# Estimation Using Suggested EM Algorithm Based on Progressively Type-II Censored Samples from a Finite Mixture of Truncated Type-I Generalized Logistic Distributions with an Application

Authors: Saieed F. Ateya; Mutua Kilai; Ramy Aldallal
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, the identifiability property has been studied for a suggested truncated type-I generalized logistic mixture model which is denoted byTTIGL. A suggested form of the EM algorithm has been applied on type-II progressive censored samples to obtain the maximum likelihood estimates MLE′s of the parameters, survival function SF, and hazard rate function HRF of the studied mixture model. Monte Carlo simulation algorithm has been applied to study the behavior of the mean squares errors MSE′s of the estimates. Also, a comparative study is conducted between the suggested EM algorithm and the ordinary algorithm of maximizing the likelihood function, which depends on the differentiation of the log likelihood function. The results of this paper have been applied on a real dataset as an application.


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## Body

## 1. Introduction

The progressive type-II censored model is a very important model in the field of reliability and life testing (see [1]). This censoring model can be shown as follows.Consider a lifetime test in whichn identical units are tested. Ri surviving units are removed randomly from the experiment once the ith failure has occurred, 1≤i≤r. Thus, if the number of observed failures is r, then R1+R2+⋯+Rr units are progressive censored, and hence n=r+R1+R2+⋯+Rr and X1:r:nM<X2:r:nM<⋯<Xr:r:nM describe the progressive censored failure times, where M=R1,R2,…,Rr and ∑i=1rRi=n−r. The likelihood function LF based on type-II progressive censored data x=x1:r:nM,x2:r:nM,…,xr:r:nM which can be written for simplicity as x=x1,x2,…,xr is given by(1)Lθ;x=c∏i=1rfΘxiSΘxiRi,where c=nn−R1−1n−R1−R2−2…n−R1−R2−⋯−Rr−1−r+1 (see [1]). The functions fΘxi and SΘxi are the probability density function PDF and the survival function SF of the studied distribution at a value xi.The importance of the mixture models appear in the theoretical and applied fields when the population under study is heterogeneous. For details about mixture models, see [2–5].The contributions of this paper are suggesting anEM algorithm suitable for estimation based on progressively type-II censored samples from a finite mixture of distributions, studying the identifiability property of a finite mixture of TTIGL distributions, and finally using a real dataset as an application.A random variableX is said to have a finite mixture of certain distributions with PDF′sfjx;θj,j=1,2,…,k and vector of parameters θj,j=1,2,…,k if its PDF is given by(2)fΘx=∑j=1kpjfjx;θj,where(3)Θ=θ1,θ2,…,θk,p1,p2,…,pk,∑j=1kpj=1and0≤pj≤1.The cumulative distribution functionCDF and SF are(4)FΘx=∑j=1kpjFjx;θj,SΘx=∑j=1kpjSjx;θj.This paper is organized as follows. In Section2, a suggested form of EM algorithm is introduced to compute the MLE′s of the parameters of a finite mixture of distributions based on progressively type-II censored samples. In Section 3, the identifiability property of the finite mixture of TTIGL distributions is studied using Chandra’s theorem in [6]. In Section 4, the suggested form of the EM algorithm is applied on a finite mixture of TTIGL distributions. In Section 5, the main results are introduced. Finally, concluding remarks are introduced in Section 6.


## 2. Maximum Likelihood Estimation Using EM Algorithm

It is generally known that theMLE of the parameter vector θ maximizes the log of the LF (1). The log-LF is challenging to optimize since it includes the log of the sum and has high number of parameters.In this section, we will employ Krishnan and Krishnan’s idea [7] of missing data. The vector which represents the missing data is U=U1,…,Un, where Ui=Ui1,…,Uik,i=1,2,…,n and(5)Uij=1ifxicomes from groupj,0otherwise.Based onX=x1,x2,…,xn and U, we can write(6)LCθ;x,u=∏i=1r∏j=1kpjfjxi;θjuijpjSjxi;θjRiuij,which is known as complete data likelihood function.In this paper, the E and M steps of the suggestedEM algorithm may be written as follows.


### 2.1. E Step

The functionQθ,θ0=EUlogLCΘ;X,U|θ0 may be rewritten as(7)Qθ,θ0=∑i=1r∑j=1klogpj1+RiEUUij;Xi=xi|θ0+logfjxi;θjEUUij;Xi=xi|θ0+RilogSjxi;θjEUUij;Xi=xi|θ0,where(8)EUUij|xi=t∗|θ0=pj0fjxi;θj0∑l=1kpl0flxi;θl0,t∗=xi,pj0Sjxi;θj0∑l=1kpl0Slxi;θl0,xi<t∗<xi+1,and θ0 is an initial value for θ.


### 2.2. M Step

By maximizing the functionQθ,θ0, we get(9)pj^=1n∑i=1rpj0fjxi,θj0∑l=1kpl0flxi;θl0+Ripj0Sjxi;θj0∑l=1kpl0Slxi;θl0,and(10)θ^j=argmaxθj∑i=1rpj0fjxi;θj0logfjxi;θj∑l=1kpl0flxi;θl0+Ripj0Sjxi;θj0logSjxi;θj∑l=1kpl0Slxi;θl0,where j=1,2,…,k.The E and M steps should be repeated until the valueLθs+1;x−Lθs;x become a small amount. In this case, θs will be the MLE of θ, denoted by θ^.


## 2.1. E Step

The functionQθ,θ0=EUlogLCΘ;X,U|θ0 may be rewritten as(7)Qθ,θ0=∑i=1r∑j=1klogpj1+RiEUUij;Xi=xi|θ0+logfjxi;θjEUUij;Xi=xi|θ0+RilogSjxi;θjEUUij;Xi=xi|θ0,where(8)EUUij|xi=t∗|θ0=pj0fjxi;θj0∑l=1kpl0flxi;θl0,t∗=xi,pj0Sjxi;θj0∑l=1kpl0Slxi;θl0,xi<t∗<xi+1,and θ0 is an initial value for θ.


## 2.2. M Step

By maximizing the functionQθ,θ0, we get(9)pj^=1n∑i=1rpj0fjxi,θj0∑l=1kpl0flxi;θl0+Ripj0Sjxi;θj0∑l=1kpl0Slxi;θl0,and(10)θ^j=argmaxθj∑i=1rpj0fjxi;θj0logfjxi;θj∑l=1kpl0flxi;θl0+Ripj0Sjxi;θj0logSjxi;θj∑l=1kpl0Slxi;θl0,where j=1,2,…,k.The E and M steps should be repeated until the valueLθs+1;x−Lθs;x become a small amount. In this case, θs will be the MLE of θ, denoted by θ^.


## 3. Identifiability of the Finite Mixture ofTTIGL Distributions

We can say that the random variableX∼ Mixture TTIGLβ,γ,α distributions with parameters θj=βj,γj,αj, j=1,2,…,k, if its PDF is given as below:(11)fΘx=∑j=1kpjfjx;θj,where θ=θ1,θ2,…,θk,p1,…,pk and for j=1,2,…,k, θj=βj,γj,αj,(12)fjx;θj=αjγj1−2−αjexp−x−βjγj1+exp−x−βjγj−αj+1,x>βj,βj≥0,γj>0,αj>0,pj≥0,∑j=1kpj=1.AL-Hussaini and Ateya [8, 9] studied the estimation problem under a finite mixture of TTIGL0,γ,α distributions using the classical and Bayes methods based on complete classified type-I censoring scheme. Ateya and Alharthi in [10] studied the estimation problem under a finite mixture of modified Weibull distributions under type-I, type-II, and type-II progressive censoring schemes using the ordinary likelihood method which depends on the differentiation of the LF with respect to the parameters and without studying the identifiability property. Also, Ateya and Alharthi in [11] studied the estimation problem under the same mixture model under type-I and type-II censoring schemes using the EM algorithm without studying the type-II progressive censoring case. Ateya in [12] studied the identifiability property and the estimation problem using EM algorithm under a finite mixture of generalized exponential distribution under type-I and type-II censoring schemes without studying type-II progressive censoring case. For more details about TTIGL distribution and its mixtures, see [8, 9, 13–15]. In our study, we will take β=0, and then the vector of parameters will be θj=γj,αj.For a valuexi of the random variable T, let(13)wji=1+e−xi/γj,εjixi,θj=1−wji−αj,ηjixi,θj=wji−αj−1wji−1εjixi,θj.So,PDF (12) and its corresponding SF and HRF can be written in the following forms (with β=0):(14)fjxi,θj=αjγj1−2−αj−1εjixi,θjηjixi,θj,xi>0,γj,αj>0,Sjxi,θj=1−2−αj−1εjixi,θj,hxi,θj=αjγjηjixi,θj.In the next section, we will writeϵji,ηji instead of ϵjixi,θj,ηjixi,θj and ϵji0,ηji0 instead of ϵjixi,θj0,ηjixi,θj0.TheSF and HRF of the finite mixture can be written as(15)SΘx=∑j=1kpjSjx;θj,and(16)hΘx=fΘxSΘx.Note that theSF of a finite mixture of distributions is a finite mixture of the SF′s corresponding to the distributions, but this is not true with respect to the HRF.It is very important to know that the statistical inference problem for the parameters in case of the mixture distributions cannot be discussed before proving the identifiability property (see [5]).The identifiability property has been explored by a variety of authors [12, 16–27].In this section, the identifiability property of the suggested mixture has been proved using Chandra’s theorem in [6].Theorem 1 (see [6]).
LetΦ be the class of all CDF′s with elements F1,F2,…,Fk and let M:F1⟶ϕ1 be a linear mapping with domain Dϕ1. Assume that there is a total ordering ≤ of Φ such that(1)
F1≤F2, F1,F2∈Φ implies Dϕ1⊆Dϕ2.(2)
For eachF1∈Φ, there exists s1 in the closure of T1=s:ϕs≠0 such that lims⟶s1ϕ2s/ϕ1s=0, for all F1<F2, F1,F2∈Φ.
Then, this class is identifiable relative toΦ.Proposition 1.
The class of all finite mixtures ofTTIGL0,γ,α distributions is identifiable.Proof.
LetX∼TTIGL0,γ,α, with PDF given by (12). Define the transform ϕ1s as a moment generating function of the TTIGL0,γ,α, which can be written as(17)ϕ1s=MXs=α1γ11−2−α1∫0∞esxexp−xγ11+exp−xγ1−α1+1dx=α11−2−α1B11−sγ1,α1+sγ1,s<1γ1,(18)ϕ2s=MXs=α2γ21−2−α2∫0∞esxexp−xγ21+exp−xγ2−α2+1dx=α21−2−α2B11−sγ2,α2+sγ2,s<1γ2,where(19)Bpa,b=∫0pua−11+u−a+bdu,a>0,b>0,is the incomplete beta (type-II) function.
Also, theCDF of TTIGL0,γ1,α1 and TTIGL0,γ2,α2 can be written in the following forms:(20)F1x;γ1,α1=1+exp−x/γ1−α1−2−α11−2−α1,x>0,and(21)F2x;γ2,α2=1+exp−x/γ2−α2−2−α21−2−α2,x>0.
From (17) and (18), we can see that Dϕ1=0,1/γ1 and Dϕ2=0,1/γ2.
Now, we will make sure that the two conditions of the previous theorem are met.Condition 1.
TheΦ family of all CDFs may be ordered as follows: F1≤F2 and α1=α2, implying that γ1>γ2. As a result, it is easy to show that Dϕ1⊆Dϕ2.Condition 2.
If we takes1=1/γ1, then lims⟶1/γ1ϕ2s/ϕ1s=lims⟶1/γ1α21−2−α1B11−γ2/γ1,α2+γ2/γ1/α11−2−α2B11−γ1/γ1,α1+1=constant/∞=0.
This means that all the mixtures ofTTIGL0,γ,α distributions are identifiable.


## 4. Maximum Likelihood Estimation under a Finite Mixture ofTTIGL0,γ,α Distributions Based on Type-II Progressive Censoring Scheme

In this section, theMLE′s of the parameters of a finite mixture of TTIGL0,γ,α distributions have been obtained using the results and the formulas obtained from Sections 2 and 3.TheMLE′s of all parameters of the finite mixture of TTIGL0,γ,α distributions can be obtained using the suggested EM algorithm as follows:(22)pj^=1n∑i=1rpj0αj0γj0−11−2−αj0−1εji0ηji0∑l=1kpl0αl0γl0−11−2−αl0−1εli0ηli0+Ripj01−2−αj0−1εji0∑l=1kpl01−2−αl0−1εli0,θ^j=argmaxθj∑i=1rpj0αj0γj0−11−2−αj0−1εji0ηji0logαjγj−11−2−αj−1εjiηji∑l=1kpl0αl0γl0−11−2−αl0−1εli0ηli0+Ripj01−2−αj0−1εji0log1−2−αj−1εji∑l=1kpl01−2−αl0−1εli0,where ϵli0 and ηli0 are defined in Section 3 and p10,p20,γ10,γ20,α10,α20 are initial values.As mentioned in Section2, after the convergence of the sequence of the likelihood values Lθs;x, θs will be the MLE′s of θ, denoted by θ^.TheMLE′s of SF and HRF, denoted by, SΘx^ and hΘx^, can be obtained by replacing each parameter by its MLE in each function.


## 5. Main Results

In this section, Monte Carlo simulation algorithm has been applied to make a comparison between the suggestedEM algorithm and the ordinary method using the MSE′s criterion. In the end of this section, the suggested EM algorithm is applied on a real dataset as an application.


### 5.1. Simulated Results

In this section, a type-II progressive censored sample from a mixture of twoTTIGL0,γ,α distributions has been generated for different schemes M=R1,R2,…,Rr as follows:(1)
Generate independent random variatesu1,u2,…,ur from U0,1.(2)
Definewi=ui1/i+∑j=r−i+1rRj,i=1,2,…,r.(3)
Definevi=1−∏j=r−i+1rwj,i=1,2,…,r, which represent a progressive type-II censored sample from U0,1.(4)
Generate a random variateyi,i=1,2,r from U0,1.(5)
Ifyi≤p, generate from F1x;γ1,α1 using vi; otherwise, generate from F2x;γ2,α2 using vi.(6)
Based on the generated type-II progressive censored sample and for different schemesM=R1,R2,…,Rr where ∑i=1rRi=n−r, the MLE′s of the all parameters and functions have been obtained using the suggested form of the EM algorithm and also using the ordinary algorithm for maximizing the likelihood function.(7)
Overm samples, the MSE′s of all estimates have been computed based on the suggested EM and ordinary algorithms of estimation.In our study, using the suggestedEM algorithm, the parameters, SF, and HRF have been estimated based on simulated type-II progressive censored samples with different values for r and n to study the behavior of the MSE′s of the estimates.The average estimates andMSE′s under the suggested EM algorithm are summarized in Tables 1 and 2. Also, the MSE′s of the all estimates using the suggested EM and ordinary algorithms are summarized in Tables 3 and 4 as comparative results.Table 1
AverageMLE′s and MSE′s of p,α1,α2,γ1, and γ2.
n,rp^MSEp^α1^MSEα1^α2^MSEα2^γ1^MSEγ1^γ2^MSEγ2^30,200.23492.01833.61751.88352.11750.01490.44020.91340.09160.060730,250.26782.21213.88561.80542.13440.01020.34760.60850.04990.049630,300.282.68624.35131.71492.10340.00360.26010.56180.02410.033940,200.24152.11373.55171.79172.21050.02170.47141.02140.10420.110550,200.23912.10533.77031.72952.22150.03720.50041.15080.15050.175160,200.24462.09133.79051.68062.3050.05130.60131.22010.21150.1905Table 2
AverageMLE′s and MSE′s of S1.3 and h1.3.
nS^1.3MSER^1.3h^1.3MSEh^1.330,200.85850.16290.00100.001430,250.83300.18680.00090.001330,300.82890.18550.00070.000940,200.82150.17730.00910.005350,200.81520.17330.01510.008260,200.82260.17050.02170.0105Table 3
MSE’s under ordinary and EM algorithms of the MSE’s of p,α1,α2,γ1, and γ2.
n,rMethodMSE’sp^MSE’sα1^MSE’sα2^MSE’sγ1^MSE’sγ2^30,20Ordinary algorithm0.02150.51081.01120.10070.1019EM algorithm0.01490.44020.91340.09160.060730,25Ordinary algorithm0.02030.41160.78080.07270.0545EM algorithm0.01020.34760.60850.04990.049630,30Ordinary algorithm0.01160.31950.61150.04480.0408EM algorithm0.00360.26010.56180.02410.033940,20Ordinary algorithm0.03310.52171.11290.13170.2005EM algorithm0.02170.47141.02140.10420.110550,20Ordinary algorithm0.04070.59011.22050.20070.2702EM algorithm0.03720.50041.15080.15050.175160,20Ordinary algorithm0.06020.63171.30650.31770.3401EM algorithm0.05130.60131.22010.21150.1905Table 4
MSE′s under ordinary and EM algorithms of the MLE′s of S1.3 and h1.3.
nMethodMSER^1.3MSEh^1.330,20Ordinary algorithm0.03160.0119EM algorithm0.00100.001430,25Ordinary algorithm0.01050.0104EM algorithm0.00090.001330,30Ordinary algorithm0.00910.0084EM algorithm0.00070.000940,20Ordinary algorithm0.04160.0152EM algorithm0.00910.005350,20Ordinary algorithm0.05220.0204EM algorithm0.01510.008260,20Ordinary algorithm0.08010.0308EM algorithm0.02170.0105From the results in Tables1–4, we can conclude that for fixed n, the MSE’s decrease by increasing r, and for fixed r, the MSE’s increase by increasing n. Also, scheme M3 represents the complete sample case in which n=r.From the results in Tables3 and 4, we can conclude that for certain values of n and r, the MSE’s using the suggested EM algorithm are less than those computed using the ordinary algorithm which means than the suggested EM algorithm is better than the ordinary algorithm.


### 5.2. An Application

In this section, two real datasets fromTTIGL0,γ,α distributions are presented which represent the waiting times (in minutes) before customer service in two different banks (see [28]).The first real dataset is as follows: 0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2, 3.3,3.5, 3.6, 4.0, 4.1, 4.2, 4.2, 4.3, 4.3, 4.4, 4.4, 4.6, 4.7, 4.7, 4.8, 4.9, 4.9, 5.0, 5.3, 5.5, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.2, 8.6, 8.6, 8.6, 8.8, 8.8, 8.9, 8.9, 9.5, 9.6, 9.7, 9.8, 10.7, 10.9, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.4, 12.5, 12.9, 13.0, 13.1, 13.3, 13.6, 13.7, 13.9, 14.1, 15.4, 15.4, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0, 19.9, 20.6, 21.3, 21.4, 21.9, 23.0, 27.0, 31.6, 33.1, and 38.5.Also, the second real dataset is as follows: 0.1, 0.2, 0.3, 0.7, 0.9, 1.1, 1.2, 1.8, 1.9, 2.0, 2.2, 2.3, 2.3, 2.3, 2.5, 2.6, 2.7, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.4, 3.5, 3.9, 4.0, 4.2, 4.5, 4.7, 5.3, 5.6, 5.6, 6.2, 6.3, 6.6, 6.8, 7.3, 7.5, 7.7, 7.7, 8.0, 8.0, 8.5, 8.5, 8.7, 9.5, 10.7, 10.9, 11.0, 12.1, 12.3, 12.8, 12.9, 13.2, 13.7, 14.5, 16.0, 16.5, and 28.0.Al-Mutairi et al. in [29] showed that Lindley distribution fits these data well, and Srinivasa in [28] proved that the TTIGL distributions fit reasonably well to the two datasets.By combining the two real datasets, the new ordered real dataset will be 0.1, 0.2, 0.3, 0.7, 0.8, 0.8, 0.9, 1.1, 1.2, 1.3, 1.5, 1.8, 1.8, 1.9, 1.9, 1.9, 2.0, 2.1, 2.2, 2.3, 2.3, 2.3, 2.5, 2.6, 2.6, 2.7, 2.7, 2.7, 2.9, 2.9, 3.1, 3.1, 3.1, 3.2, 3.2, 3.3, 3.4, 3.4, 3.5, 3.5, 3.6, 3.9, 4.0, 4.0, 4.1, 4.2, 4.2, 4.2, 4.3, 4.3, 4.4, 4.4, 4.5, 4.6, 4.7, 4.7, 4.7, 4.8, 4.9, 4.9, 5.0, 5.3, 5.3, 5.5, 5.6, 5.6, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2, 6.2, 6.3, 6.3, 6.6, 6.7, 6.8, 6.9, 7.1, 7.1, 7.1, 7.1, 7.3, 7.4, 7.5, 7.6, 7.7, 7.7, 7.7, 8.0, 8.0, 8.0, 8.2, 8.5, 8.5, 8.6, 8.6, 8.6, 8.7, 8.8, 8.8, 8.9, 8.9, 9.5, 9.5, 9.6, 9.7, 9.8, 10.7, 10.7, 10.9, 10.9, 11.0, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.1, 12.3, 12.4, 12.5, 12.8, 12.9, 12.9, 13.0, 13.1, 13.2, 13.3, 13.6, 13.7, 13.7, 13.9, 14.1, 14.5, 15.4, 15.4, 16.0, 16.5, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0, 19.9, 20.6, 21.3, 21.4, 21.9, 23.0, 27.0, 28.0, 31.6, 33.1, and 38.5.In this paper, the combined real dataset of sizen=160 is analyzed using a mixture of two TTIGL0,γ,α. The estimated parameters of the mixture model, the associated K−S test statistic, and the Pvalue are summarized in Table 5.Table 5
Estimates of the parameters of the suggested mixture model and associatedK−S test statistic based on real dataset.
Modelp^α1^α2^γ1^γ2^K−SPvalueMixture of twoTTIGL0.46723.00652.71585.90593.85950.04070.8336It is clear that the computedK−S test statistic is less than the critical value for K−S test statistic, under significance level of 0.05, which is equal to 0.108; also, the computed Pvalue is greater than the chosen significance level (0.05) which means that the suggested mixture model fits the combined real dataset quite well.For more illustration, Figure1 shows the histogram of the real data and the fitted PDF of the suggested mixture model computed at the estimated parameters.Figure 1
The histogram of the real data and the fittedPDF of the suggested mixture model.Also, Figure2 shows the fitted CDF and the empirical CDF of the suggested mixture model, where the dotted curve represents the empirical CDF curve and the continuous curve represents the fitted CDF curve computed at the estimated parameters.Figure 2
The fittedCDF and the empirical CDF of the suggested mixture model.Three type-II progressive censored samples are generated from the combined real dataset using the schemesM1 = 180, M2=160,020,110, and M3=150,040,110. The three samples, respectively, are as follows:Sample (1): 0.1, 0.3, 0.8, 0.9, 1.2, 1.5, 1.8, 1.9, 2.0, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.5, 3.6, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7, 4.9, 5.0, 5.3, 5.6, 5.7, 6.1, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.0, 8.5, 8.6, 8.6, 8.8, 8.9, 9.5, 9.6, 9.8, 10.7, 10.9, 11.0, 11.1, 11.2, 11.9, 12.3, 12.5, 12.9, 13.0, 13.2, 13.6, 13.7, 14.1, 15.4, 16.0, 17.3, 18.1, 18.4, 19.0, 20.6, 21.4, 23.0, 28.0, and 33.1Sample (2): 0.1, 0.3, 0.8, 0.9, 1.2, 1.5, 1.8, 1.9, 2.0, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.5, 3.6, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7, 4.9, 5.0, 5.3, 5.6, 5.7, 6.1, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.0, 8.5, 8.6, 8.6, 8.8, 8.9, 9.5, 9.6, 9.8, 10.7, 10.9, 11.0, 11.1, 11.2, 11.9, 12.1, 12.3, 12.4, 12.5, 12.8, 12.9, 12.9, 13.0, 13.1, 13.2, 13.3, 13.6, 13.7, 13.7, 13.9, 14.1, 14.5, 15.4, 15.4, 16.0, 16.5, 17.3, 18.1, 18.4, 19.0, 20.6, 21.4, 23.0, 28.0, and 33.1Sample (3): 0.1, 0.3, 0.8, 0.9, 1.2, 1.5, 1.8, 1.9, 2.0, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.5, 3.6, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7, 4.9, 5.0, 5.3, 5.6, 5.7, 6.1, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.0, 8.5, 8.6, 8.6, 8.8, 8.8, 8.9, 8.9, 9.5, 9.5, 9.6, 9.7, 9.8, 10.7, 10.7, 10.9, 10.9, 11.0, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.1, 12.3, 12.4, 12.5, 12.8, 12.9, 12.9, 13.0, 13.1, 13.2, 13.3, 13.6, 13.7, 13.7, 13.9, 14.1, 14.5, 15.4, 15.4, 16.0, 17.3, 18.1, 18.4, 19.0, 20.6, 21.4, 23.0, 28.0, and 33.1.Then, all parameters,SF, and HRF will be estimated. The results are given in Tables 6 and 7.Table 6
MLE′s of p,α1,α2,γ1, and γ2.
n,r,Mp^α1^α2^γ1^γ2^160,80,M10.54322.84193.86359.84044.9789160,90,M20.51923.67563.71247.29115.4526160,100,M30.53914.36613.80006.837654.7316Table 7
MLE′s of S1.2 and h1.2.
n,r,MS^1.2h^1.2160,80,M10.96540.0331160,90,M20.96640.0324160,100,M30.96750.0322


## 5.1. Simulated Results

In this section, a type-II progressive censored sample from a mixture of twoTTIGL0,γ,α distributions has been generated for different schemes M=R1,R2,…,Rr as follows:(1)
Generate independent random variatesu1,u2,…,ur from U0,1.(2)
Definewi=ui1/i+∑j=r−i+1rRj,i=1,2,…,r.(3)
Definevi=1−∏j=r−i+1rwj,i=1,2,…,r, which represent a progressive type-II censored sample from U0,1.(4)
Generate a random variateyi,i=1,2,r from U0,1.(5)
Ifyi≤p, generate from F1x;γ1,α1 using vi; otherwise, generate from F2x;γ2,α2 using vi.(6)
Based on the generated type-II progressive censored sample and for different schemesM=R1,R2,…,Rr where ∑i=1rRi=n−r, the MLE′s of the all parameters and functions have been obtained using the suggested form of the EM algorithm and also using the ordinary algorithm for maximizing the likelihood function.(7)
Overm samples, the MSE′s of all estimates have been computed based on the suggested EM and ordinary algorithms of estimation.In our study, using the suggestedEM algorithm, the parameters, SF, and HRF have been estimated based on simulated type-II progressive censored samples with different values for r and n to study the behavior of the MSE′s of the estimates.The average estimates andMSE′s under the suggested EM algorithm are summarized in Tables 1 and 2. Also, the MSE′s of the all estimates using the suggested EM and ordinary algorithms are summarized in Tables 3 and 4 as comparative results.Table 1
AverageMLE′s and MSE′s of p,α1,α2,γ1, and γ2.
n,rp^MSEp^α1^MSEα1^α2^MSEα2^γ1^MSEγ1^γ2^MSEγ2^30,200.23492.01833.61751.88352.11750.01490.44020.91340.09160.060730,250.26782.21213.88561.80542.13440.01020.34760.60850.04990.049630,300.282.68624.35131.71492.10340.00360.26010.56180.02410.033940,200.24152.11373.55171.79172.21050.02170.47141.02140.10420.110550,200.23912.10533.77031.72952.22150.03720.50041.15080.15050.175160,200.24462.09133.79051.68062.3050.05130.60131.22010.21150.1905Table 2
AverageMLE′s and MSE′s of S1.3 and h1.3.
nS^1.3MSER^1.3h^1.3MSEh^1.330,200.85850.16290.00100.001430,250.83300.18680.00090.001330,300.82890.18550.00070.000940,200.82150.17730.00910.005350,200.81520.17330.01510.008260,200.82260.17050.02170.0105Table 3
MSE’s under ordinary and EM algorithms of the MSE’s of p,α1,α2,γ1, and γ2.
n,rMethodMSE’sp^MSE’sα1^MSE’sα2^MSE’sγ1^MSE’sγ2^30,20Ordinary algorithm0.02150.51081.01120.10070.1019EM algorithm0.01490.44020.91340.09160.060730,25Ordinary algorithm0.02030.41160.78080.07270.0545EM algorithm0.01020.34760.60850.04990.049630,30Ordinary algorithm0.01160.31950.61150.04480.0408EM algorithm0.00360.26010.56180.02410.033940,20Ordinary algorithm0.03310.52171.11290.13170.2005EM algorithm0.02170.47141.02140.10420.110550,20Ordinary algorithm0.04070.59011.22050.20070.2702EM algorithm0.03720.50041.15080.15050.175160,20Ordinary algorithm0.06020.63171.30650.31770.3401EM algorithm0.05130.60131.22010.21150.1905Table 4
MSE′s under ordinary and EM algorithms of the MLE′s of S1.3 and h1.3.
nMethodMSER^1.3MSEh^1.330,20Ordinary algorithm0.03160.0119EM algorithm0.00100.001430,25Ordinary algorithm0.01050.0104EM algorithm0.00090.001330,30Ordinary algorithm0.00910.0084EM algorithm0.00070.000940,20Ordinary algorithm0.04160.0152EM algorithm0.00910.005350,20Ordinary algorithm0.05220.0204EM algorithm0.01510.008260,20Ordinary algorithm0.08010.0308EM algorithm0.02170.0105From the results in Tables1–4, we can conclude that for fixed n, the MSE’s decrease by increasing r, and for fixed r, the MSE’s increase by increasing n. Also, scheme M3 represents the complete sample case in which n=r.From the results in Tables3 and 4, we can conclude that for certain values of n and r, the MSE’s using the suggested EM algorithm are less than those computed using the ordinary algorithm which means than the suggested EM algorithm is better than the ordinary algorithm.


## 5.2. An Application

In this section, two real datasets fromTTIGL0,γ,α distributions are presented which represent the waiting times (in minutes) before customer service in two different banks (see [28]).The first real dataset is as follows: 0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2, 3.3,3.5, 3.6, 4.0, 4.1, 4.2, 4.2, 4.3, 4.3, 4.4, 4.4, 4.6, 4.7, 4.7, 4.8, 4.9, 4.9, 5.0, 5.3, 5.5, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.2, 8.6, 8.6, 8.6, 8.8, 8.8, 8.9, 8.9, 9.5, 9.6, 9.7, 9.8, 10.7, 10.9, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.4, 12.5, 12.9, 13.0, 13.1, 13.3, 13.6, 13.7, 13.9, 14.1, 15.4, 15.4, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0, 19.9, 20.6, 21.3, 21.4, 21.9, 23.0, 27.0, 31.6, 33.1, and 38.5.Also, the second real dataset is as follows: 0.1, 0.2, 0.3, 0.7, 0.9, 1.1, 1.2, 1.8, 1.9, 2.0, 2.2, 2.3, 2.3, 2.3, 2.5, 2.6, 2.7, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.4, 3.5, 3.9, 4.0, 4.2, 4.5, 4.7, 5.3, 5.6, 5.6, 6.2, 6.3, 6.6, 6.8, 7.3, 7.5, 7.7, 7.7, 8.0, 8.0, 8.5, 8.5, 8.7, 9.5, 10.7, 10.9, 11.0, 12.1, 12.3, 12.8, 12.9, 13.2, 13.7, 14.5, 16.0, 16.5, and 28.0.Al-Mutairi et al. in [29] showed that Lindley distribution fits these data well, and Srinivasa in [28] proved that the TTIGL distributions fit reasonably well to the two datasets.By combining the two real datasets, the new ordered real dataset will be 0.1, 0.2, 0.3, 0.7, 0.8, 0.8, 0.9, 1.1, 1.2, 1.3, 1.5, 1.8, 1.8, 1.9, 1.9, 1.9, 2.0, 2.1, 2.2, 2.3, 2.3, 2.3, 2.5, 2.6, 2.6, 2.7, 2.7, 2.7, 2.9, 2.9, 3.1, 3.1, 3.1, 3.2, 3.2, 3.3, 3.4, 3.4, 3.5, 3.5, 3.6, 3.9, 4.0, 4.0, 4.1, 4.2, 4.2, 4.2, 4.3, 4.3, 4.4, 4.4, 4.5, 4.6, 4.7, 4.7, 4.7, 4.8, 4.9, 4.9, 5.0, 5.3, 5.3, 5.5, 5.6, 5.6, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2, 6.2, 6.3, 6.3, 6.6, 6.7, 6.8, 6.9, 7.1, 7.1, 7.1, 7.1, 7.3, 7.4, 7.5, 7.6, 7.7, 7.7, 7.7, 8.0, 8.0, 8.0, 8.2, 8.5, 8.5, 8.6, 8.6, 8.6, 8.7, 8.8, 8.8, 8.9, 8.9, 9.5, 9.5, 9.6, 9.7, 9.8, 10.7, 10.7, 10.9, 10.9, 11.0, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.1, 12.3, 12.4, 12.5, 12.8, 12.9, 12.9, 13.0, 13.1, 13.2, 13.3, 13.6, 13.7, 13.7, 13.9, 14.1, 14.5, 15.4, 15.4, 16.0, 16.5, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0, 19.9, 20.6, 21.3, 21.4, 21.9, 23.0, 27.0, 28.0, 31.6, 33.1, and 38.5.In this paper, the combined real dataset of sizen=160 is analyzed using a mixture of two TTIGL0,γ,α. The estimated parameters of the mixture model, the associated K−S test statistic, and the Pvalue are summarized in Table 5.Table 5
Estimates of the parameters of the suggested mixture model and associatedK−S test statistic based on real dataset.
Modelp^α1^α2^γ1^γ2^K−SPvalueMixture of twoTTIGL0.46723.00652.71585.90593.85950.04070.8336It is clear that the computedK−S test statistic is less than the critical value for K−S test statistic, under significance level of 0.05, which is equal to 0.108; also, the computed Pvalue is greater than the chosen significance level (0.05) which means that the suggested mixture model fits the combined real dataset quite well.For more illustration, Figure1 shows the histogram of the real data and the fitted PDF of the suggested mixture model computed at the estimated parameters.Figure 1
The histogram of the real data and the fittedPDF of the suggested mixture model.Also, Figure2 shows the fitted CDF and the empirical CDF of the suggested mixture model, where the dotted curve represents the empirical CDF curve and the continuous curve represents the fitted CDF curve computed at the estimated parameters.Figure 2
The fittedCDF and the empirical CDF of the suggested mixture model.Three type-II progressive censored samples are generated from the combined real dataset using the schemesM1 = 180, M2=160,020,110, and M3=150,040,110. The three samples, respectively, are as follows:Sample (1): 0.1, 0.3, 0.8, 0.9, 1.2, 1.5, 1.8, 1.9, 2.0, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.5, 3.6, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7, 4.9, 5.0, 5.3, 5.6, 5.7, 6.1, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.0, 8.5, 8.6, 8.6, 8.8, 8.9, 9.5, 9.6, 9.8, 10.7, 10.9, 11.0, 11.1, 11.2, 11.9, 12.3, 12.5, 12.9, 13.0, 13.2, 13.6, 13.7, 14.1, 15.4, 16.0, 17.3, 18.1, 18.4, 19.0, 20.6, 21.4, 23.0, 28.0, and 33.1Sample (2): 0.1, 0.3, 0.8, 0.9, 1.2, 1.5, 1.8, 1.9, 2.0, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.5, 3.6, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7, 4.9, 5.0, 5.3, 5.6, 5.7, 6.1, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.0, 8.5, 8.6, 8.6, 8.8, 8.9, 9.5, 9.6, 9.8, 10.7, 10.9, 11.0, 11.1, 11.2, 11.9, 12.1, 12.3, 12.4, 12.5, 12.8, 12.9, 12.9, 13.0, 13.1, 13.2, 13.3, 13.6, 13.7, 13.7, 13.9, 14.1, 14.5, 15.4, 15.4, 16.0, 16.5, 17.3, 18.1, 18.4, 19.0, 20.6, 21.4, 23.0, 28.0, and 33.1Sample (3): 0.1, 0.3, 0.8, 0.9, 1.2, 1.5, 1.8, 1.9, 2.0, 2.2, 2.3, 2.5, 2.6, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.5, 3.6, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7, 4.9, 5.0, 5.3, 5.6, 5.7, 6.1, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.0, 8.5, 8.6, 8.6, 8.8, 8.8, 8.9, 8.9, 9.5, 9.5, 9.6, 9.7, 9.8, 10.7, 10.7, 10.9, 10.9, 11.0, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.1, 12.3, 12.4, 12.5, 12.8, 12.9, 12.9, 13.0, 13.1, 13.2, 13.3, 13.6, 13.7, 13.7, 13.9, 14.1, 14.5, 15.4, 15.4, 16.0, 17.3, 18.1, 18.4, 19.0, 20.6, 21.4, 23.0, 28.0, and 33.1.Then, all parameters,SF, and HRF will be estimated. The results are given in Tables 6 and 7.Table 6
MLE′s of p,α1,α2,γ1, and γ2.
n,r,Mp^α1^α2^γ1^γ2^160,80,M10.54322.84193.86359.84044.9789160,90,M20.51923.67563.71247.29115.4526160,100,M30.53914.36613.80006.837654.7316Table 7
MLE′s of S1.2 and h1.2.
n,r,MS^1.2h^1.2160,80,M10.96540.0331160,90,M20.96640.0324160,100,M30.96750.0322


## 6. Conclusions

In this paper, based on type-II progressive censoring samples, the estimation problem of the parameters of a finite mixture ofTTIGL distributions has been studied, after studying the identifiability property of the mixture, using a suggested EM algorithm. A comparative study has been carried out between the suggested EM algorithm and the ordinary algorithm for maximizing the LF, and it is found that the suggested EM algorithm is better than the ordinary algorithm which can be interpreted as follows: the accuracy of the estimates using the ordinary algorithm decreases in case of high number of parameters (like our case). Finally, the results of the paper are applied on simulated and real data.



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