# Mixture of Lindley and Lognormal Distributions: Properties, Estimation, and

Authors: A. S. Al-Moisheer
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Finite mixture models provide a flexible tool for handling heterogeneous data. This paper
introduces a new mixture model which is the mixture of Lindley and lognormal distributions
(MLLND). First, the model is formulated, and some of its statistical properties are
studied. Next, maximum likelihood estimation of the parameters of the model is considered,
and the performance of the estimators of the parameters of the proposed models is
evaluated via simulation. Also, the flexibility of the proposed mixture distribution is
demonstrated by showing its superiority to fit a well-known real data set of 128 bladder
cancer patients compared to several mixture and nonmixture distributions. The Kolmogorov
Smirnov test and some information criteria are used to compare the fitted models to the
real dataset. Finally, the results are verified using several graphical methods.


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## Body

## 1. Introduction

In most reliability applications, data is modeled by a single parametric distribution.
However, in many situations, a population can be divided into a number of subpopulations
each representing a different type of failure. Finite mixture models play an important role
in modeling such heterogeneous data. Applications of mixture models are especially in
clustering and classification, see, for example, Everitt and Hand [1], McLachlan and Peel [2],
McLachlan and Basford [3], Titterington et al. [4], Lindsay [5],
McLachlan and Krishnan [6], Al-Moisheer et al. [7, 8], and
Al-Moisheer [9, 10]. In this paper, we will introduce a finite mixture of Lindley and lognormal
distributions (MLLND). The motivation of suggesting this mixture comes from the importance
of its component distributions. The one-parameter Lindley distribution was introduced by
Lindley [11, 12], and then Ghitany et al. [13]
illustrated its importance in lifetesting and reliability applications. With regards to one
component lognormal distribution, it has found important applications in a wide variety of
fields; (see, Kim and Yum [14] and Lin et al. [15]). In the literature, work has been done on mixture
models having the Lindley distribution as one of its components, see, for example,
Al-Moisheer et al. (Al-Moisheer et al. [16],
Al-Moisheer et al. [17]) for the mixture of two
one-parameter Lindley distribution and the mixture of Lindley and inverse Weibull
distributions, respectively. Also, Daghestani et al. [18] considered the mixture of Lindley and Weibull distributions. In this paper, we
will introduce a new mixture distribution, namely, the finite mixture of Lindley and
lognormal distribution (MLLND). This paper is organized as follows. In Section 2, we obtain the new model and derive some of its
properties. In Sections 3 and 4, we derive the probability density function of the order statistics
and the equations required to obtain the maximum likelihood estimation of the model
parameters. In Section 5, the flexibility of the
proposed model is illustrated by showing its ability to provide the best fit for a
well-known real data set compared to six competitive models. Finally, In Section 6, we draw a conclusion.


## 2. Model Formulation and Some Properties

The MLLND has the following probability density function (pdf)(1)fx;Θ=p1f1x;Θ1+p2f2x;Θ2,0<p1,p2<1,p1+p2=1,x≥0,whereas the pdf of the Lindley component is given
by(2)f1x;Θ1=θ2θ+11+xe−θx,x≥0,θ>0.The pdf of the lognormal component is given by(3)f2x;Θ2=12πσxe−12logx−μσ2,x≥0,−∞<μ<∞,σ>0,Θ=p1,θ,μ,σ, Θ1=θ, and Θ2=μ,σ.Evidently, the cumulative distribution function (cdf) of the MLLND is given
by(4)Fx;Θ=p1F1x;Θ1+p2F2x;Θ2,where(5)F1x;Θ1=1−θ+1+θxθ+1e−θx,x≥0,θ>0,F2x;Θ2=Φlogx−μσ,x≥0,−∞<μ<∞,σ>0,with Φ. referring to the cdf of the standard normal distribution.Ghitany et al. [13] and Shanker et al. [19] displayed some properties of the LD in (2), while properties of the LND in (3) were given, for example, by Crow and Shimizu
[20] and Johnson et al. [21]. In this section, we introduce some properties of the MLLND by mixing
the results of the LD and LND.


### 2.1. Mean and Variance

The mean of the MLLND in (1) is simply
given by(6)EX=p1θ+2θθ+1+p2eμ+σ22,forx≥0,θ,σ>0,−∞<μ<∞,while the variance is given by(7)VarX=p1θ2θ+12θ+3−p1θ+2θ+p2e2μ+σ2eσ2−p2−2p1p2θ+2θθ+1eμ+σ22,forx≥0,θ,σ>0,−∞<μ<∞.Also, therth moments of the MLLND is given by(8)EXr=p1θ+r+1Γr+1θrθ+1+p2erμ+r2σ22,forr=1,2,3,⋯.


### 2.2. Mode and Median

It can be shown that the equations for obtaining the modes and median of the MLLND,
respectively, are(9)p1θ2e−θxθ+11−θ1+x−p212πσ3x2e−12logx−μσ2σ2+logx−μ=0,and(10)p11−θ+1+θxe−θxθ+1+p2Φlogx−μσ=0.5.Figure1(a) shows the pdf of the MLLND unimodal
case at the choice of parameters Θ=p1=0.5,θ=0.95,μ=0.25,σ=0.25 with the values of mode and median (1.1623, 1.2712),
respectively. Also, Figure 1(b) shows the shape of
the pdf in the MLLND bimodal case at the choice of parameters Θ=p1=0.5,θ=2.95,μ=0.85,σ=0.25 with the values of mode and median ((1.4370, 2.1430),
1.4542), respectively. For plotting the pdf of the LD and LND in R, we use the function dlindley() and dlnorm(), respectively.
The package rootSolve() in R is used for modes and median of the MLLND.Figure 1
pdf plot of MLLND with (a)Θ=p1=0.5,θ=0.95,μ=0.25,σ=0.25 (b) Θ=p1=0.5,θ=2.95,μ=0.85,σ=0.25.
(a)(b)


### 2.3. Reliability and Failure Rate Functions

The reliability function of the MLLND is given by(11)Rx=p1θ+1+θxθ+1e−θx+p21−Φlogx−μσ,x≥0.By using (3) and (4), the hazard rate function (HRF) of the
MLLND is given by(12)rx=p1θ2/θ+11+xe−θx+p21/2πσxe−1/2logx−μ/σ2p1θ+1+θx/θ+1e−θx+p21−Φlogx−μ/σ,x≥0,which can be written by using the result in AL-Hussaini and
Sultan [22], as(13)rx=hxr1x+1−hxr2x,where(14)hx=11+p2R2x/p1R1x,rix=fixRix,i=1,2,R1x=θ+1+θxθ+1e−θxandR2x=1−Φlogx−μσ.The HRF of the MLLND in (12) achieves the
following limits.Lemma 1.
(15)limx⟶0rx=p1θ2θ+1,(16)limx⟶∞rx=0.Proof.
To prove the first part of the limits, using the equation (13), we get(17)limx⟶0hx=p1,andlimx⟶01−hx=p2.Then, we have(18)limx⟶0r1x=p1θ2θ+1,andlimx⟶0r2x=0,and thus (15)
is proved.Also, to prove (16),(19)limx⟶∞rx=limx⟶∞fx1−Fx=−limx⟶∞f′xfx,f′x=dfxdx,(20)where,f′x=p1f′1x+p2f′2x.It follows from (19) and (20) that(21)limx⟶∞rx=−limx⟶∞p1f′1x+p2f′2xp1f1x+p2f2x.It follows that(22)limx⟶∞rx=0.For more details, see Sultan [23], Sultan and
Al-Moisheer [24], and Al-Moisheer et al.
(Al-Moisheer et al. [16]).


### 2.4. Skewness, Kurtosis, and the Coefficient of Variation

The coefficient of skewnessSk, the coefficient of kurtosis Ku, and the coefficient of variation Cv of the MLLND distribution are given by,
respectively,(23)Sk=EX3−3μσ2−μ3σ3,=6p1θ+4θ3θ+1−p1θ+2θθ+1+p2eμ+σ223+p2e3μ+9σ22−32p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ2p1θ+2θθ+1+p2eμ+σ22/2p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ232,Ku=EX4−4μEX3+6μ2σ2+3μ4σ4,=24p1θ+5θ4θ+1−46p1θ+4θ3θ+1+p2e3μ+9σ22p1θ+2θθ+1+p2eμ+σ22+3p1θ+2θθ+1+p2eμ+σ224+p2e4μ+8σ2+6p1θ+2θθ+1+p2eμ+σ2222p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ2/2p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ22,(24)Cv=σμ=2p1θ+3/θ2θ+1−p1θ+2/θθ+1+p2eμ+σ2/22+p2e2μ+σ2p1θ+2/θθ+1+p2eμ+σ2/2.Some values of the mean, standard deviation, coefficient of variation, coefficient
skewness, and coefficient kurtosis for the MLLND distributions are obtained for the two
choices of the parameterθ and different values of the parameter
μ. From the results which are presented in Tables 1 and 2, we
note that as μ increases, both the mean and the standard deviation
increase, whereas the values of the other measures remain fairly stable.Table 1
Results of the mean, standard deviation, coefficient of variation, skewness, and
kurtosis for the MLLND at the choice (p1=0.5;θ=0.1;σ=3).
μMeanStand.Dev.CvSkKu1124.01315574.6125.5891.03165×1068.62356×10152334.23742336.1126.6651.03165×1068.62354×10153905.688115081127.0651.03165×1068.62353×101542459.05312824127.2131.03165×1068.62353×101556681.53850343127.2681.03165×1068.62353×1015618159.42.31147×106127.2881.03165×1068.62353×1015749359.66.28324×106127.2951.03165×1068.62353×101581341701.70796×107127.2981.03165×1068.62353×101593647104.64272×107127.2991.03165×1068.62353×1015109913811.26202×108127.2991.03165×1068.62353×1015Table 2
Results of the mean, standard deviation, coefficient of variation, skewness, and
kurtosis for the MLLND at the choice (p1=0.5;θ=0.5;σ=3).
μMeanStand.Dev.CvSkKu1131.89115574.5118.0861.03166×1068.62368×10152342.11642336.0123.7471.03165×1068.62359×10153913.567115081125.9691.03165×1068.62355×101542466.93312824126.8071.03165×1068.62354×101556689.41850343127.1181.03165×1068.62353×1015618167.32.31147×106127.2331.03165×1068.62353×1015749367.46.28324×106127.2751.03165×1068.62353×101581341781.70796×107127.2901.03165×1068.62353×101593647184.64272×107127.2961.03165×1068.62353×1015109913891.26202×108127.2981.03165×1068.62353×1015


## 2.1. Mean and Variance

The mean of the MLLND in (1) is simply
given by(6)EX=p1θ+2θθ+1+p2eμ+σ22,forx≥0,θ,σ>0,−∞<μ<∞,while the variance is given by(7)VarX=p1θ2θ+12θ+3−p1θ+2θ+p2e2μ+σ2eσ2−p2−2p1p2θ+2θθ+1eμ+σ22,forx≥0,θ,σ>0,−∞<μ<∞.Also, therth moments of the MLLND is given by(8)EXr=p1θ+r+1Γr+1θrθ+1+p2erμ+r2σ22,forr=1,2,3,⋯.


## 2.2. Mode and Median

It can be shown that the equations for obtaining the modes and median of the MLLND,
respectively, are(9)p1θ2e−θxθ+11−θ1+x−p212πσ3x2e−12logx−μσ2σ2+logx−μ=0,and(10)p11−θ+1+θxe−θxθ+1+p2Φlogx−μσ=0.5.Figure1(a) shows the pdf of the MLLND unimodal
case at the choice of parameters Θ=p1=0.5,θ=0.95,μ=0.25,σ=0.25 with the values of mode and median (1.1623, 1.2712),
respectively. Also, Figure 1(b) shows the shape of
the pdf in the MLLND bimodal case at the choice of parameters Θ=p1=0.5,θ=2.95,μ=0.85,σ=0.25 with the values of mode and median ((1.4370, 2.1430),
1.4542), respectively. For plotting the pdf of the LD and LND in R, we use the function dlindley() and dlnorm(), respectively.
The package rootSolve() in R is used for modes and median of the MLLND.Figure 1
pdf plot of MLLND with (a)Θ=p1=0.5,θ=0.95,μ=0.25,σ=0.25 (b) Θ=p1=0.5,θ=2.95,μ=0.85,σ=0.25.
(a)(b)


## 2.3. Reliability and Failure Rate Functions

The reliability function of the MLLND is given by(11)Rx=p1θ+1+θxθ+1e−θx+p21−Φlogx−μσ,x≥0.By using (3) and (4), the hazard rate function (HRF) of the
MLLND is given by(12)rx=p1θ2/θ+11+xe−θx+p21/2πσxe−1/2logx−μ/σ2p1θ+1+θx/θ+1e−θx+p21−Φlogx−μ/σ,x≥0,which can be written by using the result in AL-Hussaini and
Sultan [22], as(13)rx=hxr1x+1−hxr2x,where(14)hx=11+p2R2x/p1R1x,rix=fixRix,i=1,2,R1x=θ+1+θxθ+1e−θxandR2x=1−Φlogx−μσ.The HRF of the MLLND in (12) achieves the
following limits.Lemma 1.
(15)limx⟶0rx=p1θ2θ+1,(16)limx⟶∞rx=0.Proof.
To prove the first part of the limits, using the equation (13), we get(17)limx⟶0hx=p1,andlimx⟶01−hx=p2.Then, we have(18)limx⟶0r1x=p1θ2θ+1,andlimx⟶0r2x=0,and thus (15)
is proved.Also, to prove (16),(19)limx⟶∞rx=limx⟶∞fx1−Fx=−limx⟶∞f′xfx,f′x=dfxdx,(20)where,f′x=p1f′1x+p2f′2x.It follows from (19) and (20) that(21)limx⟶∞rx=−limx⟶∞p1f′1x+p2f′2xp1f1x+p2f2x.It follows that(22)limx⟶∞rx=0.For more details, see Sultan [23], Sultan and
Al-Moisheer [24], and Al-Moisheer et al.
(Al-Moisheer et al. [16]).


## 2.4. Skewness, Kurtosis, and the Coefficient of Variation

The coefficient of skewnessSk, the coefficient of kurtosis Ku, and the coefficient of variation Cv of the MLLND distribution are given by,
respectively,(23)Sk=EX3−3μσ2−μ3σ3,=6p1θ+4θ3θ+1−p1θ+2θθ+1+p2eμ+σ223+p2e3μ+9σ22−32p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ2p1θ+2θθ+1+p2eμ+σ22/2p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ232,Ku=EX4−4μEX3+6μ2σ2+3μ4σ4,=24p1θ+5θ4θ+1−46p1θ+4θ3θ+1+p2e3μ+9σ22p1θ+2θθ+1+p2eμ+σ22+3p1θ+2θθ+1+p2eμ+σ224+p2e4μ+8σ2+6p1θ+2θθ+1+p2eμ+σ2222p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ2/2p1θ+3θ2θ+1−p1θ+2θθ+1+p2eμ+σ222+p2e2μ+σ22,(24)Cv=σμ=2p1θ+3/θ2θ+1−p1θ+2/θθ+1+p2eμ+σ2/22+p2e2μ+σ2p1θ+2/θθ+1+p2eμ+σ2/2.Some values of the mean, standard deviation, coefficient of variation, coefficient
skewness, and coefficient kurtosis for the MLLND distributions are obtained for the two
choices of the parameterθ and different values of the parameter
μ. From the results which are presented in Tables 1 and 2, we
note that as μ increases, both the mean and the standard deviation
increase, whereas the values of the other measures remain fairly stable.Table 1
Results of the mean, standard deviation, coefficient of variation, skewness, and
kurtosis for the MLLND at the choice (p1=0.5;θ=0.1;σ=3).
μMeanStand.Dev.CvSkKu1124.01315574.6125.5891.03165×1068.62356×10152334.23742336.1126.6651.03165×1068.62354×10153905.688115081127.0651.03165×1068.62353×101542459.05312824127.2131.03165×1068.62353×101556681.53850343127.2681.03165×1068.62353×1015618159.42.31147×106127.2881.03165×1068.62353×1015749359.66.28324×106127.2951.03165×1068.62353×101581341701.70796×107127.2981.03165×1068.62353×101593647104.64272×107127.2991.03165×1068.62353×1015109913811.26202×108127.2991.03165×1068.62353×1015Table 2
Results of the mean, standard deviation, coefficient of variation, skewness, and
kurtosis for the MLLND at the choice (p1=0.5;θ=0.5;σ=3).
μMeanStand.Dev.CvSkKu1131.89115574.5118.0861.03166×1068.62368×10152342.11642336.0123.7471.03165×1068.62359×10153913.567115081125.9691.03165×1068.62355×101542466.93312824126.8071.03165×1068.62354×101556689.41850343127.1181.03165×1068.62353×1015618167.32.31147×106127.2331.03165×1068.62353×1015749367.46.28324×106127.2751.03165×1068.62353×101581341781.70796×107127.2901.03165×1068.62353×101593647184.64272×107127.2961.03165×1068.62353×1015109913891.26202×108127.2981.03165×1068.62353×1015


## 3. Order Statistics

LetX1,X2,⋯,Xm be a random sample of size m selected from a distribution with pdf
fx and cdf Fx, and also let X1:m<X2:m<⋯<Xm:m be the corresponding order statistics. The pdf of the
rth order statistics that say Xr:m is given by(25)fr:mx=m!r−1!m−r!Fxr−11−Fxm−rfx,−∞<x<∞,and the corresponding cdf is given by(26)Fr:mx=∑j=rmmjFjx1-Fxm-j=∑j=rm∑k=0m-jmjm−jk-1kFj+kx.Therefore, using (25) and (26), the pdf and cdf of the
rth order statistics, are, respectively, given
by(27)fr:mx,θ,σ=m!p1θ2x+1e−θx/θ+1+p2e−1/2logx−μ/σ2/2πσxWΓjΓm−j+1,where(28)W=p22erfcμ−logx2σ+p11−e−θxθ+θx+1θ+1j−1×−p22erfcμ−logx2σ+p1e−θxθ+θx+1θ+1−1+1m−j,(29)Fr:mx,θ,σ=m!δj−mp2/2erfcμ−logx/2σ+p11−e−θxθ+θx+1/θ+1j+1j+1ΓjΓm−j+1.Accordingly, the density functions of the minimum and maximum order statistics,
respectively, are given by(30)f1:mx=mp1θ2x+1e−θxθ+1+p2e−1/2logx−μ/σ22πσx,×−p22erfcμ−logx2σ+p1−1−e−θxθ+θx+1θ+1+1m+1,(31)fm:mx=mp1θ2x+1e−θxθ+1+p2e−1/2logx−μ/σ22πσx,×p22erfcμ−logx2σ+p11−e−θxθ+θx+1θ+1m−1.


## 4. Maximum Likelihood Estimation

The likelihood function (LF) for the MLLND in (1) is given by(32)LΘ=∏j=1np1f1xj;Θ1+p2f2xj;Θ2,where Θ1=θ and Θ2=μ,σ. By differentiating the log LF with respect to the model parameters
Θ=p1,θ,μ,σ, respectively, we get the following
equations(33)∑j=1nωxj;Θ=0,∑j=1np1ψ1xj;Θη1xj;Θ=0,∑j=1np2ϕ1xj;Θη2xj;Θ=0,∑j=1np2ϕ2xj;Θη2xj;Θ=0,where ωxj;Θ, ψ1xj;Θ, η1xj;Θ, η2xj;Θ, ϕ1xj;Θ, and ϕ2xj;Θ are as follows:(34)ωxj;Θ=f1xj;Θ1−f2xj;Θ2fxj;Θ,ψ1xj;Θ=2θ−1−xj−1,η1xj;Θ=f1xj;Θ1fxj;Θ,η2xj;Θ=f2xj;Θ2fxj;Θ,ϕ1xj;Θ=logxj−μσ2,ϕ2xj;Θ=logxj−μ2σ−3−σ−1,and fxj;Θ, f1xj;Θ1, and f2xj;Θ2 are as in 1,2,3, respectively. The MLEs of the parameters can be obtained by
solving systems of nonlinear Eqs. given in (33) using the package nleqslv() in R.The numerical results are obtained in Table3 for
two different combinations of the parameters. The first one corresponds to a unimodal
distribution ,whereas the second choice is for bimodal distribution.Table 3
MLEs for the MLLND parameters with their averages, biases, MSEs, and CIs.
Θ=p1,θ,μ,σnp∧1θ∧μ∧σ∧(0.50, 0.95, 0.25, 0.25)Aveg.300.420981781.38808920.260073140.26373918Bias-0.07901800.43808900.010073140.01373918MSE0.043144942.45383430.012614920.0198413390%CIs(0.06875, 0.71539)(0.63975, 4.21360)(0.09383, 0.43337)(0.09522, 0.49725)95%CIs(0.02500, 0.76579)(0.57416, 7.27137)(0.05603, 0.47826)(0.06583, 0.57054)Aveg.500.452017521.15321980.2561105760.256695006Bias-0.047982480.20321980.0061105760.006695006MSE0.025781891.02428920.0064858390.00952193590%CIs(0.16097, 0.68077)(0.71706, 1.71599)(0.12595, 0.38932)(0.12420, 0.43820)95%CIs(0.09463, 0.72194)(0.66882, 3.49642)(0.09611, 0.41885)(0.10079, 0.49013)Aveg.750.474606791.020757860.2532594730.252181908Bias-0.025393210.070757860.0032594730.002181908MSE0.015647270.288499040.004146850.005414490%CIs(0.26408, 0.66297)(0.76299, 1.25958)(0.14861, 0.35969)(0.14830, 0.38325)95%CIs(0.19820, 0.69821)(0.72714,1.46520)(0.12957,0.38270)(0.12761,0.42832)Aveg.1000.479891550.988212250.2528829860.251725793Bias-0.020108450.038212250.0028829860.001725793MSE0.011016420.132575660.0029253080.00365751090%CIs(0.30911, 0.64102)(0.79201, 1.18252)(0.16472, 0.34125)(0.16431, 0.35621)95%CIs(0.26617, 0.67108)(0.76341, 1.26408)(0.14856, 0.36121)(0.14669, 0.38801)(0.50, 2.95, 0.85, 0.25)Aveg.300.510855943.16593820.8549103540.244567051Bias0.010855940.21593820.004910354-0.00543290MSE0.012873781.34309480.0510542500.03286265890%CIs(0.33493, 0.69426)(1.65368, 5.12052)(0.71319, 0.98390)(0.13525, 0.33944)95%CIs(0.30361, 0.73304)(1.44724, 5.71634)(0.67448, 1.01376)(0.11017, 0.33944)Aveg.500.5075760543.07071050.8524703820.240278966Bias0.0075760540.12071050.002470382-0.00972100MSE0.0070659070.67096170.0100099870.00434002990%CIs(0.37350, 0.64665)(1.89865, 4.48536)(0.74975, 0.95241)(0.16456, 0.31975)95%CIs(0.35033, 0.67656)(1.71581, 4.85850)(0.72685, 0.97424)(0.14466, 0.33891)Aveg.750.5055537563.024066670.85052068470.24284061Bias0.0055537560.074066670.0005206847-0.00715939MSE0.0045230980.400535000.0025602090.00211485790%CIs(0.39807, 0.61738)(1.94383, 4.39174)(0.77062, 0.93072)(0.18319, 0.30590)95%CIs(0.37909, 0.63999)(2.08437, 4.12198)(0.75182, 0.94835)(0.17228, 0.32117)Aveg.1000.5025086082.998519360.85072974080.24475867Bias0.0025086080.048519360.0007297408-0.0052413MSE0.0032102670.276511390.0017792170.0010800990%CIs(0.41022, 0.59639)(2.20464, 3.91893)(0.78248, 0.91894)(0.19393, 0.29928)95%CIs(0.39236, 0.61599)(2.08727, 4.15153)(0.76705, 0.93227)(0.18413, 0.31123)In each case, the averages of the MLEs, biases, mean squared errors (MSE), and the lower
and upper limits of the95% nd 90% confidence intervals (CIs) for the parameters are computed at
different sample sizes.It is clear from Table3 that the MSE decreases as
the sample size increases for all estimates parameters. Furthermore, the values of the bias
decrease. Also, as the sample size increases, the width of the confidence intervals (CIs)
for the parameters decreases. The number of replications of the simulation results is taken
to 10000.


## 5. Application

The flexibility of the proposed model is illustrated by applying it on a real data set
given in Shanker et al. [25]. It represents the
remission times (in months) of sample size n=128 bladder cancer patients as reported in Lee and Wang [26]. This data was previously analyzed by Daghestani et
al. [18], who compared their proposed model, mixture
of Lindley and Weibull distribution MLWD to two other models; mixture of two one- parameter
Lindley distribution MTLD and mixture of two Weibull distribution MTWD. They showed that
their model provides the best fit as it has the lowest values of the KS statistic and AIC
criterion and the highest p value.In this paper, we compared our proposed model with six other models including the three
models given in Daghestani et al. [18] and there
other models, namely, the mixture of Lindley inverse Weibull distributions (MLIWD), mixture
of inverse Weibull and Weibull distributions (MIWWD), one component Lindley distribution,
and one component lognormal distribution. All the seven models are fitted to the real data.
The results are listed in Table 4. Table 4 shows the MLEs of the parameters of the seven models
with their standard errors and values of the KS statistic which is used to assess the
similarity between the actual data and the fitted distributions. In the
R software, the packages (MASS) and (fitdistrplus) are used to
calculate the values of KS statistics and their corresponding R values for the seven distributions. The loglikelihood
function and some criteria that measure the quality of the fitted models such as AIC and BIC
criteria are computed. Table 5 gives the results of
the variance covariance matrices for the competitive models calculated by the function
vcov() in R which depends on the package fitdist(). Lower and upper
limits of the 95% nd 90% confidence intervals (CIs) for the parameters of the
different distributions are also provided. Figure 2
displays the plots of the pdfs of the seven fitted models superimposed on the histogram of
the real data set by using the function denscomp() in R. The figure shows that the MLLND provides a very good fit for
these data compared to other mixtures and one component models. Figure 3 shows the comparisons of the plots of the theoretical cdfs of the
fitted distributions to the empirical cdf of the data using the function cdfcomp() in
R. Again, it is clear that the cdf of the MLLND is closer to
the empirical distribution than any other model. Figures 4 and 5 show the pp plots and qq plots for
the real data to those of the compared models using the functions ppcomp() and qqcomp(),
respectively, in R. The plots show the adequency of the proposed model to fit
the real data compared to other models. In short, all the above figures indicate that the
MLLND is the perfect fit for the real data set compared to all the competitive models.Table 4
Results for fitting the MLLND comparing with their components and others models.
ModelsMLEsStd.ErrorLoglikelihoodAICBICKSp valueMLLNDp∧1=0.16733 (0.0525)-399.423806.8461818.25420.03440.9981θ∧=1.8819 (0.7011)μ∧=1.9453 (0.1119)σ∧=0.8455 (0.0852)MTLDp∧1=0.2022 (0.1150)-402.4761810.9522819.50830.07040.5495θ∧1=0.0912 (0.0264)θ∧2=0.3239 (0.0483)MLIWDp∧1=0.2374 (0.0701)-399.8651807.7303819.13840.04280.9730θ∧=1.2975 (0.4920)α∧=0.1804 (0.0221)β∧=1.5473 (0.1528)MTWDp∧1=0.9554 (0.0438)-401.2711812.5422826.80230.05840.7746α∧1=8.8542 (1.0315)α∧2=0.4747 (0.2526)β∧1=0.9751 (0.0811)β∧2=2.5650 (1.9307)MIWWDp∧1=0.7344 (0.0680)-398.7583807.5166821.77680.04390.9655α∧1=0.1879 (0.0166)α∧2=1.7452 (0.5879)β∧1=1.5208 (0.1446)β∧2=0.8896 (0.1764)LNDμ=1.5111∧ (0.1133)-406.8025817.6050823.30910.09980.1556σ=1.2818∧ (0.0801)LDθ=0.2129∧ (0.0134)-417.9239837.8477840.69970.13350.0207Table 5
CIs real data of bladder cancer patients for the MLLND and others models.
ModelsVariance-covariance matrixParametersConfidence level95%90%LowerUpperLowerUpperMLLND0.002769817−0.023773800.003083012−0.002329422−0.0237738050.49176060−0.0407584080.0275291510.003083012−0.040758410.012532830−0.004612044−0.0023294220.02752915−0.0046120440.007266494p∧10.06420.27040.08080.2538θ∧0.50763.25620.72853.0352μ∧1.72592.16471.76112.1294σ∧0.67841.01270.70520.9858MTLD0.0132340950.00250075240.00444309000.0025007520.00069872470.00087883810.0044430900.00087883810.0023397156p∧1-0.0230.42770.1300.3914θ∧10.03940.14300.04780.1347θ∧20.2290.41870.24440.4035MLIWD0.004920991−0.026698250−0.00107112780.004868889−0.0266982500.2421005120.0079738493−0.034270165−0.0010711280.0079738490.0004890109−0.0011123350.004868889−0.034270165−0.00111233520.023372212p∧10.10000.37490.12210.3528θ∧0.3312.26190.48822.1068α∧0.13710.22370.14400.2168β∧1.24771.84701.29591.7988MTWD0.001923592−0.02215016−0.005512600−0.0018795670.04563591−0.0221501611.064195670.0667825660.044848532−0.61971134−0.0055126000.066782570.0638298720.004559112−0.24653248−0.0018795670.044848530.0045591120.006593381−0.051259900.045635907−0.61971134−0.246532478−0.0512599043.72766949p∧10.86941.04130.88321.0275α∧16.832310.8767.157410.551α∧2-0.0200.96980.05910.8902β∧10.81591.13420.84151.1086β∧2-1.2196.3491-0.6105.7407MIWWD0.00462427963.494765e−040.025746042−2.986093e−030.00686686310.00034947652.762975e−040.002350313−3.977995e−050.00039934530.02574604242.350313e−030.345648000−2.312262e−020.0740712470−0.0029860928−3.977995e−05−0.0231226242.091868e−02−0.00572836730.00686686313.993453e−040.074071247−5.728367e−030.0311504072p∧10.60110.86770.62250.8462α∧10.15530.22040.16050.2152α∧20.59292.89750.77822.7122β∧11.23731.80431.28291.7587β∧20.54361.23550.59931.1799LND1.283697e−022.541429e−062.541429e−066.417786e−03μ∧1.28901.73311.32471.6974σ∧1.12481.43881.15001.4136LD0.0001798515θ∧0.18660.23920.19080.2349Figure 2
Comparison of different pdfs of the data application.Figure 3
Comparison of different CDFs of the data application.Figure 4
Comparison of different pp plots of the data application.Figure 5
Comparison of different qq plots of the data application.


## 6. Concluding Remarks

This paper introduces a new mixture model which is the MLLND to handle heterogeneous data.
This model was proposed due to the importance of each of the Lindley and lognormal
distributions and their great applications, and so it was expected that mixing these two
distributions together would lead to a more flexible model than its components
distributions. Some properties of the MLLND were obtained such as the expectation, the mean,
variance, the mode (s), median, reliability function, HRF, skewness, kurtosis, and
coefficient of variation. The pdf for the minimum and maximum order statistics of the MLLND
is presented. Maximum likelihood estimation of the parameters of the model was discussed and
estimated via simulation with number of replications 10000 runs. The main objective of this
paper was to illustrate the applicability of the proposed distribution compared to six
competitive distributions. This was achieved by showing the ability of the model to fit a
well-known real data set better than the compared models. This was done by using the formal
test likeK-S statistic as well as information criteria and also through
graphical procedures such as plots of theoretical and empirical cdfs, pp plots, and qq
plots.



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