BAYESIAN RELIABILITY ANALYSIS FOR THE GUMBEL FAILURE MODEL. 1. Introduction

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1 BAYESIAN RELIABILITY ANALYSIS FOR THE GUMBEL FAILURE MODEL CHRIS P. TSOKOS AND BRANKO MILADINOVIC Department of Mathematics Statistics, University of South Florida, Tampa, FL 3362 ABSTRACT. The Gumbel or double exponential probability distribution is used to characterize the failure times of a given system. The ordinary Bayesian settings of the reliability functions are being studied. In the Bayesian reliability analysis, we utilize Jeffrey s non-informative prior to obtain estimates of the target time of a system to achieve a desired reliability. Lindley s approximation procedure is used to obtain numerical estimates of the target time. 1. Introduction Extreme value probability distributions have been used effectively to model various problems in engineering, environment, business, etc. Some key references are (Burton Markopoulos, 1985; Naess, 1998; Osella et al., 1992; Ramachran, 1982; Rao et al., 1997; Sastry Pi, 1991; Silbergleit, 1996; Suzuki Ozaha, 1994; Tsokos, 1999). The object of the present study is to use the classical Gumbel or double exponential probability distribution to characterize the failure times of a given system, both in the ordinary Bayesian settings. In the Bayesian setting, we assume that the prior probability density function is the Jeffrey s non-informative prior under the mean square error loss function. We are interested in obtaining ordinary Bayesian estimates of a target time t α, subject to a desired specified reliability. That is, for a given system what is the time to failure, t α, with at least (1 - α)% assurance. For example, we want to be at least 95% certain that the system will be operable to time t.5. We develop both ordinary Bayesian estimates of t α introduce Lindley s approximation procedure that is used to obtain numerical results that illustrate the usefulness of the study. 2. The Gumbel Model For the Gumbel model, the probability distribution function (p.d.f) the cumulative distribution function (c.d.f) of the failure time T are given, respectively, by (2.1) f(t) = 1 e t µ e t µ,

2 2 CHRIS P. TSOKOS AND BRANKO MILADINOVIC < t <, < µ <, > (t µ) (2.2) F(t) = exp{ exp{( }} where µ are the location scale parameters, respectively. This model has been used in fire protection, insurance problems, prediction of earthquake magnitudes, carbon dioxide levels in the atmosphere, high return levels of wind speeds in the design of structures among others. In the present study, we shall apply the subject model in reliability analysis, more specifically, Bayesian reliability modeling. 3. Reliability modeling Let t 1, t 2, t 3,..., t n be the failure times that follow the Gumbel p.d.f. given by (2.1). The likelihood function L(µ, ), is given by L(µ, ) = n t i µ exp{ its logarithmic form is (3.1) LogL = nln ( t i µ ) exp( t i µ )} ) The maximum likelihood estimates (MLE) for µ can be obtained from the likelihood functions by solving the following equations (3.2) ˆ + Σt ie t i/ˆ Σe t i/ˆ = t (3.3) ˆµ = ˆln{ 1 n Σe t i/ˆ } Equations (3.2) (3.3) are not analytically tractable must be solved numerically to obtain approximate MLE s of µ, that is ˆµ ˆ. By taking the natural logarithm of both sides of equation (2.2) solving for t we obtain the expression for the target time t α under the desired reliability 1 - α given by (3.4) t α = µ (ln( ln(α)) Thus, by the invariance property of the MLE s we can obtain the MLE of the target time t α (3.5) ˆt α = ˆµ ˆ(ln( ln(α)) Classical estimates confidence intervals for t α can be obtained using the method of maximum likelihood the normal approximation for different extreme value models. In the present study, we shall examine the estimation of t α for an extreme

3 GUMBEL FAILURE MODEL 3 value model in a Bayesian setting under a specified prior mean square error loss function. 4. Bayesian approach to the Gumbel Model In the Bayesian approach we regard µ behaving as rom variables with a joint p.d.f. π(µ, ). We shall investigate the point estimator of t α for Jeffrey s non-informative prior Jeffrey s non-informative prior. Jeffrey s non-informative prior chooses the prior π(µ, ) to be proportional to deti(θ), where I(θ) is the expected Fisher information matrix. That is, 2 lnl µ I(ˆµ, ˆ) = E 2 2 lnl µ 2 2 lnl 2 lnl µ 2 2 Using logl as given in (3.1), we obtain I(θ) as n n (1 γ) (4.1) I(θ) = 2 2 n n (1 γ) {γ 2 2γ η(2, 2)} 2 2 where γ is the Euler s constant Hence, η(p, q) = 1 Γ(p) det(i(θ)) = K 4 t p 1 e qt dt. 1 e t implying that the Jeffrey s non-informative prior is given by.. (4.2) π(µ, ) = 1 2 We remark that π is an improper prior p.d.f. Consistent with the aim of the present study in identifying the target time t α, we proceed to obtain its analytical form Posterior distribution. The posterior probability density function of (µ, ) given the failure times t 1,..., t n is given by π(µ, t 1, t 2,..., t n ) = L(µ, t 1,..., t n )π(µ, ) L(µ, t 1, t 2,..., t n )π(µ, )dµd where L(µ, t 1,..., t n ) is given by (3.1). We shall first compute the marginal probability density function, that is, L(µ, t 1,..., t n )π(µ, ) dµ d. Using the prior π(µ, ) = 1, > letting x = n t 2 i, we obtain

4 4 CHRIS P. TSOKOS AND BRANKO MILADINOVIC L(µ, t 1,..., t n )π(µ, )dµd (4.3) = n 2 + e x/ e (nµ/) e e(µ/θ) n e t i / dµd. Let u = e, a = n e (t 1/), the expression (4.3) can be written as (4.4) L(µ, t 1,..., t n )π(µ, )dµd = n 1 e x/ u n 1 e au dud = Γ(n) = Γ(n) = Γ(n) where x = n t i = n t. n 1 e x/ a n d v n 1 e xv ( v n 1 ( e tiv ) n dv e v(ti+ t) ) n dv 4.3. Bayesian estimation of t α for Jeffrey s prior. The Bayes estimate of t α = µ ln( ln(α)) for squared error loss is given by ˆt B = E(t α t 1, t 2,..., t n ) [µ ln( ln(α)]l(µ, t 1, t 2,..., t n )π(µ, )dµd or (4.5) ˆt B = [µ ln( lnα)]l(µ, t 1, t 2,..., t n )π(µ, )dµd Γ(n) v n 1 ( e (t i+ t)v ) n dv Proceeding as we did before for obtaining the marginal probability distribution we can write (4.6) E(µ t) = v n 2 e xv (lnu)u n 1 e au dudv where a = e tiv,

5 (4.7) E( t) = Γ(n) Hence, where E(ˆt α t 1,..., t n ) = +( ln( lnα)) GUMBEL FAILURE MODEL 5 v n 2 ( e v(ti+ t) ) n dv. v n 2 e xv (lnu)un 1 e au dudv Γ(n) v n 1 [ n e v(t i+ t) ] n dv v n 2 [ e v(t i+ t) ] n dv v n 1 [ n e v(t i+ t) ] n dv a = e t iv t = n. To evaluate the above expression to obtain approximate Bayesian estimates of t α, we shall use Lindley s approximation method The Lindley Approximation. Similar to our work in the previous chapter, let u(θ)v(θ)e L(θ) dθ I = v(θ)e L(θ) dθ where θ = (θ 1, θ 2,..., θ k ), a vector of parameters. Also, let L=log(likelihood function) Note that I is the posterior expectation of u( θ) given the failure data, for a prior v(θ). Denote by Furthermore, u 1 = u θ 1 u 11 = 2 u θ 2 1 t i u 2 = u θ 2 u 22 = 2 u θ2 2 p = π(θ 1, θ 2 ) p 1 = p θ 1 ; p 2 = p θ 2 L 2 = 2 L ; L θ1 2 2 = 2 L θ2 2 L 3 = 3 L ; L θ1 3 3 = 3 L θ = ( L 2 ) 1 22 = ( L 2 ) 1 E(u(θ) t) = u(ˆθ 1, ˆθ 2 ) (u u ) + P 1 u P 2 u (L 3u L 3 u L 21 u L 12 u )

6 6 CHRIS P. TSOKOS AND BRANKO MILADINOVIC evaluated at (ˆθ 1, ˆθ 2 ), where ˆθ 1 ˆθ 2 are the MLEs of θ 1 θ 2. The target time for the Gumbel model given by t B = µ b = u(µ, ), where θ 1 = µ θ 2 =. Also, u 1 = 1 u 2 = b where b = ln( lnα) where u 11 = u 22 =. Thus, we can write P(θ 1, θ 2 ) = π(µ, ) = 1 2 P 1 = P 2 = 2 3. Let ˆµ ˆ be the classical MLEs for µ, respectively. Furthermore, we have { } L = n exp ( t i µ ) exp( t i µ ) or Thus, Also, L,2 = 2 lnl 2 lnl lnl = nln lnl µ = n 1 ( t i µ L 2, = 2 lnl µ 2 = 1 2 = n + (t i µ) = n 2[ (t 2 i µ)] which can be expressed as or n ( (t 2 i µ)) 1 + ( 3 n [ 2(t 2 i µ)[1 e (t i µ) We proceed to find L 3, L,3, that is ) e t i µ e t i µ. e t i µ e t i µ ) (t i µ) ]] ( L 3, = 3 lnl µ 3 = 1 3 e t i µ. ) (t i µ) (ti µ) 1 4 e t i µ (ti µ) 2 ) (t i µ) 2 ) 1 4. e t i µ ) (t i µ) 2

7 Also, or GUMBEL FAILURE MODEL 7 L,3 = 3 lnl = 2n + 6[ [ (t i µ) 2 ] 1 1[ 5 e (t i µ) L 21 = ( 2 lnl µ 2 ) = [ (t i µ)[1 e (t i µ) ]] 1 4 e (t i µ) (t i µ) 3 ] 1 6. ) ] 2 3 [ L 12 = µ ( 2 lnl 2 ) )(t i µ) ] 1 4 L 12 = 2 3(n e t i µ 4 ) + (ti µ)e t i µ 4 1 (t 5 i µ) 2 e t i µ Thus, a Bayesian approximate estimate for t α is given by (4.8) ˆt B = ˆt α (MLE) + P 2 u (L L 3 u L 21 u L ) evaluated at the MLE of µ, ˆµ ˆ. 5. Numerical Analysis In this section we present a numerical study in order to compare the maximum likelihood Bayes estimates for determining the target time of the Gumbel failure model subject to specified reliability. Our numerical simulation was conducted in the following manner: 1. Under the assumption that the location parameter µ the scale parameter behave romly independently, we simulated m (m = 5, 1, 2) location parameters from the normal distribution. In order to study the effects of the prior variance on our estimates, we simulated location parameters from the normal distribution with mean 25 variances equal to 1, 4, 9 respectively. 2. We assumed the scale parameter follows the uniform distribution. However, in order to see what effects the increase of variance has on our estimates, we let equal to 1, 2 4 respectively. 3. Using the obtained m pairs of µ, we generated n (n = 5, 1, 2) observations from the Gumbel p.d.f. calculated both the maximum likelihood Bayes estimates of the target time. 4. For comparison purposes, we calculated the absolute value of the difference between the true target time the corresponding ML Bayes estimates for 99% reliability. A schematic diagram of the complete step-by-step process of the numerical analysis is presented in Figure 1.

8 8 CHRIS P. TSOKOS AND BRANKO MILADINOVIC Figure 1. Numerical Study of the Gumbel Failure Time m n µ B, ˆµ B, ˆ t α ˆt α t α ˆt B , , , , , , , , , , , , , , , , , , 3, Table 1. Comparison between ML Bayesian Estimates of Reliability Time: µ N(25, 1), = 1,2,4, α =.1 Due to the size of our simulation some of the numerical results are given in Tables 1-3 under 99% reliability. In each table we present the size of the prior sample m used to calculate the Bayes estimate µ B, while ˆµ ˆ are the ML estimates of the location scale parameters. t α ˆt α t α ˆt B represent the absolute value of the

9 GUMBEL FAILURE MODEL 9 difference between the true target time, maximum likelihood Bayes target time estimates respectively. As we can see from Table 1, by keeping the prior sample m n µ B, ˆµ B, ˆ t α ˆt α t α ˆt B , , , , , , , , , , , , , , , , , , Table 2. Comparison between ML Bayesian Estimates of Reliability Time: µ N(25, 2), = 1,2,4, α =.1 size m = 5 prior variance fixed varying the sample size of the failure model from n = 5 to n = 2, the absolute value of the difference t α ˆt α t α ˆt B decreases. This behavior is consistent as we increase n, except that we notice m n µ B, ˆµ B, ˆ t α ˆt α t α ˆt B , , , , , , , , , , , , , , , , , , Table 3. Comparison between ML Bayesian Estimates of Reliability Time: µ N(25, 3), = 1,2,4, α =.1 a significant improvement in the ML estimate. In Table 2 Table 3 we increase the prior sample size m to 1 2 prior variance to 4 9 respectively also observe that the absolute value of the difference t α ˆt α t α ˆt B decreases. The increase in the prior variance has no effect on the behavior of our estimates. This is consistent as we increase n, we again notice a significant

10 1 CHRIS P. TSOKOS AND BRANKO MILADINOVIC improvement in the ML estimate. In almost every case the Bayes estimate is closer to the true target time than its maximum likelihood counterpart. 6. Conclusion As expected, the Monte Carlo simulation indicates that the Bayes estimate under the non-informative prior is closer to the true reliability time than its maximum likelihood counterpart. However, the following findings are in order: 1. An increase in the prior sample size for the location parameter has no effect on the behavior of the estimates. 2. An increase in the sample size of the simulated Gumbel data results in the improvement of both the maximum likelihood Bayesian estimates. 3. When we increase the variance of the prior distribution from 1 to 4 to 9 the variance of the simulated Gumbel data from 1 to 2 to 4, we notice a significant improvement in the maximum likelihood estimate. We therefore conclude that for large sample size high variance there is very little difference between the maximum likelihood the Bayes estimates of the target time subject to specified reliability.

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