Simulation studies of stress- strength model for ELED
M.A.El-Damcesea M.O.Mohamedb Gh.mareib,c aMathematical Department, Faculty of science, Tanta University, Tanta, Egypt. bMathematical Department, Faculty of Science, Zagazig University, Zagazig, Egypt.
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Abstract
In this paper, we study the estimation of R = P [Y <X], so-called the stress- strength model, when both X and Y are two independent random variables with the extended linear exponential distribution, under different assumptions about their parameters.Maximum likelihood estimator in the case of definition two parameters, commonknown scale parameters and unknown scale andshape parameters can also be obtained in explicit form.Estimating R with Bayes estimator with non-informativepriorin the same previous cases, the same parameters , We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.
Key words: Extendedlinear exponential distribution (ELED), Stress –Strength, Bayes, Maximum Likelihood.
1. Introduction
Estimation of R = P (Y < X), when X and Y are random variables following the specified Distributions have been extensively discussed in the literature, including quality control, engineering statistics, reliability, medicine, psychology, biostatistics.This quantity can be obviously seen as a function of the parameters of the distribution of the random vector (X, Y ) and could be calculated in the closed formfor a limited number of cases (Kotz et al., 2003, Nadarajah, 2005, Cordeiro et al, 2011) For instance, the estimation of R when X and Y are independent and normally distributed has been considered by several authors including Milan and Vesna(2014). Recently, Greco and venture (2011) reported a list of papers related to the estimation problem Of R when X and Y are independent and follow a class of life-time distributions including Exponential, bivariate Exponential, generalized exponential, Gamma distributions, Burr type X model, Weibull distribution, and among others.
In this paper, the mainobjective of this paper is to focus on the inference of R= P[ Y< X], where X and Y follow the Extended Linear Exponential distributions are independently distributed. In Section 3 we study the estimation of R when the scale parameters of both distributions are common and known. In this section, we derive the ML estimator of the stress-strength modeland non-informative Bayes estimator of R. In Section 4, we carry out similar inference, made in the previous section, about R when the common scale parameters are unknown. We consider inference about R for the general case when the parameters of both distributions are not known and non-common, we derive maximum likelihood estimators of R and non-informative Bayes estimator in Section 5.Simulation results will study in sections 6.
2.On Extension of Generalized Linear Failure Rate Distribution
It is well known that the exponential, generalized exponential or Rayleigh distribution are among the most commonly used distributions for analyzing lifetime data.They can be used quite effectively in modeling strength and general lifetime data. Kundu and Raqab (2005) used different methods to estimate the parameters of the generalized Rayleigh on the observeddata. In analyzing lifetime data, the exponential, Rayleigh, linear failure rate or generalized exponential distributions are normally used. Extension of linear Exponential distribution (ELED) with three parameters, referred to as the Poisson distribution is used to add a new parameter to the linear exponential distribution which developed by El-Damcese andMarei (2014)
(1) (2)
3. Estimation ofR with Known Scale Parameters
In this section, the main aim is the estimation of R = P [Y < X ], where independent random Variables X and Y follow the extension of the linear exponential distributionwith the known common scale parameters, that is, X ∼ELED(1, 2, α) and Y ∼ELED(1, 2, β).
The stress-strength parameter, R is defined as:
, (3)
3.1. MLE of R
Suppose that and a random sample of andunits are observed fromELED. The MLE estimator of denoted by .To compute the MLE of R, the corresponding log-likelihood of the observed sample is given by:
(4)
Where, t= and .
The MLEs of (α, β) denoted by ()can be derived by solving the following equations:
, .
The maximum likelihood estimates can be obtained by solving the non-linear equations numerically for α, β. This can be done using MATHCAD15. The relatively large number of parameters can cause problems especially when the sample size is not large. A good set of initial values is essential. Oncewe geton the values ofthe parameterscan be evaluate it.
(5)
3.2. Bayes based on non-informative prior with known parameter.
In this section, the Bayes estimator of denoted by is obtained with non-informative prior, where the equation to find fisher Information as follow :
I () = -E () (6)
The Fisher Information measures the sensitivity of an estimator.Jeffrey's (1961)be considered as a prior for the likelihood function L(θ). The Jeffrey's prior is justified on the grounds of its invariance under parameterization according to Sinha (1986).
The prior density of () and () respectively
g() ; (7)
and
g() ; (8)
Combining the prior densities of () , () and the likelihood function to obtain the joint posterior density of () as:
= (9)
(10) Where,
Therefore, the Bayesian estimator of R under squared error loss function is given by
(11)
The Bayes estimate of R under squared error loss cannot be computedanalytically. Alternatively, numerical solution based on MATHCAD15 program is employed to evaluate for different values of the parameters.
4.1. Maximum likelihood estimator of R with unknown scale parameters
In this section, we wish to make inference about, when the common scale parameters of and, that is, () are unknown, and then investigate its properties. Let be a random sample from ELED() and () be a random sample from ELED ().
, (12)
To compute the MLE of, the corresponding log-likelihood of the observed Sample is given by
()= (13)
Where,
(14)
(15)
(16)
(17)
Fromnon-linear Equations (16) and (17), we can obtain the maximum likelihood estimates of and as function of and ,by replacing in Equations (14) and (15), the MLEs of a and b can be then achieved.
Finally, due to the invariance property of the ML estimators, the MLE of R which denoted by will be asFollows
(18)
4.2 Asymptoticdistributions
As the exact distribution of does not exist, it is essential to investigate the asymptoticbehavior of the derived MLE of R which is considered in this section. We first derive the asymptotic distribution of and then the asymptotic distribution of will be accordingly obtained. We then, based on the asymptotic distribution of, calculate the asymptotic confidence interval of R.We denote the observed information matrix of as
I()=
=
=
Theorem 1. As n,m
Where
(19)
And
, = , ,
, = , ,
Proof : The proof follows from the asymptotic normality of MLE (See Ferguson (1967) and references therein.
Theorem 2. As (n,m
Where
= (20)
The proof (FatemehShahsanaei and AlirezaDaneshkhah(2013)) and references therein.
The motivation behind the asymptotic distribution presented above for is to construct an asymptotic confidence interval for R. In order to construct this confidence interval, we first need to estimate Duo to the invariance property of the ML estimator, we can estimateby estimating its elements via replacing (,, by their MLEs, (,, , ) We will calculate this asymptotic confidence interval in Section 6 where the simulation results.
4.3. Bayes based on Non-Informative Prior with unknown scale parameters ().
According toJeffrey's (1961), the Fisher Information measuresof and be considered as a prior pdf for each parameter . The Jeffrey's prior is justified on the grounds of its invariance under parameterization according to Sinha (1986).
The prior density of, respectively
g() ; g() ; ,g(, g() (21) where all of them are independent then
g() . ; (22)
Combining the joint prior density of () and the likelihood function to obtain the joint posterior density of () as
, (23)
(24)
Where,
Therefore, the Bayesian estimator of Rdenoted by under squared error loss function is given by
(25)
The Bayes estimate of R under squared error loss cannot be computed analytically. Alternatively, numerical solution based on MATHCAD15 program is employed to evaluate for different values of the parameters.
5.Estimation of R in the General Case
In this section, we present the estimations of the stress-strength model,When X ∼ELED(, , ) and Y ∼ELED(,, β) We present the MLE of R and its associated confidence intervals in the next subsection.
5.1 Maximum likelihood estimator of R
Suppose further ()is a random sample fromELED(, , ) and () is another random sample from ELED(,, β).Thestress-strength parameter, R is defined as
=]dx (26)
The log-likelihood function of the observed samples is presented as
Where,
Then,
(27)
(28)
(29)
(30)
(31)
(32)
The MLEs of denoted by and the MLEs of denoted by ,Due to the invariance property of the ML estimators, the MLE of R will be as follows
(33)
5.2. Bayes estimator with non-informative priorsin general case.
From pervious sections we can introduce the non-informative prior density of () is:
g();(34) Combining the joint prior density of () and the likelihood function to obtain the joint posterior density of () as
(35)
(36)
Where,
,
Therefore, the Bayesian estimator of R under squared error loss function is given by
(37)
The Bayes estimate of R under squared error loss cannot be computed analytically. Alternatively, numerical solution based on MATHCAD15 program is employed to evaluate for different values of the parameters.
6. Simulation Results
In this section, Monte Carlo simulation is performed to test the behavior of theproposed estimators for different sample sizes and for different parameter values. The
Performances of the MLEs and the Bayes estimates are compared in terms of biases andmean squares errors (MSEs). Bayes estimators are computed based non-informative priors, where we have three cases for ELED.
Case1: when the scale parameter are fixed where andfor and , respectively.
Case2: when the scale parameters are unknown whereand
For and , respectively.
Case3: when the scale parameters are unknown and different For and , respectively.
We will obtain the MLE of the unknown parameters of the ELEDto obtain the MSE of the reliability function and the Bayes estimators of the reliabilityfunction of Weibull distribution will be obtained by the same way. The following steps will be considered to obtain the estimators:
Step (1): Generate random samples and from ELEDwith sample sizes 5, 10, 15 , 25 and 50, then we have three cases:
case1, the scale parameters are fixed whereand where parameter , and are unknown for , respectively.where their results shown in tables 1and table2, case2, the scale parameters are unknown where and also, the shape parameter are unknown for , respectively where the initial values can be taken as follow:
,and .their results for MLE ,Bayes results also the approximate confidence interval shown in tables 3,4 and last case is case 3, the scale parameters are unknown and different where and where the parameter are unknown for , respectively. So we take different initial values such as:,and .their results shown in table 5and table 6.
For each values of the sample size and, we will generate 1000 random samples from ELED.
Step (2): Using the Eq. (5) to find the MLE of R and use Eq.(11) to find Bayes estimators of R using non-informative prior for the first case . Also , Using the Eq. (18) to find the MLE of R and use Eq.(25) to find Bayes estimators of R using non-informative prior for the second case .Finally, Using the Eq. (33) to find the MLE of R and use Eq.(37) to find Bayes estimators of R using non-informative prior for the third case
Step (3): we take the average of their 1000 values then calculate the bias and the mean square error of R in different cases where for first case, for second case and for the last case.
It can be noted that for large ample sizes, the performance of the MLEestimators are better than the Bayes estimators of R in terms of biases and MSEs. It is also observedthat when () increases, the MSE and biases decrease for MLE and increase for Bayesianestimators expect in some points in large sample only. In addition, it is noted that for the small samples the MSE of both MLE and Bayes estimator of Rincrease little bit than it recorded for large samples.
The estimated values of R by using two methods MLE and Bayes estimator are recorded only for small values of parameters .
The confidence intervals CLAS, performs quite well as the sample sizes increases, have large interval length.
Table 1:ML Estimation of R in the case of common unknown parameter where are unknown and =1,/ / / Α / Β / / / / / / Bias() / MSE
(5 , 5) / 1 / 2 / 0.01 / 0.5 / 0.911 / 2.166 / 0.01 / 0.408 / 0.36 / -0.133 / 0.018
1 / 2 / 0.02 / 0.3 / 0.97 / 1.881 / 1.99*10-2 / 0.289 / 0.41 / 0.207 / 0.043
1 / 2 / 0.01 / 0.2 / 1.039 / 2.021 / 1.1*10-2 / 0.201 / 0.438 / 0.012 / 1.488*10-4
(5 , 10) / 1 / 2 / 0.01 / 0.5 / 1.227 / 1.842 / 0.01 / 0.761 / 0.36 / 0.479 / 0.23
1 / 2 / 0.02 / 0.3 / 1.069 / 2.1 / 0.19 / 0.437 / 0.41 / 0.553 / 0.305
1 / 2 / 0.01 / 0.2 / 1.144 / 1.965 / 1.002*10-2 / 0.187 / 0.438 / 0.384 / 0.147
(10,10) / 1 / 2 / 0.01 / 0.5 / 1.108 / 1.348 / 0.01 / 0.522 / 0.365 / 5.504* / 3.029*10-5
1 / 2 / 0.02 / 0.3 / 1.159 / 1.933 / 1.98*10-2 / 0.488 / 0.41 / -0.043 / 1.819*10-3
1 / 2 / 0.01 / 0.2 / 1.042 / 2.02 / 1.004*10-2 / 0.195 / 0.438 / -0.105 / 0.011
Table 2:Bayes Estimation of R in the case of common unknown parameter where are unknown and =1,
/ / / α / Β / / Bias() / MSE(5 , 5) / 1 / 2 / 0.01 / 0.5 / 0.36 / -0.36 / 0.129
1 / 2 / 0.02 / 0.3 / 0.41 / -0.41 / 0.168
1 / 2 / 0.01 / 0.2 / 0.438 / 0.438 / 0.192
(5 , 10) / 1 / 2 / 0.01 / 0.5 / 0.36 / -0.36 / 0.129
1 / 2 / 0.02 / 0.3 / 0.41 / -0.41 / 0.168
1 / 2 / 0.01 / 0.2 / 0.438 / -0.438 / 0.192
(10,10) / 1 / 2 / 0.01 / 0.5 / 0.365 / -0.36 / 0.129
1 / 2 / 0.02 / 0.3 / 0.41 / -0.41 / 0.168
1 / 2 / 0.01 / 0.2 / 0.438 / -0.438 / 0.192
Table 3: ML Estimation of R in the case of common unknown parameter where are unknown and asymptotic confidence intervals of R based on MLEs at significance level 0.05
/ / / α / Β / / / / / / CIAS / Bias() / MSE
(15 , 15) / 0.5 / 0.5 / 1 / 1 / 0.484 / 0.572 / 1.161 / 1.08 / 0.164 / ( 0.206 , 0.222 ) / 0.03 / 9.298*
0.5 / 0.2 / 3 / 4 / 0.555 / 0.2 / 2.212 / 3.424 / 0.03 / ( 0.006 , 0.679 ) / -0.037 / 1.347*
0.3 / 0.6 / 2 / 3 / 0.315 / 0.64 / 2.019 / 2.463 / 0.087 / ( 0.042 , 0.124 ) / -0.059 / 3.532*
(15 , 25) / 0.5 / 0.5 / 1 / 1 / 0.533 / 0.547 / 1.179 / 1.318 / 0.164 / ( 0.206 , 0.326 ) / 0.037 / 1.39*
0.5 / 0.2 / 3 / 4 / 0.537 / 0.215 / 1.215 / 1.9 / 0.03 / ( 0.0064 , 0.819 ) / -0.037 / 1.345*
0.3 / 0.6 / 2 / 3 / 0.377 / 0.639 / 2.206 / 3.095 / 0.087 / ( 0.019 , 0.8 ) / -0.058 / 3.388*
(25, 25) / 0.5 / 0.5 / 1 / 1 / 0.519 / 0.538 / 0.904 / 0.921 / 0.164 / ( 0.136 , 0.347 ) / 0.033 / 1.006*
0.5 / 0.2 / 3 / 4 / 0.414 / 0.22 / 2.606 / 3.666 / 0.03 / ( 0.0076 , 0.111 ) / -0.032 / 1.049*
0.3 / 0.6 / 2 / 3 / 0.294 / 0.68 / 2.533 / 2.867 / 0.045 / (0.018 , 0.108 ) / -0.053 / 2.848*
Table 4: ML. Estimation of R in the case of common unknown parameter where are unknown
/ / / α / Β / / Bias() / MSE
(25 , 25) / 0.5 / 0.5 / 1 / 1 / 0.216 / -0.216 / 0.047
0.5 / 0.2 / 3 / 4 / 0.64 / -0.064 / 4.104*10-8
0.3 / 0.6 / 2 / 3 / 0.087 / -0.087 / 7.593*10-8
(25 , 50) / 0.5 / 0.5 / 1 / 1 / 0.216 / -0.216 / 0.047
0.5 / 0.2 / 3 / 4 / 0.64 / -0.064 / 4.104*10-8
0.3 / 0.6 / 2 / 3 / 0.087 / -0.087 / 7.593*10-8
(50,50) / 0.5 / 0.5 / 1 / 1 / 0.216 / -0.216 / 0.047
0.5 / 0.2 / 3 / 4 / 0.64 / -0.064 / 4.104*10-8
0.3 / 0.6 / 2 / 3 / 0.087 / -0.087 / 7.593*10-8
Table5:ML.Estimation of R in the general case where are unknown
/ / / / / α / β / / / / / / / / Bias() / MSE
(25, 25) / 1 / 0.5 / 1.5 / 0.5 / 1.5 / 1 / 1.023 / 0.418 / 1.549 / 0.668 / 1.672 / 1.116 / 0.229 / 3.10*10-2 / 9.87*10-4
1 / 0.5 / 1.5 / 0.5 / 1.5 / 1.5 / 1.057 / 0.759 / 1.517 / 0.495 / 1.482 / 1.884 / 0.117 / -3.30*10-2 / 1.22*10-3
0.5 / 0.7 / 2 / 2 / 1.5 / 1.5 / 0.953 / 0.867 / 1.795 / 2.654 / 1.438 / 1.419 / 0.208 / -5.72*10-3 / 3.27*10-5
(25 ,50) / 1 / 0.5 / 1.5 / 0.5 / 1.5 / 1 / 1.039 / 0.968 / 1.553 / 0.658 / 1.106 / 1.378 / 0.229 / 2.50*10-2 / 6.214*10-4
1 / 0.5 / 1.5 / 0.5 / 1.5 / 1.5 / 0.925 / 0.654 / 1.385 / 0.48 / 1.713 / 1.282 / 0.177 / -3.30*10-2 / 1.121*10-3
0.5 / 0.7 / 2 / 2 / 1.5 / 1.5 / 0.563 / 0.751 / 2.294 / 1.894 / 1.586 / 1.498 / 0.208 / -1.00*10-2 / 1.056*10-4
(50,50) / 1 / 0.5 / 1.5 / 0.5 / 1.5 / 1 / 1.196 / 0.516 / 1.744 / 0.418 / 1.484 / 1.047 / 0.229 / -2.219*10-2 / 4.925*10-3
1 / 0.5 / 1.5 / 0.5 / 1.5 / 1.5 / 1.204 / 0.512 / 1.602 / 0.898 / 1.477 / 1.529 / 0.177 / -3.843*10-2 / 1.477*10-5
0.5 / 0.7 / 2 / 2 / 1.5 / 1.5 / 0.615 / 0.897 / 1.921 / 1.745 / 1.481 / 1.531 / 0.208 / 0.022 / 4.851*10-4
Table6 :Bayes estimation of R in the general case where are unknown
/ / / / / α / Β / / Bias() / MSE(25 , 25) / 1 / 0.5 / 1.5 / 0.5 / 1.5 / 1 / 0.229 / -0.229 / 0.053
1 / 0.5 / 1.5 / 0.5 / 1.5 / 1.5 / 0.177 / -0.177 / 0.031
0.5 / 0.7 / 2 / 2 / 1.5 / 1.5 / 0.208 / -0.208 / 0.043
(25 , 50) / 1 / 0.5 / 1.5 / 0.5 / 1.5 / 1 / 0.229 / -0.229 / 0.053
1 / 0.5 / 1.5 / 0.5 / 1.5 / 1.5 / 0.177 / -0.177 / 0.031
0.5 / 0.7 / 2 / 2 / 1.5 / 1.5 / 0.208 / -0.208 / 0.043
(50,50) / 1 / 0.5 / 1.5 / 0.5 / 1.5 / 1 / 0.229 / -0.0229 / 0.053
1 / 0.5 / 1.5 / 0.5 / 1.5 / 1.5 / 0.177 / -0.177 / 0.031
0.5 / 0.7 / 2 / 2 / 1.5 / 1.5 / 0.208 / -0.208 / 0.043
7. Conclusion
In this paper, the problem of estimating P(Y <X) for ELED has been addressed. The asymptotic distribution of the maximum likelihood estimator has been used to construct confidence intervals which function well even for small sample sizes. It has been observed that the Bayes estimators behave quite converge to zero. Moreover, the MSE of the estimates of R by two ways of estimators decrease as the sample sizes increase. The performance of theMLE estimators is also quite well and the MSEs of MLE estimators are smallerthan the MSEs of Bayes. Finally, the average lengths of all intervalsdecrease rapidly as increase.
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