Statistical Inference of the Odd Generalized ExponentialExponential Distribution based on Type I and Type II Censored Samples
Samia Adham α & Amani Alsalafi σ
____________________________________________
ABSTRACT
In this paper, the graphical of the probability density, cumulative and hazard function of the OGEE(ζ) distribution, limit of the cumulative distribution function of the proposed distribution are presented. Moreover, the maximum likelihood estimation for the parameters of the OGEE(ζ) distribution based on Type I and Type II censored samples are obtained. Finally, the performance of the obtained estimators from the considered method s of Type II censored samples estimation is compared using numerical study through Monte Carlo simulation for different sample sizes.
Keywords: odd generalized exponential exponential distribution, maximum likelihood estimation, type i and type ii censored samples, monte carlo simulation.
Author α : Department of Statistics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia.
σ: MSc. Student of Statistics, king Abdulaziz University, Jeddah, Kingdom of Saudi Arabia.
INTRODUCTION
The generalization of the lifetime distribution is of great interest for many statisticians as it provides more flexibility to the distribution. The generalized exponential (GE) distribution suggested by [4] is an important lifetime distribution in survival analysis. In fact, odd generalized exponentialexponential distribution is one of the generalization forms of the exponential distribution. [3] have projected the family of continuous distributions referred to as the odd generalized exponential family, whose hazard rate function might be increasing, decreasing. They derived specific expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Rényi entropies and order statistics for the primary time obtained the generating function of the Fréchet distribution and projected characterizations of the family.
[10] introduced the odd generalized exponential Gompertz distribution. [5] suggested the odd generalized exponential linear failure rate distribution. They obtained some properties and mentioned the estimation of the model parameters by maximum likelihood method. [8] projected the generalization of the Rayleigh distribution referred to as odd generalized exponential Rayleigh distribution. [6] considered the odd loglogistic logarithmic generated family of distributions. They presented some special models and investigate the asymptotes and shapes. [9] considered the odd Burr Lindley (OBUL) distribution. [7] defined the odd generalized exponential flexible Weibull extension (OGEFEW) distribution. [2] introduced the odd generalized exponentialexponential distribution (OGEE(ϴ)) for modeling lifetime data. A comprehensive study of the cumulative distribution function, probability density function, survival and hazard function of the OGEE(ϴ) are presented. Moreover, the maximum likelihood estimation of the parameters of the OGEE(ϴ) distribution is considered for both simulated and real data sets.
[11] studied the odd generalized exponentialexponential distribution. They studied different properties of the proposed probability model comprised moment, moment generated function, quantile function, survival function, and hazard function. For illustrative purposes, two datasets to proposed probability distribution with the existed of odd generalized exponentialexponential distribution and observed the proposed model fits with two data sets better. [1] studied the OGEE(ϴ) distribution. Some of the mathematical properties and graphical description of the new distribution are obtained and discussed. In addition, the density functions of the smallest and largest order statistics of the odd generalized exponentialexponential distribution are obtained.
This paper is outlined as follows. In Section 2, the odd generalized exponential exponential distribution, construction of the OGEE(ζ) distribution, graphical description of the OGEE(ζ) distribution, limit of CDF function of the OGEE(ζ) distribution is obtained. Then, Section 3, deals with the maximum likelihood estimation based on Type I censored samples. However, the maximum likelihood estimation based on Type II censored samples are presented in Section 4. In Section 5, simulation study for Type II censored samples are provided. Finally, the results are concluded in Section 6.
ODD GENERALIZED EXPONENTIALEXPONENTIAL DISTRIBUTION
2.1 Construction of the OGEE(ζ) Ddistribution
A nonnegative continuous random variable X is said to have the OGEE(ζ) distribution with parameters α, λ and γ written as OGEE(ζ) where the vector ζ is defined in the form ζ = (α, λ, γ), if its CDF is given as
 (1) 
where F(x) =1 , , so that
   (2) 
the corresponding PDF has the form
(3)
where the parameters are shape parameters and is a scale parameter.
2.2 Graphical Description of the OGEE(ζ) Distribution
In this section, the PDF, CDF and hazard function of the OGEE(ζ) distribution is graphically illustrated, when considering different parameters. Assuming three cases, in each case the values of the three parameters will be determined and the effect of this choice will be explained.
2.2.1 PDF of the OGEE distribution
Case 1: The parameters are assumed to be equal, while the parameter takes different values. That is, in Figure (1) the parameter takes the values (0.2, 0.6, 1, 2, 4). While are assumed to be equal and fixed to the values 0.5, 1, 3 in Figures (1a), (1b) and (1c), respectively.
Figure 1: PDF of various OGEE distribution for different values of when
Figure (1) represents the behavior of the density function and explains the tractability and flexibility of the model graphically with its subfamilies and shows the effect of the parameter when it takes different values. In Figures (1a) and (1b), when are ≤ 1, the density is decreasing when, then after that it becomes right skewed when . Whereas, when the PDF curve reduces to be symmetric. Also, when the PDF curve in Figure (1a) becomes approximately left skewed and when the PDF curve in Figure (1b) reduces to be symmetric. In Figure (1c), when and are , the density is decreasing when and the density is unimodal with different kurtosis and right skewed when .
Case 2: The parameters and are assumed to be equal to each other, while the parameter takes different values. That is, in Figure (2) the parameter takes the values (0.2, 0.6, 1, 2, 4). While and are both fixed to the values 0.5, 1 and 3.
Figure 2: PDF of various OGEE distribution for different values of when =.
Figure (2) shows the effect of the parameter when it takes different values. In Figure (2a), the PDF curves are decreasing for different values of and the parameters and are both In Figure (2b), the PDF plot shows that for , the newly developed model has exponentially decreasing behavior. For the curves are unimodal with different kurtosis and right skewed. In Figure (2c), when the parameters and are both 1, the PDF curves are right skewed when and almost symmetric when
Case 3: The parameters and are assumed to be equal to each other, while the parameter takes different values. That is, in Figure (3) the parameter takes the values (0.2, 0.6, 1, 2, 4). While and are both fixed to the values 0.5, 1and 3.
Figure 3: PDF of various OGEE distribution for different values of when =
Figure (3) shows the effect of the scale parameterwhen it takes different values. In Figure (3a) the distribution has decreasing curves for all values of, and are both 1. Figure (3b) the density is positively skewed for all values of and the parameters and are equal 1. In Figure (3c), when the parameter and are both the density curves are bell shaped for the different values of However, when the parameter decreases the curve kurtosis increases.
2.2.2 CDF of the OGEE distribution
Case 1: The parameters are assumed to be equal to each other, while the parameter takes different values. That is, in Figure (4) the parameter takes the values (0.2, 0.6, 1, 2, 4). While are both fixed to the values 0.5, 1, 3.
Figure 4: CDF of various OGEE distribution for different values of when
Case 2: The parameters and are assumed to be equal, while the parameter takes different values. That is, in Figure (5) the parameter takes the values (0.2, 0.6, 1, 2, 4). While and are assumed to be equal and fixed to the values 0.5, 1 and 3 in Figures (5a), (5b) and (5c), respectively.
Figure 5: CDF of various OGEE distribution for different values of when =.
Case 3: The parameters and are assumed to be equal to each other, while the parameter takes different values. That is, in Figure (6) the parameter takes the values (0.2, 0.6, 1, 2, 4). While and are both fixed to the values 0.5, 1, 2 and 3.
Figure 6: CDF of various OGEE distribution for different values of when =
Figures (4), (5) and (6) show the CDF curves for different values of the three parameters. It is clear that all the CDF curves are increasing.
2.2.3 HRF of the OGEE distribution
Case 1: The parameter are assumed to be equal, while the parameters takes different values. That is, in Figure (7) the parameter takes the values (0.2, 0.6, 1, 2, 4). Whileare assumed to be equal and fixed to the values (0.5, 0.5), (1, 1) and (3, 3) respectively.
Figure 7: Hazard rate function of various OGEE distribution for different values of when
Figure (7) shows the effect of the parameter when it takes different values. In Figures (7a), (7b) and (7c), when the parameters and are both the hazard rate function takes the bathtub shape when However, when the HRF curve is increasing.
Case 2: The parameters and are assumed to be equal to each other, while the parameter takes different values. That is, in Figure (8) the parameter takes the values (0.2, 0.6, 1, 2, 4). While and are both fixed to the values 0.5, 1, 2 and 3.
Figure 8: Hazard rate function of various OGEE distribution for different values of when =.
In Figure (8a), when and are both the HRF curve is a bathtub for all values of Whereas, when and are the HRF curve is exponentially increases when increases; this is clear in Figures (8b) and (8c).
Case 3: The parameters and are assumed to be equal, while the parameter takes different values. That is, in Figure (3.9) the parameter takes the values (0.2, 0.6, 1, 2, 4). While and are assumed to be equal and fixed to the values 0.5, 1, 2 and 3 in Figures (3.9a), (3.9b) and (3.9c), respectively.
Figure 9: Hazard rate function of various OGEE distribution for different values of when =
In Figure (9a), when and are both the HRF curve takes the bathtub shape when However, it decreases when . Whereas, when and are both the HRF curve is exponentially increases when increases; this clear in Figure (9b) and (9c).
2.3 Limit of CDF function of OGEE distribution
Since the CDF of OGEE distribution in equation (2) is:
MAXIMUM LIKELIHOOD ESTIMATION OF THE OGEE DISTRIBUTION BASED ON TYPE Ӏ CENSORED SAMPLES
The likelihood function of Type Ӏ censoring is given by
(4)
Substituting the PDF and CDF given by from (2) and (3) respectively into (4), we get the likelihood function of the Type Ӏ censored samples from the OGEE distribution is given by
(5)
The loglikelihood function is given by
(6)
Partially differentiating (6) with respect to respectively, gives
(7)
(8)

(9) 
The corresponding likelihood functions cannot be solved easily, in order to obtain the ML estimators of the parameters. Therefore, the numerical methods are then required to solve these equations.
MAXIMUM LIKELIHOOD ESTIMATION OF THE OGEE DISTRIBUTION BASED ON TYPE ӀI CENSORED SAMPLES
Let X1, X2,…,Xn are a random sample from the OGEE distribution. Assume that X(1) ≤ X(2) ≤…≤ X(n) represent the order statistics for this sample. For the Type ӀӀ censoring the smallest r observations have been chosen, where r˂ n; the likelihood function of Type ӀӀ censoring is given by
Where and are the PDF and CDF of the OGEE distribution respectively. Compensation in equation (10) by equations (2) and (3) then the likelihood function of the Type ӀӀ censored samples from the OGEE(ζ) distribution is given by
 (11)

The loglikelihood function is given as follow:
(12)
Partial differentiating of (12) with respect to, respectively, we get
, (13)
(14)
(15)
Similarly, as the cases of the MLE in the complete and Type I censored samples, the maximum likelihood estimates based on Type II censoring are only can be obtained by considering numerical methods. An R program is constructed to solve the ML equations numerically.
4.1 Simulation study
The method of the ML estimation for the parameters of the OGEE distribution based on Type II censored samples is obtained through a Monte Carlo simulation study. The sample size n=100, 150, are considered, and m the number of samples is set to be 1000. The ML estimates of the parameters α, λ and γ are obtained numerically as the following steps:
 The ML estimates of parameters α, λ and γ based on Type II censored samples are computed by solving the system of nonlinear equations (13), (14) and (15), simultaneously, by using NewtonRaphson method held in the (nlminb) function in R package.
 Bias and MSE's of the estimates are calculated using the following:
(16)
 (17) 
Where), and is the mean of . Table (I) and Table (II) summarizes the results of the simulation study for different parameters values.
Tables (I) and (II) the ML estimation of the parameters of the OGEE distribution when considering Type II censored simulated samples. The censoring percentage of 95% and 75% are applied for the sample sizes of 100 and 150.
The bias and MSE’s of the estimation are computed in order to deal with performance of the estimation when censoring is considered.
Table 1: Estimates, Bias and MSE for Type ӀӀ censored simulated samples (n =100)
r  α, λ, γ  Estimate  Bias  MSE 
        
95  2,1,0.5  2.1133  1.5106  0.5184  0.1133  0.5106  0.0184  0.4554  3.4161  0.0413 
1.5,1,0.5  1.5717  1.5221  0.5294  0.0678  0.5221  0.0294  0.1728  3.6689  0.0474 
0.5,1,1  0.5068  1.8663  1.1275  0.0068  0.8663  0.1275  0.0067  10.7302  0.3346 
2,1,1  2.1183  1.5439  1.0541  0.1183  0.5439  0.0541  0.4355  5.5769  0.1709 
75  2,1,0.5  2.1669  2.2065  0.5367  0.1669  1.2065  0.0367  0.5671  9.7226  0.0948 
1.5,1,0.5  1.6092  2.4179  0.5242  0.1092  1.4179  0.0242  0.2332  11.1185  0.1009 
0.5,1,1  0.5045  3.6412  1.3559  0.0045  2.6411  0.3559  0.0073  39.6786  1.2138 
2,1,1  2.2134  3.4968  1.0438  0.2133  2.4968  0.0438  0.7020  44.0228  0.4186 
Table 2: ML estimates, Bias and MSE for Type ӀӀ censored simulated samples (n =150).
r  α, λ, γ  Estimate  Bias  MSE 
        
143  2,1,0.5  2.0644  1.2423  0.5275  0.0644  0.2422  0.0275  0.2861  1.4332  0.0297 
1.5,1,0.5  1.5516  1.2959  0.5141  0.0516  0.2959  0.0141  0.1103  1.3891  0.0282 
0.5,1,1  0.5043  1.4929  1.0673  0.0043  0.4929  0.0673  0.0041  5.0079  0.2089 
2,1,1  2.0633  1.2737  1.0348  0.0633  0.2737  0.0348  0.2543  1.6479  0.1116 
113  2,1,0.5  2.1225  1.9523  0.5214  0.1225  0.9523  0.0214  0.4183  7.2767  0.0702 
1.5,1,0.5  1.5422  1.7742  0.5317  0.0422  0.7742  0.0317  0.1235  5.5154  0.0699 
0.5,1,1  0.5013  2.8849  1.2598  0.0013  1.8849  0.2598  0.0045  25.7341  0.7872 
2,1,1  2.1604  2.6111  1.0283  0.1605  1.6111  0.0283  0.4798  24.7245  0.2983 
It is clear from Tables (5.6 and 5.7) the ML estimates, when 95% censoring is better than the ML estimates when 75% censoring is considered. In general, one can see that when the sample sizes increase, the estimates are getting better.
V. CONCLUSION
In this article, the graphical description of the odd generalized exponential exponential distribution was obtained and discussed to show the impact of changing of the parameters on the shape of the PDF, CDF and HRF of the OGEE distribution. The ML estimates of the parameters of the OGEE distribution are computed considering simulation for Type II censored under different sizes and different values of the parameters and the ML estimates are found considering simulated samples and 95% and 75% censoring percentages. It was clear from the computed results that when the sample sizes increase, the ML estimates are getting better for all parameters.
REFERENCES
 S. . A. Adhama and A. A. ALSalafi, "Properties of the odd Generalized Exponential Exponential Distribution," American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS), pp. Volume 49, No 1, pp 8694, 2018.
 A. . A. AlSalafi and S. A. Adham, "Maximum Likelihood Estimation in the Odd Generalized ExponentialExponential Distribution," International Journal of Contemporary Mathematical Sciences, pp. Vol. 13, no. 3, 111  123, 2018.
 Tahir, Muhammad H; Cordeiro, Gauss M; Alizadeh, Morad; Mansoor, Muhammad; Zubair, Muhammad; Hamedani, Gholamhossein G;, "The odd generalized exponential family of distributions with applications," Journal of Statistical Distributions and Applications, pp. 128, 2015.
 Gupta, R D; Kundu, D;, "Theory & methods: Generalized exponential distribution," Australian and New Zealand Journal of Statistics, vol. 41, no. 2, pp. 173188, 1999.
 M. A. ElDamcese, A. Mustafa, B. S. ElDesouky and M. E. Mustafa, "The Odd Generalized Exponential Linear Failure Rate Distribution," math.ST,p.arXiv:1510. 06395v1, 2015.
 M. Alizadeh, S. M. T. K. MirMostafee, E. M. M. Ortega, T. G. Ramires and G. M. Cordeiro, "The odd loglogistic logarithmic generated family of distributions with applications in different areas," Journal of Statistical Distributions and Applications, pp. 46, 2017.
 A. Mustafa, B. S. ElDesouky and S. ALGarash, "Odd Generalized Exponential Flexible Weibull Extension Distribution," Journal of Statistical Theory and Applications, vol. 17, no. 1, p. 77–90, 2018.
 A. Luguterah, "odd Generalized Exponential Rayleigh Distribution," Advances and Applications in Statistics, vol. 48, no. 1, pp. 3348, 2016.
 G. Altun, M. Alizadehz, E. Altun and G. Ozel, "Odd Burr Lindley distribution with properties and applications," Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 2, pp. 255 276, 2017.
 M. A. ElDamcese1, A. Mustafa, B. S. ElDesouky and M. . E. Mustafa, "The Odd Generalized Exponential Gompertz Distribution," Scientific Research Publishing Applied Mathematics, vol. 6, no. 23402353, 2015.
 B. Abba and S. VV, "New Odd Generalized Exponential  Exponential Distribution: Its Properties and Application," Biostatistics and Biometrics Open Access Journal, vol. 6, no. 3, 2018.