. Censored Samples Bayesian Analyses of the Stress-Strength Gompertz Reliability Model under Singly Type

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Bayesian Analyses of the Stress-Strength Gompertz Reliability Model under Singly Type Censored Samples

Ass.P.Dr. Nada S. Karam^* Abthal F. Sabea

AL-Mustanseriya University, College of Education, Mathematical Department, Baghdad, Iraq

Abstract When X and Y are in depended. Gompertz random variable parameters (β,α) and (λ ,α) respectively . A stress –strength model defines life of a component with strength X and is subjected to stress Y the system fails if and only if, at any time, the applied stress is greater than its strength. In this paper , we consider the estimation problem of when X Gompertz ( β ,α) and Y Gompertz (λ ,α ). by Bayesian analysis has been considered in the paper. The Gamma and Quasi prior have been assumed for posterior analysis. The estimation has been made under singly type ΙΙ censored samples. The Bayes estimator for the reliability function (R) has been obtained under four different loss functions (Weighted, Quadratic, Entropy, Squared error). The simulation study has been conducted to compare by mean square error (MSE) for the performance of various estimators

.

Keywords: Stress–Strength Reliability, Gompertz Distribution ,Bayesian Estimation, Prior (Gamma and Quasi), (Squared error, Quadratic, Weighted ,Entropy) loss functions.

1. INTRODUCTION

The Gompertz distribution was originally introduced by Gompertz (1825).This distribution is used to model survival times, human mortality and actuarial tables. It has many real life applications, especially in medical and actuarial studies. The Gompertz distribution is also used as a survival model in reliability. It has an increasing hazard rate for the life of the systems. Due to its complicated form it has not received enough attention in past. However, recently, this distribution has received considerable attention from demographers and actuaries. Gordon (1990) examined the feasibility of maximum likelihood estimation of a mixture of two Gompertz distributions when censoring occurs. Pollard and Valkovics (1992) were the first to deal with the Gompertz distribution thoroughly. However, their results are true only in cases where the initial level of mortality is very close to zero. Chen (1997) developed an exact confidence interval and a joint confidence region for the parameters of Gompertz distribution. Willemse and Koppelaar (2000) reformulated the Gompertz force of mortality and derived relationships for this new formulation. this distribution has been studied by some authors. For example, see Wu et al. [12], Jaheen [7], Wu et al. [13, 14], Ismail [6] and Al-Khedhair & El-Gohary[8].

aim of this paper is to use Beyes method to estimate the Reliability (R) of the Gompertz distribution for the stress- strength models by two priors ( Gamma and Quasi ) distribution under four loss function (Squared error, Quadratic, Weighted ,Entropy) and for realization this aim we treat the subject as follows :

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It is assumed that the univariate Gompertz distribution with the shape parameter β>0 ,λ>0 and the scale parameter α>0 has the following probability density function, cumulative distribution function and survival function for x > 0 , y>0 ;

(1)

(2)

(3)

(4)

The problem of estimating R = P(Y < X) arises in the context of mechanical reliability of a system with strength X and stress Y , the reliability, R, is chosen as a measure of system reliability. In a stress strength model, Let be the strength of a system and be the stress acting on it .where and are two random variables from with parameters (β,α) and (λ,α) respectively. That is, the probability density functions and the cumulative distribution functions of and are, respectively

2.Likelihood function.

Consider a random sample of size (n)and (m) from Gompertz Type X and Y distribution, and let ( … ) ,( … we want the work of ( r ) units where ( r < n)and ( r < m) , so the time in this case a random variable cannot be determined and thus stop the test up to get r units censoring , the likelihood function for β and λ given the Type II single censored sample x = ( … ) and y=( … are :[5] (5) Substitute equation (1) in

=

Substitute equation (3) in λ

λ λ

(3)

=

… Substitute equations (6) and (7) in equation (5) : L(β,α\x)=

…

Substitute equations (8) and (9) in equation (5) we get:

L(λ,α\Y) =

λ …

2 . Bayesian Estimations under single type II censored samples using different priors and loss function .

In this section Bayesian Estimators of the shape parameter for two different prior functions and under four different loss functions has been determine.

● Types of loss functions using in this paper

If λ represent of estimator for the shape parameter θ (β,λ) , then for 1-Weighted loss function : the weighted loss function defined as:

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2-Quadratic loss function : the quadratic loss function defined as: [11]

(13)

3-Entropy loss function : the entropy loss function defined as: [6]

(14)

4-Squared error loss function : the squared error loss function defined as:[11]

(15)

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● The Posterior distributions with different priors

For the given two random variable X and Y, the posterior density function of the shape parameters β and λ is well known as :

For Bayesian estimation, we specify two different prior distributions for the shape parameter, and which can be obtained two different posterior distributionsunder single type ΙΙ censored samples, as follows :

1-The Gamma prior:

The most widely used prior distribution of the parameters (β ,λ) is the gamma distribution with hyper-parameters ‘ ’ and ‘ ’ with probability density function given by :

g(β)=

g (λ)=

combining the prior densities of λ ,and the likelihood functions given in equations and ,to obtain the joint posterior density of ( λ

where

where

=

(5)

=

λ (16) Where

2-The Quasi prior:

The Quasi prior of the parameters (β, λ ) with hyper-parameter ‘ ’ is defined as;

p(β)=

p(λ)= ,λ > 0

combining the prior densities of λ ,and the likelihood functions given in equations and ,to obtain the joint posterior density of ( λ

where

λ

where

(6)

=

λ

=

λ ^λ (17)

where

3-1. Bayesian Estimators under Single Type censored samples under Gamma Prior using different loss functions.

Weighted loss function 1.

- 1 - 3

The Bayesian estimator for , denoted by equation(16), for Gamma prior information under weighted loss function and , from equation(12),is given by:

are:

^[

Where

(7)

loss function

Quadratic .

2 - 1 - 3

The Bayesian estimator for , denoted by equation(16), for Gamma prior information under Quadratic loss function and , from equation(13),is given by:

loss function Entropy

. 3 - 1 - 3

The Bayesian estimator for , denoted by equation(16), for Gamma prior information under Entropy loss function and , from equation(14),is given by:

(8)

loss function

Squared error .

4 - 1 - 3

The Bayesian estimator for , denoted by equation(16), for Gamma prior information under Squared error loss function and , from equation(4),(5),is given by:

3-2. Bayesian Estimators under Single Type censored samples under Quasi Prior using different loss functions.

1. Weighted loss function -

2 - 3

The Bayesian estimator for , denoted by equation(17), for Quasi prior information under weighted loss function and , from equation(12),is given by:

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λ

tion

loss func Quadratic

. 2 - 2 - 3

The Bayesian estimator for , denoted by equation(17), for Quasi prior information under Quadratic loss function and , from equation(13),is given by:

loss function Entropy

. 3 - 2 - 3

The Bayesian estimator for , denoted by equation(17), for Quasi prior information under Entropy loss function and , from equation(14),is given by:

(10)

λ

loss function

rror Squared e .

4 - 1 - 3

The Bayesian estimator for , denoted by equation(17), for Quasi prior information under Squared error loss function and , from equation(15),is given by:

λ

(11)

4 .Simulation results and Conclusions.

In this section, the results presented of some of numerical experiments to compare the performance of the Bayes estimators Reliability function for shape parameter under two prior distributions and four loss functions proposed in the previous sections, applying Monte Carlo simulations to comer the performance of different estimators, mainly with respect to their mean squared error (MSE) for different sample size (n=

15, 25, 30, 50, 100,50, 80) and (m=16,25,30,50,100,60,100) and four values of the shape parameters (β ,λ,α)=(6,2,4),(2,2,3),(3.5,3,3),(2,4,2). The results of (MSE) are computed

over (1000) replications for one different case ( case I ; , , , and recorded in tables (1),(2),(3) ,(4).

The random number has been generated by inverse function method, which is for uniform random U:

… …

The random number has been generated by inverse function method, which is for uniform random U:

… …

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best Real =0.2500 β=6 ,λ=2 ,α=4

sample

gamma prior ,b1=1 ,b2=4 m

n

Rs RE

RQ Rw

15 15

0.2077 0.2002

0.2007 0.2197

mean

Rw 0.0018

0.0025 0.0024

0.00090129 mse

Quasi prior k1=5 ,k2=0.1

Rs RE

RQ Rw

0.3579 0.2952

0.2960 0.3200

mean

RE 0.0116

0.0020 0.0021

0.0049 mse

gamma prior ,b1=1 ,b2=4

25 25

Rs RE

RQ Rw

0.2101 0.2189

0.2191 0.2309

mean

Rs 0.0016

0.00096538 0.00095329

0.00036494 mse

Rs RE

RQ Rw

0.3075 0.2739

0.2741 0.2877

mean

RE 0.0033

0.00057088 0.00058241

0.0014 mse

Rs RE

RQ Rw

30 30

0.2774 0.2199

0.2200 0.2301

mean

Rw 0.00075000

0.00090624 0.00089784

0.00039699 mse

Rs RE

RQ Rw

0.2992 0.2712

0.2713 0.2827

mean

RE 0.0024

0.00044846 0.00045571

0.0011 mse

Rs RE

RQ Rw

50 50

0.2453 0.2313

0.2314 0.2374

mean

Rs 0.000021689

0.00035053 0.00028926

0.00015820 mse

Rs RE

RQ Rw

0.2788 0.2630

0.2630 0.2696

mean

RE 0.00083148

0.00016849 0.00016998

0.00038526 mse

Rs RE

RQ Rw

100 100

0.2546 0.2399

0.2397 0.2430

mean

Rs 0.000021614

0.00010169 0.00010143

0.000048863 mse

Quasi prior k1=5,k2=0.1

Rs RE

RQ Rw

0.2647 0.2571

0.2572 0.2603

mean

RE 0.00021544

0.000049902 0.000050096

0.00010591 mse

60 50

Rs RE

RQ Rw

0.2365 0.2325

0.2326 0.2387

mean

Rw 0.00018302

0.00030451 0.00030273

0.00012750 mse

Rs RE

RQ Rw

0.2784 0.2625

0.2626 0.2692

mean

RE 0.00080518

0.00015693 0.00015838

0.00036768 mse

100 80

RS RE

RQ RW

0142.0 014200

014208 0.2448

mean

RW 8.408

0.0000 84320

0.0000 8.004

0.0000 000047068

0.

MSE

RS RE

RQ RW

014706 01470.

01470.

014724 mean

RE 38034

0.000 .0.62

0.000 .04.0

0.000 40020

0.000 mse

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Table (2). The value of (MSE) for Bayesian est.

best Real = 0.5000 β=2 ,λ=2 ,α=3

sample

gamma prior ,b1=1 ,b2=4 m

n

Rs RE

RQ Rw

15 15

0.4341 0.3726

0.3734 0.3945

Rs 0.0043

0.0162 0.0160

0.0111 mse

Rs RE

RQ Rw

0.6176 0.5582

0.5589 0.5770

mean

RE 0.0138

0.0034 0.0035

0.0059 mse

25 25

Rs RE

RQ Rw

0.5516 0.4140

0.4143 0.4266

mean

Rs 0.0027

0.0074 0.0073

0.0054 mse

Rs RE

RQ Rw

0.5664 0.5318

0.5320 0.5430

mean

RE 0.0044

0.0010 0.0011

0.0018 mse

Rs RE

RQ Rw

30 30

0.5408 0.4244

0.4246 0.4349

mean

RS 0.0017

0.0058 0.0057

0.0042 mse

Rs RE

RQ Rw

0.5566 0.5275

0.5276 0.5370

mean

RE 0.0032

0.00075412 0.00076423

0.0014 mse

Rs RE

RQ Rw

50 50

0.5299 0.4524

0.4525 0.4585

mean

Rs 0.00089633

0.0023 0.0023

0.0017 mse

Rs RE

RQ Rw

0.5332 0.5161

0.5161 0.5217

mean

RE 0.0011

0.00025784 0.00025990

0.00047078 mse

Rs RE

RQ Rw

100 100

0.5047 0.4751

0.4751 0.4780

mean

Rs 0.000021875

0.00062049 0.00061968

0.00048360 mse

Rs RE

RQ Rw

0.5163 0.5078

0.5078 0.5106

mean

RE 0.00026498

0.000061288 0.000061535

0.00011292 mse

gamma prior b1=1 ,b2=4

60 50

Rs RE

RQ Rw

0.5165 0.4521

0.4521 0.4582

mean

Rs 0.00027320

0.0023 0.0023

0.0018 mse

Rs RE

RQ Rw

0.5328 0.5157

0.5158 0.5213

mean

RE 0.0011

0.00024652 0.00024854

0.00045543 mse

100 80

RS RE

RQ RW

0.5204 0.4691

0.4692 0.4728

mean

RS 0.000079680

0.00095174 0.00095015

0.00073782 mse

RS RE

RQ RW

0.5204 0.5098

0.5098 0.5133

mean

RE 0.00041615

0.000096090 0.000096575

0.00017703 mse

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Best Real =0.4615 β=3.5 ,λ=3 ,α=3

sample

gamma prior b1=1 ,b2=4 m

n

Rs RE

RQ Rw

0.4015 0.3192

0.3200 0.3415

mean

15 15

Rs 0.0036

0.0203 0.0200

0.0144 mse

Rs RE

RQ Rw

0.5796 0.5175

0.5183 0.5381

mean

RE 0.0139

0.0031 0.0032

0.0059 mse

25 25

Rs RE

RQ Rw

0.4890 0.3627

0.3629 0.3758

mean

Rs 0.00075649

0.0098 0.0097

0.0074 mse

Rs RE

RQ Rw

0.5316 0.4959

0.4962 0.5080

mean

RQ, RE 0.0049

0.0012 0.0012

0.0022 mse

Rs3 RE3

RQ3 Rw3

30 30

0.4373 0.3736

0.3738 0.3846

mean

Rs 0.00058791

0.0077 0.0077

0.0059 mse

Rs RE

RQ Rw

0.5219 0.4919

0.4921 0.5021

mean

RE 0.0036

0.00092046 0.00093242

0.0016 mse

gamma prior β=0.6 ,λ=3 ,α=3 ,b1=1 ,b2=4

Rs RE

RQ Rw

50 50

0.4500 0.4026

0.4026 0.4090

mean

Rs 0.00013249

0.0036 0.0035

0.0028 mse

Rs RE

RQ Rw

0.4941 0.4764

0.4765 0.4825

mean

RE 0.0011

0.00022126 0.00022331

0.00043940 mse

gamma prior β=0.6 ,λ=3 ,α=3 ,b1=1 ,b2=4

Rs RE

RQ Rw

100 100

0.4525 0.4286

0.4285 0.4316

mean

RS 0.000081155

0.0011 0.0012

0.0008.9574 mse

Rs RE

RQ4 Rw4

0.4801 0.4715

0.4715 0.4745

mean

RE 34529

0.000098852 0.000099185

0.00016735 mse

60 50

Rs RE

RQ Rw

0.4085 0.4030

0.4030 0.4094

mean

Rw 0.0028

0.0034 0.0034

0.0027 mse

Rs RE

RQ Rw

0.4952 0.4776

0.4777 0.4837

mean

RE 0.0011

0.00025907 0.00026128

0.00049119 mse

.00 80

RS RE

RQ RW

012360 012443

012442 012473

mean

RS 22607

0.000 0100.2

0100.2 0100.4

mse

RS RE

RQ RW

0128.6 012600

012600 012627

mean

RE 20684

0.000 86.70

0.0000 086722

0.000 .6.70

0.000 mse

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beast β=4 ,λ=2 ,α=4

Real =0.3333 sample

gamma prior b1=1 ,b2=4 m

n

Rs RE

RQ Rw

15 15

0.3608 0.2552

0.2558 0.2765

mean

Rs 0.00072020

0.0061 0.0061

0.0032 mse

Rs RE

RQ Rw

0.4597 0.3937

0.3946 0.4182

mean

RE 0.0160

0.0036 0.0038

0.0072 mse

25 25

Rs RE

RQ Rw

0.3071 0.2767

0.2769 0.2896

mean

Rs 0.00068853

0.0032 0.0032

0.0019 mse

Rs RE

RQ Rw

0.3960 0.3595

0.3597 0.3735

mean

RE 0.0039

0.00068271 0.00069781

0.0016 mse

Rs RE

RQ Rw

30 30

0.3147 0.2872

0.2873 0.2982

mean

Rs 0.00034593

0.0021 0.0021

0.0012 mse

Rs4 RE4

RQ4 Rw4

0.3890 0.3584

0.3587 0.3703

mean

RE 0.0031

0.00063085e 0.00064118

0.0014 mse

Rs RE3

RQ3 Rw3

50 50

0.3165 0.3024

0.3025 0.3090

mean

Rs 0.0002846

0.00095565 0.00095170

0.00059097 mse

Rs RE

RQ Rw

0.3654 0.3479

0.3479 0.3547

mean

RE 0.0010

0.00021129 0.00021333

0.00045837 mse

Rs RE

RQ Rw

100 100

0.3240 0.3174

0.3175 0.3208

mean

RS 0.000087369

0.00025192 0.00025141

0.00015813 mse

Rs RE

RQ Rw

0.3509 0.3424

0.3424 0.3457

mean

RE 0.00030810

0.000081937 0.000082244

0.00015411 mse

60 50

Rs RE

RQ Rw

0.3435 0.3024

0.3024 0.3090

mean

Rs 0.00010291

0.00095947 0.00095551

0.00059397 mse

Rs4 RE4

RQ4 Rw4

0.3652 0.3477

0.3478 0.3546

mean

RE 0.0010

0.00020627 0.00020829

0.00045135 mse

.00 80

RS RE

RQ RW

013040 013.4.

013.44 013.73

mean

RW 0.00092653

22886 0.000 2268.

0.000 40.42

0.000 mse

RS RE

RQ RW

0132.7 013200

013200 01322.

mean

RW 33344

0.000 26342

0.000 0.0005727

0.00013921 mse

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It appears in the tables the MSE values for parameters and different methods and by table (1) we get that the best way was when we approximate Gompertz distribution under Gamma distribution in squared loss function when the sample size (100,100).

At the table (2) we get that the best way was when under Gamma distribution in squared loss function when the sample size (100,100).

At the table (3) we get that the best way was when under Gamma distribution in squared error loss function when the sample size (100,100).

At the table (4) we get that the best way was when under Quasi distribution in squared loss function when the sample size (100,100).

5 CONCLUSIONS

The above study suggests that in order to estimate the parameter of Burr type X and Y distribution under a Bayesian framework, when( β, λ, α =6 ,2, 4 ) that the performance of Bayes estimator R under Gamma prior with( Squared error loss function, records full appearance "for all sample sizes", as the best prior distribution and using different loss functions and when( β, λ, α =6 ,2, 4 ) , that the performance of Bayes estimator R under Quasi prior with( Entropy loss function , records full appearance "for all sample sizes", as the best loss function and prior distribution. Can be preferred for the single type II censored sample.

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