Stress-strength reliability and its estimation for a component which is exposed two independent stresses

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Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 131-135, 2011 Applied Mathematics

Stress-Strength Reliability and Its Estimation for a Component Which Is Exposed Two Independent Stresses

Neriman Karadayı, Bu˘gra Saraço˘glu, Ahmet Pekgör

Selcuk University, Faculty of Science, Department of Statistics, 42003 Campus, Konya, Turkey

e-mail: nkaradayi@ selcuk.edu.tr; bugrasarac@selcuk.edu.tr ; ap ekgor@ selcuk.edu.tr

Abstract. In this study, the stress-strength reliability, R = P [M ax(Y1, Y2) <

X], and its maksimum likelihood estimation (MLE) are obtained when a com-ponent which has X strength with Gamma distribution is exposed to two inde-pendent stresses having exponential distributions with diﬀerent parameters. In addition, A simulation study is performed to see how the MLE behaves.

Key words: Stress-Strength Reliability, Maksimum Likelihood Estimator (MLE), Mean Square Eror (MSE).

2000 Mathematics Subject Classification: 62N05. 1. Introduction

Stress-strength model define the life of a component being exposed to Y stress and X having strength. According to this, If stress exceeds strength (Y > X), it will be impossible for a component to live. The reliability of such a sys-tem with one component which consist of stress and strength is expressed as R = P (Y < X) . In the majority of studies in this field, the distributions of X and Y random variables have been accepted to be independent variables com-ing from the same family and estimation of system reliability in stress-strength models for various distribution families has been studied. There are important contributions from Saraço˘glu et all (2005), Saraço˘glu et all (2007), Saraço˘glu et all (2009) for Gombertz distribution, Tong (1974, 1975), Beg (1980), Saraço˘glu et all (2011) ) for exponential distributions familiy, McCool (1991) for Weibull distributions familiy, Kundu ve Gupta (2005) for genaralized exponential dis-tribution and Hanagal (2003) for stress-strength models with multi components to the problem of system reliability estimation in stress-strength models.In this study, the reliability R = P [M ax(Y1, Y2) < X] of a component being exposed

to Y1 and Y2 stresses and having strength and MLE for this reliability have

been obtained. In addition, a simulation study on bias and MSE of MLE has been done.

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1.1. Stress-Strength Reliability

When a component having X strength is exposed to two independent Y1 and

Y2 stresses at the same time, stress-strength reliability is obtained as follows,

R = P [M ax(Y1, Y2) < X] = ∞ Z x=0 P (Y1< x) P (Y2< x) fX(x) dx R = ∞ Z x=0 [1 − exp (−x/θ1)][1 − exp (−x/θ2)] 1 Γ (α) βαx α−1_{exp (−x/β) dx} (1) _{R = 1 −} θ α 1 (β + θ1)α− θα₂ (β + θ2)α + θ α 1θα2 (θ1β + θ2β + θ1θ2)α

2. MLE of Stress-Strength Reliability

Let sample taken from Gamma distribution with parameters (α, β), (α is known) is X=(X1, X2, . . . , Xn) . Y1=(Y11, Y12, . . . , Y1n) and Y2=(Y21, Y22, . . . , Y2n) are

taken from with parameters θ1and θ2exponential distribution respectively and

regarding these samples, likelihood and log-likelihood functions are as follows.

(2) L =³_Γ(α)β1 α ńYn i=1x α−1 i exp Ã −β1 n X i=1 xi !_³ 1 θ1 ń × × exp Ã −θ11 n X i=1 y1i !_³ 1 θ2 ń exp Ã −θ12 n X i=1 y2i ! (3) ln L = −n ln Γ (α) − n ln β + (α − 1) n X i=1 ln xi− 1 β n X i=1 xi− 1 θ1 n X i=1 y1i− 1 θ2 n X i=1 y2i

And then as a result of taking diﬀerential lnL with respect to β, θ1 and θ2

parameters and equation to zero, the MLE’s of β, θ1and θ2have been obtained

as follows; (4) β =ˆ n X i=1 xiÁnα (5) ˆθ1= n X i=1 y1iÁn 132

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(6) ˆθ2= n

X

i=1

y2iÁn

By benefiting from invariant property of MLE, the MLE of stress-strength reli-ability is as follows; (7) _{R = 1 − 1}ˆ ˆθ α 1 ³ ˆ β + ˆθ1 ´α − ˆ θα₂ ³ ˆ β + ˆθ2 ´α + ˆ θα₁ˆθα₂ ³ ˆ θ1β + ˆˆ θ2β + ˆˆ θ1ˆθ2 ´α 3. Simulation Study

The MLE of the reliability, R = P [M ax(Y1, Y2) < X], of a component being

exposed to Y1 and Y2 stresses and having X strength, for various values of n,

θ1, θ2 and β simulation study with 1000 trials has been done. As a result of

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Table 3.1.For diﬀerent values of n, θ1,θ2,α and β

the MSE and bias of the MLE of R

4. Conclusion

In this study, the reliability, R = P [M ax(Y1, Y2) < X], of a component having

X strength coming from Gamma distribution with α and β parameters (α is known) and being exposed to Y1 and Y2 stresses which have exponential

distri-butions with θ1and θ2parameters has been obtained. Furthermore, acccording

to the table 3.1., when n is increased MSE and bias of MLE are decreased. When α is increased, MSE and bias of MLE don’t change much. When n, θ1, θ2

and α are fixed and β is increased, MSE and bias of MLE are decreased. When θ2 and α together are increased and others are fixed, MSE value is decreased

while bias value is increased. Also, MSE of MLE for R has been observed as quite small.

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References

1. Beg, M.A. 1980, On the Estimation ofP (Y < X)for the Two Parameter Expo-nential Distribution. Metrika, 27 : 29-34.

2. Hanagal D. 2003, Estimation of system reliability in multicomponent stress strength models; Journal of the Indian Statistical Association, 41, 1-7.

3. Kundu, D. and Gupta, R.D. (2005), "Estimation ofR = P (Y < X) for the gen-eralized exponential distribution", Metrika, vol. 61, 291 - 308.

4. McCool, J.I. 1991 Inference onP (Y < X)in the Weibull Case. Communications in Statistics: Simulation And Computation, 20: 129-148.

5. Saraço˘glu, B., Kınacı, ˙I., Kundu, D. (2011) On estimation of R = P (Y < X) for exponential distribution under progressive type-II censoring, Journal of Statistical Computation and Simulation (in press)

6. Saraço˘glu, B., Kaya, M.F. (2007), Maximum likelihood estimation and confidence intervals of system reliability for Gompertz distribution in stress-strength models", Selçuk Journal of Applied Mathematics, vol. 8, 25 - 36.

7. Saraço˘glu, B., Kaya, M.F., Abd-Elfattah, A.M. (2009), "Comparison of estimators for stress-strength reliability in Gompertz Case", Hacettepe Journal of Mathematics and Statistics, vol. 38, 339 - 349.

8. Saraço˘glu, B., Kaya, M.F., 2005. Reliability and Its Estimation for Gompertz Dis-tribution, Ordered Statistical Data: Approximations, Bounds and Characterizations, International Conference, ˙Izmir, Turkey.

9. Tong, H. 1974. A Note on the Estimation ofP [Y < X] in the Exponential Case. Technometrics, 16: 625.