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

In this paper, the identifiability property has been studied for a suggested truncated type-I generalized logistic mixture model which is denoted by ( TTIGL ) . 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.


Introduction
e progressive type-II censored model is a very important model in the eld of reliability and life testing (see [1]). is censoring model can be shown as follows.
Consider a lifetime test in which n identical units are tested. R i surviving units are removed randomly from the experiment once the i th failure has occurred, 1 ≤ i ≤ r. us, if the number of observed failures is r, then R 1 + R 2 + · · · + R r units are progressive censored, and hence n r + R 1 + R 2 + · · · + R r and X M 1: r: n < X M 2: r: n < · · · < X M r: r: n describe the progressive censored failure times, where M (R 1 , R 2 , . . . , R r ) and r i 1 R i n − r. e likelihood function (LF) based on type-II progressive censored data x (x M 1: r: n , x M 2: r: n , . . . , x M r: r: n ) which can be written for simplicity as x (x 1 , x 2 , . . . , x r ) is given by where c n(n − R 1 − 1)(n − R 1 − R 2 − 2) . . . (n − R 1 − R 2 − · · · − R (r− 1) − r + 1) (see [1]). e functions f Θ (x i ) and S Θ (x i ) are the probability density function (PDF) and the survival function (SF) of the studied distribution at a value x i . e importance of the mixture models appear in the theoretical and applied elds when the population under study is heterogeneous. For details about mixture models, see [2][3][4][5]. e contributions of this paper are suggesting an EM algorithm suitable for estimation based on progressively type-II censored samples from a nite mixture of distributions, studying the identi ability property of a nite mixture of TTIGL distributions, and nally using a real dataset as an application.
e cumulative distribution function (CDF) and SF are is paper is organized as follows. In Section 2, 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.

Maximum Likelihood Estimation Using EM Algorithm
It is generally known that the MLE of the parameter vector θ maximizes the log of the LF (1). e 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. e vector which represents the missing data is Based on X � (x 1 , x 2 , . . . , x n ) and U, we can write (6) which is known as complete data likelihood function. In this paper, the E and M steps of the suggested EM algorithm may be written as follows.

E
Step. where and θ 0 is an initial value for θ.
e E and M steps should be repeated until the value |L(θ s+1 ; x) − L(θ s ; x)| become a small amount. In this case, θ s will be the MLE of θ, denoted by θ.

Identifiability of the Finite Mixture of TTIGL Distributions
We can say that the random variable X ∼ Mixture TTIGL(β, c, α) distributions with parameters θ j � (β j , c j , α j ), j � 1, 2, . . . , k, if its PDF is given as below: where θ � (θ 1 , θ 2 , . . . , θ k , p 1 , . . . , p k ) and for j � 1, 2, . . . , k, (12) AL-Hussaini and Ateya [8,9] studied the estimation problem under a finite mixture of TTIGL(0, c, 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][14][15]. In our study, we will take β � 0, and then the vector of parameters will be θ j � (c j , α j ).
For a value x i of the random variable T, let So, PDF (12) and its corresponding SF and HRF can be written in the following forms (with β � 0): In the next section, we will write ϵ ji , η ji instead of . e SF and HRF of the finite mixture can be written as and Note that the SF 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]). e identifiability property has been explored by a variety of authors [12,[16][17][18][19][20][21][22][23][24][25][26][27].
In this section, the identifiability property of the suggested mixture has been proved using Chandra's theorem in [6].
Theorem 1 (see [6]). Let Φ be the class of all CDF ′ s with elements F 1 , F 2 , . . . , F k and let M: en, this class is identifiable relative to Φ.

Condition 1.
e Φ family of all CDFs may be ordered as follows: F 1 ≤ F 2 and α 1 � α 2 , implying that c 1 > c 2 . As a result, it is easy to show that D ϕ 1 ⊆D ϕ 2 .

Maximum Likelihood Estimation under a Finite Mixture of TTIGL(0, γ, α) Distributions Based on Type-II Progressive Censoring Scheme
In this section, the MLE ′ s of the parameters of a finite mixture of TTIGL(0, c, α) distributions have been obtained using the results and the formulas obtained from Sections 2 and 3. e MLE ′ s of all parameters of the finite mixture of TTIGL(0, c, α) distributions can be obtained using the suggested EM algorithm as follows: where ϵ 0 li and η 0 li are defined in Section 3 and (p 0 1 , p 0 2 , c 0 1 , c 0 2 , α 0 1 , α 0 2 ) are initial values. As mentioned in Section 2, after the convergence of the sequence of the likelihood values L(θ s ; x) { }, θ s will be the MLE ′ s of θ, denoted by θ.
e MLE ′ s of SF and HRF, denoted by, S Θ (x) and h Θ (x), can be obtained by replacing each parameter by its MLE in each function.

Main Results
In this section, Monte Carlo simulation algorithm has been applied to make a comparison between the suggested EM 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.
. . , r, which represent a progressive type-II censored sample from U(0, 1). (4) Generate a random variate y i , i � 1, 2, r from U(0, 1). (5) If y i ≤ p, generate from F 1 (x; c 1 , α 1 ) using v i ; otherwise, generate from F 2 (x; c 2 , α 2 ) using v i . In our study, using the suggested EM 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. e average estimates and MSE ′ 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.
From the results in Tables 1-4, 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 M 3 represents the complete sample case in which n � r.
From the results in Tables 3 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.
It is clear that the computed (K − 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 (P value) is greater than the chosen significance level (0.05) which means that the suggested mixture model fits the combined real dataset quite well.

Mathematical Problems in Engineering
For more illustration, Figure 1 shows the histogram of the real data and the tted PDF of the suggested mixture model computed at the estimated parameters.
Also, Figure 2 shows the tted 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 tted CDF curve computed at the estimated parameters.

Conclusions
In this paper, based on type-II progressive censoring samples, the estimation problem of the parameters of a nite mixture of TTIGL distributions has been studied, after studying the identi ability 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.
Data Availability e data used to support the ndings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no con icts of interest.