributions. We refer to the new model as the Five Parameter Lindley Distribution (FPLD) and study some of its mathematical and statistical properties. We provide the pdf, the cdf and the hazard rate function of the new model and explicit expressions for the moments. The model parameters are estimated by the method of maximum likelihood. The new model is compared with three lifetime models and provides consistently better fit than them. We hope that the proposed distribution will serve as an alternative model to other models available in the literature for modeling positive real data in many areas such as engineering, survival analysis, hydrology and economics.
©2021 Ahmed, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work noncommercially.
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