Application of a Sequential Life-Testing to an ...

Proceedings of the 2007 Industrial Engineering Research Conference G. Bayraksan, W. Lin, Y. Son, and R. Wysk, eds.

Application of a Sequential Life-Testing to an Accelerated LifeTesting with an Underlying Three-Parameter Inverse Weibull Model. A Maximum Likelihood Approach Assed N. Haddad Construction Science Engineering Department, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. Daniel I. De Souza Jr. Department of Production Engineering Fluminense Federal University, Niterói, RJ, Brazil and North Fluminense State University, Campos, RJ, Brazil. Abstract In this work we will be life-test a new industrial product. We will be assuming an underlying three-parameter Inverse Weibull distribution and a linear acceleration condition. To estimate the three parameters of the Inverse Weibull model we will use a maximum likelihood approach for censored failure data. To evaluate the accuracy (significance) of the parameter values obtained under normal conditions for the underlying three-parameter Inverse Weibull model we will apply to the expected normal failure times a sequential life test using a truncation mechanism. An example will illustrate the application of the proposed approach.

Keywords Inverted three-parameter Weibull model, maximum likelihood, sequential life-testing, accelerated life-testing.

1. Introduction Sometimes the amount of time available for testing could be considerably less than the expected lifetime of the component; so it is not possible to test the component under normal stress conditions. To overcome such a problem, there is the accelerated life-testing alternative aimed at forcing components to fail by testing them at much higherthan-intended application conditions. These models are known as acceleration models. In this work we will be lifetest a new industrial product. To estimate the three parameters of the Inverse Weibull model we will use a maximum likelihood approach for censored failure data, since the life-testing will be terminated at the moment the truncation point is reached. We will be assuming a linear acceleration condition. To evaluate the accuracy (significance) of the parameter values obtained under normal conditions for the underlying three-parameter Inverse Weibull model we will apply to the expected normal failure times a sequential life test using a truncation mechanism developed by De Souza [1]. The two-parameter Inverse Weibull distribution has been used in Bayesian reliability estimation to represent the information available about the shape parameter of an underlying Weibull sampling distribution, see Erto [2], De Souza and Lamberson [3]. It has also been used previously in sequential life testing situations by De Souza [4] [5]. The three-parameter Inverse Weibull distribution has a location or minimal life parameter φ, a scale parameter θ and a shape parameter β. Its density function is given by: β (θ t − ϕ)β+1 exp ⎡⎢− (θ t − ϕ)β ⎤⎥ ; t ≥ 0; θ, β, ϕ > 0 f (t ) = (1) θ ⎣ ⎦ Since the first decade of the twentieth century, when only thermal stresses were significant, an empirical model, known as the Arrhenius model, has been used with relative success as an accelerated model. This model is given by: R rate = e − A GTn + N (2) Here, Rrate is the rate of reaction, A represents the energy of activation of the reaction, G the gas constant (1.986 calories per mole), Tn the temperature in degrees Kelvin (273.16 plus the degrees Centigrade) at normal condition of use, and N a constant.

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Haddad and De Souza This empirical model first presented in 1898, could probably be connected to the “Maxwell Distribution Law.” This law, which expresses the distribution of kinetic energies of molecules, is represented by the following equation: NK TE = M tot e − A GT (3) Equation (3) expresses the probability of a molecule having energy in excess of A, where NKTE represents the number of molecules at a particular absolute Kelvin temperature T, that passes a kinetic energy greater than A among the total number of molecules present, Mtot., A is the energy of activation of the reaction and G represents the gas constant (1.986 calories per mole). The acceleration factor AF2/1 (or the ratio of the specific rates of reaction R2/R1), at two different stress temperatures, T2 and T1, will be given by: NK TE (2) e − A GT2 = = exp [A G (1 T1 − 1 T2 )] NK TE (1) e − A GT1 Applying natural logarithm to both sides of Equation (4) and after some algebraic manipulation, we will obtain: ln (AF 2 / 1 ) = ln (NK TE (2) NK TE (1) ) = A G (1 T1 − 1 T2 ) From Equation (5) we can estimate the term A/G by testing at two different stress temperatures and computing acceleration factor on the basis of the fitted distributions. Then: ln AF2 1 A = G (1 T1 − 1 T2 ) AF 2 / 1 =

(

)

(4) (5) the (6)

The acceleration factor AF2/1 will be given by the relationship θ1/θ2, with θi representing a scale parameter or a percentile at a stress level corresponding to Ti. Once the term A/G is determined, the acceleration factor AF2/n to be applied at the normal stress temperature is found in Equation (4) by replacing the stress temperature T1 by the temperature at normal conditions of use Tn. Then: AF 2 / n = exp [A G (1 Tn − 1 T2 )] (7)

2. The Accelerating Condition De Souza [6] has shown that under a linear acceleration assumption, if a three-parameter Inverse Weibull model represents the life distribution at one stress level, a three-parameter Inverse Weibull model also represents the life distribution at any other stress level. We will be assuming a linear acceleration condition. In general, the scale parameter and the minimum life can be estimated by using two different stress levels (temperature or cycles or miles, etc.), and their ratios will provide the desired value for the acceleration factors AFθ and AFϕ. Then: AFθ = θ n θ a (8) AFϕ = ϕ n ϕ a (9) According to De Souza [6], for the Inverse Weibull model the cumulative distribution function at normal testing condition Fn(tn−ϕn) for a certain testing time t = tn, will be given by: β ⎡ ⎧ ⎫ n ⎤⎥ θ n AF ⎛ t ⎞ Fn (t ) = Fa ⎜ (10) ⎟ = exp ⎢− ⎨ ⎬ ⎢ ⎩ AF (t − ϕ n AF) ⎭ ⎥ ⎝ AF ⎠ ⎣ ⎦ Equation (10) tells us that, under a linear acceleration assumption, if the life distribution at one stress level is Inverse Weibull, the life distribution at any other stress level is also an Inverse Weibull model. The shape parameter remains the same while the accelerated scale parameter and the accelerated minimum life are multiplied by the acceleration factor. The equal shape parameter is a necessary mathematical consequence to the other two assumptions; assuming a linear acceleration model and an Inverse Weibull sampling distribution. If different stress levels yield data with very different shape parameters, then either the Inverse Weibull sampling distribution is the wrong model for the data or we do not have a linear acceleration condition.

3. MLE for the Inverse Weibull Model for Censored Type II Data (Failure Censored) The standard maximum likelihood method for estimating the parameters of the three-parameter Weibull model can have problems since the regularity conditions are not met; see Murthy et al. [7], Blischk, [8], Zanakis and Kyparisis [9]. To overcome the resulting “no-regularity” problem above mentioned, we will apply a modification proposed by Cohen et al. [10]. De Souza [6] presents a complete discussion on this mater for the three-parameter Inverse Weibull model. According to De Souza [6], the Maximum Likelihood Equations for the three-parameter Inverse Weibull Model are given by:

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Haddad and De Souza ⎡ r β β⎤ 1β θ = ⎢r ∑ (1 t i − ϕ) + (n − r )(1 t r − ϕ) ⎥ ⎢⎣ i =1 ⎥⎦ β β ⎡ r ⎛ ⎤ ⎛ 1 ⎞ 1 ⎞ ⎟⎟ ln(t i − ϕ) + (n − r ) ⎜⎜ ⎟⎟ ln(t r − ϕ)⎥ r × ⎢ ∑ ⎜⎜ r ⎢i =1⎝ t i − ϕ ⎠ ⎥ ⎝ tr − ϕ ⎠ r ⎦= ln (t i − ϕ) + ⎣ – β β β r ⎛ 1 ⎞ ⎛ 1 ⎞ i =1 ∑ ⎜⎜ t − ϕ ⎟⎟ + (n − r )⎜⎜ t − ϕ ⎟⎟ ⎠ ⎝ r ⎠ i =1⎝ i

(11)

∑

r

1 – ( ) i =1 t i − ϕ

(β + 1) ∑

β +1 β +1 ⎤ ⎡ r ⎛ ⎛ 1 ⎞ 1 ⎞ ⎢ ⎥ ⎟ ⎟ β r ∑ ⎜⎜ + (n − r ) ⎜⎜ ⎢i =1⎝ t i − ϕ ⎟⎠ ⎥ t r − ϕ ⎟⎠ ⎝ ⎣ ⎦ β

r ⎛

⎛ 1 ⎞ 1 ⎞ ∑ ⎜⎜ t − ϕ ⎟⎟ + (n − r )⎜⎜ t − ϕ ⎟⎟ ⎠ ⎝ r ⎠ i =1⎝ i

β

0 (12)

=0

(13)

To overcome the “no-regularity problem,” one of the approaches proposed by Cohen et al. [10] is to replace Equation (13) with the equation n n ⎧ ⎧ ⎤⎫ ⎤⎫ g ⎪k +1 ⎡ −1 β ⎞ ⎛ g ⎪ k +1 ⎡ ⎪ ⎪ −U −U n × θ× × ⎨ ∑ ⎢⎛⎜ U i ⎟ ⎜1 − e i ⎞⎟ × (1, 2 or 4 )⎥ ⎬ + n × ϕ j × × ⎨ ∑ ⎢⎛⎜1 − e i ⎞⎟ × (1, 2 or 4)⎥ ⎬ = t1 (14) ⎠ ⎠ ⎠⎝ 3 ⎪ i =1 ⎢⎝ 3 ⎪ i =1 ⎢⎝ ⎥⎪ ⎥⎪ ⎣ ⎦⎭ ⎣ ⎦⎭ ⎩ ⎩

(

)

U i = θ t i − ϕ j β ; j = 1, 2, …, k Here, t1 is the first order statistic in a sample of size n. In solving the maximum likelihood equations, we will use the approach proposed by Cohen et al. [10]. De Souza [6] shows the derivation of Equation (14).

4. Hypothesis Testing The hypothesis testing situations will be given by: 1. For the scale parameter θ: H 0 : θ ≥ θ 0 ; H 1 : θ < θ 0 . The probability of accepting H0 will be set at (1-α) if θ = θ0. Now, if θ = θ1 where θ1 < θ0, the probability of accepting H0 will be set at a low level γ. 2. For the shape parameter β: H 0 : β ≥ β 0 ; H 1 : β < β 0 . The probability of accepting H0 will be set at (1-α) if β = β0. Now, if β = β1 where β1 < β0, the probability of accepting H0 will also be set at a low level γ. 3. For the location parameter ϕ: H 0 : ϕ ≥ ϕ 0 ; H 1 : ϕ < ϕ 0 .Again, the probability of accepting H0 will be set at (1-α) if ϕ = ϕ0. Now, if ϕ = ϕ1 where ϕ < ϕ0, then the probability of accepting H0 will be once more set at a low level γ. When the decisions about θ0, θ1, β0, β1, ϕ0, ϕ1, α and γ are made, the sequential test is totally defined.

5. The Sequential Testing According to Kapur and Lamberson [11] and De Souza [5], the development of a sequential test uses the likelihood ratio given by the following relationship: SPR = L1,1,1,n / L0,0,0,n. According to De Souza [12], for the Inverse Weibull model, the sequential probability ratio (SPR) will be given by: β0 β0 β1 β1 ⎛ ⎛ ⎞ ⎞ ⎡ (1 − α ) ⎤ ⎡ (1 − γ ) ⎤ n ln ⎜⎜ β1 θ 0 × θ 1 β 0 ⎟⎟ ln ⎢ < X < n ln ⎜⎜ β1 θ 0 × θ 1 β 0 ⎟⎟ + ln ⎢ (15) ⎥ ⎥ ⎣ α ⎦ ⎝ ⎝ ⎠ ⎠ ⎣ γ ⎦

−

X=

n ⎡ β1

∑ ⎢θ1

i =1 ⎣

(t i − ϕ 1 )β1 − θ 0β0 (t i − ϕ 0 )β0 ⎤⎥ − (β 0 + 1) ∑ ln (t i − φ 0 ) + (β 1 + 1) n

⎦

i =1

n

∑ ln (t i

i =1

− φ1

)

(16)

6. Expected Sample Size of a Sequential Life Testing for Truncation Purpose According toMood and Graybill [13], an approximate expression for the expected sample size E(n) of a sequential life testing for truncation purpose will be given by: P(θ, β ) ln A + [1 − P(θ, β)] ln B (1 − γ ) γ E(n ) = ; A= ; B= (17) E(w ) (1 − α ) α

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According to De Souza [12], for the three-parameter Inverse Weibull model we will have: β0 β1 ⎛ ⎞ ⎡ ⎤ ⎡ E(w ) = ln ⎜⎜ β1 θ 0 × θ 1 β 0 ⎟⎟ + β 0 + 1 E ⎢ln t i − ϕ 0 ⎥ − β 1 + 1 E ⎢ln t i − ϕ 1 ⎣ ⎦ ⎣ ⎝ ⎠ β β ⎡ ⎡ ⎤ − θ 1 E ⎢1 t i − ϕ 1 β1 ⎥ + θ 0 E ⎢1 0 1 ⎣ ⎣ ⎦ The solution for the components of Equation (18) can be found in De Souza [12].

(

)

(

)

(

(

)

(

)

)

⎤ ⎥− ⎦

(t i − ϕ 0 )β0

⎤ ⎥ ⎦

(18)

7. Example We are trying to determine the values of the shape, scale and minimum life parameters of an underlying Weibull model, representing the life cycle of a new electronic component. Once a life curve for this component is determined, we will be able to verify using sequential life testing, if new units produced will have the necessary required characteristics. It happens that the amount of time available for testing is considerably less than the expected lifetime of the component. So, we will have to rely on an accelerated life testing procedure to obtain failure times used on the parameters estimation procedure. The electronic component has a normal operating temperature of 296 K (about 23 degrees Centigrade). Under stress testing at 480 K, 16 electronic components were subjected to testing, with the testing being truncated at the moment of occurrence of the twelfth failure. Table 1 shows these failure time data (hours). Table 1: Failure times (hours) of electronic parts tested under accelerated temperature conditions (480K) 765.1 843.6 850.4 862.2 877.3 891.0 909.4 930.9 952.4 973.2 1,014.7 1,123.6 Now, under stress testing at 520 K, 16 electronic components were again subjected to testing, with the testing being truncated at the moment of occurrence of the twelfth failure. Table 2 shows these failure time data (hours). Table 2: Failure times (hours) of electronic parts tested under accelerated temperature conditions (520K) 52.5 673.6 683.1 692.9 705.1 725.4 738.2 769.2 776.6 784.9 816.0 981.9 Using the maximum likelihood estimator approach for the shape parameter β, for the scale parameter θ and for the minimum life ϕ of the Inverse Weibull model for censored Type II data (failure censored), we obtain the following values for these three parameters under accelerated conditions of testing. At 480 K. β1 = βn = β = 8.4; θ1 = 642.3 hours; ϕ1 = 117.9 hours At 520 K. β2 = βn = β = 8.4; θ2 = 548.0 hours; ϕ2 = 100.2 hours The shape parameter did not change with β ≈ 8.4 The acceleration factor for the scale parameter AFθ2/1 will be given by AFθ 2 1 = θ1 θ 2 = 642.3 548.0

(

)

ln AF2 1 ln(642.3 548.0 ) A = = = 990.8 G (1 T1 − 1 T2 ) (1 480 − 1 520) Using now Equation (7), the acceleration factor for the scale parameter, to be applied at the normal stress temperature AFθ2/n, will be AF 2 / n = exp [A G (1 Tn − 1 T2 )] = exp [990.8 (1 296 − 1 520 )] = 4.23. Then, the scale

Using Equation (6), we can estimate the term A/G. Then:

parameter at normal operating temperatures is estimated to be: θ n = AF 2 / n × θ2 = 4.23 × 548.0 = 2,318.0 hours. The acceleration factor for the minimum life parameter will be given by AFφ2/1 = ϕ1 ϕ 2 = 117.9 100.2 ln AF2 1 ln (117.9 100.2 ) A Applying Equation (6), we can again estimate the term A/G. Then: = = = 1,015.1 G (1 T1 − 1 T2 ) (1 480 − 1 520) Using once more Equation (7), the acceleration factor for the minimum life parameter, to be applied at the normal stress temperature AFφ2/n, will be AFϕ 2 / n = exp [A G (1 Tn − 1 T2 )] = exp [1,015.1 (1 296 − 1 520 )] = 4.38 Then, as we expected, AFθ = 4.23 ≈AFϕ = 4.38 ≈ AF = 4.3. Finally, the minimum life parameter of the component at normal operating temperatures is estimated to be ϕ n = AFϕ 2 / n × ϕ2 = 4.3 × 100.2 = 430.9 hours.

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Then, the electronic component life when operating at normal use conditions could be represented by a threeparameter Inverse Weibull model having a shape parameter β of 8.4; a scale parameter θ of 2,318.0 hours and a minimum life φ of 430.9 hours. To evaluate the accuracy (significance) of the three-parameter values obtained under normal conditions for the underlying Inverse Weibull model we will apply, to the expected normal failure times, a sequential life testing using a truncation mechanism developed by De Souza [1]. These expected normal failure times will be acquired by multiplying the nine failure times obtained under accelerated testing conditions at 520 K given by Table 2 by the accelerating factor AF of 4.3. It was decided that the value of α was 0.05 and γ was 0.10. In this example, the following values for the alternative and null parameters were chosen: alternative scale parameter θ1 = 2,100 hours, alternative shape parameter β1 = 7.8 and alternative location parameter ϕ1 = 380 hours; null scale parameter θ0 = 2,320 hours, null shape parameter β0 = 8.4 and null minimum life parameter ϕ0 = 430 hours. Now electing P(θ,β,ϕ) to be 0.01, we can calculate the expected sample size E(n) of this sequential life testing under analysis. Using now Equations (17) and (18), the expression for the expected sample size of the sequential life testing for truncation purpose E(n), we will have: E(w ) = – 5.50081 + 9.4 × 7.802064 – 8.8 × 7.749322 – 0.033074 + 1.0 = 0.6115 Now, with P (θ,β,ϕ) = 0.01, ln (B) = ln [(1 − 0.10 ) 0.05] = 2.8904; ln (A ) = ln [0.10 1 − 0.05] = −2.2513, we get: P (θ, β) ln (A ) + [1 − P (θ, β)] ln (B) = −0.01 × 2.2513 + 0.99 × 2.8904 = 2.8390

Then: E(n ) = 2.8390 0.6115 = 4.6427 ≈ 5 items So, we could make a decision about accepting or rejecting the null hypothesis H0 after the analysis of observation number 5. Using Equations (15) and (16) and the twelve failure times obtained under accelerated conditions at 520 K given by Table 2, multiplied by the accelerating factor AF of 4.3, we calculate the sequential life testing limits. Figure 1 shows the sequential life-testing for the three-parameter Inverse Weibull model. Then, since we were able to make a decision about accepting or rejecting the null hypothesis H0 after the analysis of observation number 4, we do not have to analyze a number of observations corresponding to the truncation point (5 observations). As we can see in Figure 1, the null hypothesis H0 should be accepted since the final observation (observation number 4) lays on the region related to the acceptance of H0. NUMBER OF ITEMS 0 0

V A L U E S

1

2

3

4

5

6

-5 -10 ACCEPT Ho

-15 -20 -25

O F X

REJECT Ho

-30 -35 -40

Figure 1: Sequential test graph for the three-parameter Inverse Weibull model

7. Conclusions There are two key limitations to the use of the Arrhenius equation: first, at all the temperatures used, linear specific rates of change must be obtained. This requires that the rate of reaction, regardless of whether or not it is measured or represented, must be constant over the period of time at which the aging process is evaluated. Now, if the expected rate of reaction should vary over the time of the test, then one would not be able to identify a specific rate that is assignable to a specific temperature. If the mechanism of reaction at higher or lower temperatures should differ, this, too, would alter the slope of the curve. Second, it is necessary that the energy activation be independent of temperature, that is, constant over the range of temperatures of interest. It happens that, according to Chornet and Roy [14], “the apparent energy of activation is not always constant, particularly when there is more than one process

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going on.” Further comments on the limitations of the use of the Arrhenius equation can be found in Feller [15]. In this work we life-tested a new industrial product using an accelerated mechanism. We assumed a linear acceleration condition. To estimate the parameters of the three-parameter Inverse Weibull model we used a maximum likelihood approach for censored failure data. The shape parameter remained the same while the accelerated scale parameter and the accelerated minimum life parameter were multiplied by the acceleration factor. The equal shape parameter is a necessary mathematical consequence of the other two assumptions; that is, assuming a linear acceleration model and a three-parameter Inverse Weibull sampling distribution. In order to translate test results obtained under accelerated conditions to normal using conditions we applied some reasoning given by the “Maxwell Distribution Law.” To evaluate the accuracy (significance) of the three-parameter values estimated under normal conditions for the underlying Inverse Weibull model we employed, to the expected normal failure times, a sequential life testing using a truncation mechanism developed by De Souza [12]. These expected normal failure times were acquired by multiplying the twelve failure times obtained under accelerated testing conditions at 520 K given by Table 2, by the accelerating factor AF of 4.3. Since we were able to make a decision about accepting or rejecting the null hypothesis H0 after the analysis of observation number 4, we did not have to analyze a number of observations corresponding to the truncation point (5 observations). As we saw in Figure 1, the null hypothesis H0 should be accepted since the final observation (observation number 4) lays on the region related to the acceptance of H0. Therefore, we accept the hypothesis that the electronic component life when operating at normal use conditions could be represented by a three-parameter Inverse Weibull model having a shape parameter β of 8.4; a scale parameter θ of 2,320 hours and a minimum life φ of 430 hours.

References 1 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15.

De Souza, Daniel I., 2004, “Application of a Sequential Life Testing with a Truncation Mechanism for an Underlying Three-Parameter Weibull Model,” Proceedings of the ESREL - PSAM 7 Conference, Berlin, Germany, June 14-18, 3, 1674-1680; Spitzer, Schmoker and Dang, Eds, Springer-Verlag publishers. Erto, Pasquale., 1982. “New Practical Bayes Estimators for the 2-Parameter Weibull Distribution,” IEEE Transactions on Reliability, R-31, (2), 194-197. De Souza, Daniel I. and Lamberson, Leonard R., 1995, “Bayesian Weibull Reliability Estimation,” IIE Transactions, 27 (3), 311-320. De Souza, Daniel I., 2001, “Truncation Mechanism in a Sequential Life Testing Approach with an Underlying Two-Parameter Inverse Weibull Model,” COMADEM Conference, Manchester, U.K, Sept 4-6, 809-816, Andrew G. Starr and Raj B. K. Rao, Eds., Elsevier publisher. De Souza, Daniel I., 2002, “Sequential Life Testing with Underlying Weibull and Inverse Weibull Sampling Distributions,” COMADEM Conference, Birmingham, UK, Sept 2-4, 120-128, Raj B. K. Rao and Asoke K. Nandi Eds., Comadem International publisher. De Souza, Daniel I., 2005, “A Maximum Likelihood Approach Applied to an Accelerated Life-Testing with an Underlying Three-Parameter Inverse Weibull Model,” Proceedings of the COMADEM Conference, Cranfield, UK, August 31-September 07, 1, 63-72, Mba and Rao, Eds., Cranfield University Press. Murthy, D. N. P.; Xie, M. and Hang, R., 2004, Weibull Models. In Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New Jersey. Blischke, W. R., 1974, “On non-regular estimation II. Estimation of the Location Parameter of the Gamma and Weibull Distributions,” Communications in Statistics, 3, 1109-1129. Zanakis, S. H. and Kyparisis, J. A., 1986, “Review of Maximum Likelihood Estimation Methods for the Three Parameter Weibull Distribution,” Journal of Statistical Computation and Simulation, 25, 53-73. Cohen, A. C.; Whitten, B. J. and Ding, Y., 1984, “Modified Moment Estimation for the Three-Parameter Weibull Distribution,” Journal of Quality Technology, 16, 159-167. Kapur, Kailash & Lamberson, Leonard R., 1977, Reliability in Engineering Design, John Willey & Sons. De Souza, Daniel I., 2004, “Sequential Life-Testing with Truncation Mechanisms for Underlying ThreeParameter Weibull and Inverse Weibull Models,” COMADEM Conference, Cambridge, U.K., August, 260-271, Raj B. K. Rao, B. E. Jones and R. I. Grosvenor Eds.; Comadem International publisher. Mood, A. M. and Graybill, F. A., 1963, Introduction to the Theory of Statistics, 2nd Edition, McGraw-Hill. Chornet and Roy, 1980, “Compensation of Temperature on Peroxide Initiated Cross linking of Polypropylene,” European Polymer Journal, 20, 81-84. Feller, Robert L. Accelerated Aging, Photochemical and Thermal Aspects. The Getty Conservation Institute, Eds.; Printer: Edwards Bross., Ann Harbor, Michigan, 1994.

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