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Applied Mathematical Sciences Vol

3 2009 no 11 541 550 Parameters Estimation of the Modi64257ed Weibull Distribution Mazen Zaindin Department of Statistics and Operations Research PO Box 2455 Riyadh 11451 Saudi Arabia mazenzaindinyahoocom Ammar M Sarhan Department of

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Applied Mathematical Sciences Vol

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Applied Mathematical Sciences, Vol. 3, 2009, no. 11, 541 - 550 Parameters Estimation of the Modiﬁed Weibull Distribution Mazen Zaindin Department of Statistics and Operations Research P.O. Box 2455, Riyadh 11451, Saudi Arabia mazenzaindin@yahoo.com Ammar M. Sarhan Department of Mathematics and Statistics Faculty of Science, Dalhousie University Halifax NS B3H 3J5, Canada asarhan@mathstat.dal.ca, asarhan0@yahoo.com Abstract Recently, Sarhan and Zaindin (2008) introduced a generalization of the Weibull distribution and named it as modiﬁed Weibull distribution. In

this paper, we deal with the problem of estimating the parameters of this distribution based on Type II censored data. The maximum likelihood and least square techniques are used. For illustrative purpose, the results obtained are applied on sets of real data. Also, simulation is used to study the properties of the estimators derived. Keywords: Maximum likelihood, least square, linear failure rate, Rayleigh, exponential distribution 1 Introduction The exponential, Rayleigh, linear failure rate and Weibull distributions are the most commonly used distributions in reliability and life testing,

Lawless (2003). These distributions have several desirable properties and nice physical interpretations. Unfortunately the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The lin- ear failure rate distribution generalizes both these distributions which may have non-increasing hazard function also. Also, the Weibull distribution gen- eralizes exponential and Rayleigh distributions. It may have an increasing or
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542 M. Zaindin and A. M. Sarhan decreasing failure rate. For the case when it has an increasing failure

rate, its failure rate function starts from the origin. This property limits its applica- tion in practice. The modiﬁed Weibull distribution introduced by Sarhan and Zaindin (2008) generalizes all the above mentioned distributions. It can be used to describe several reliability models. It has three parameters, two scale and one shape parameters. They used MWD( α,β, ) to denote the modiﬁed Weibull distribution with three parameters α,β, The traditional Weibull distribution with two parameters and , denoted as WD( β, ), has the following cdf: β, )=1

exp { βx ,x ,β,γ> (1.1) It is very well known that WD( β, ) generalized exponential distribution (when = 1) and Rayleigh distribution (when = 2). The traditional linear failure rate distribution with two parameters and , denoted as LFRD( α, ), has the following cdf, Bain (1974): α, )=1 exp αx βx ,x (1.2) where α, 0 such that β> 0. Also, the LFRD( α, ) generalizes expo- nential distribution (when = 0) and Rayleigh distribution (when = 0). The cdf of MWD( α,β, ) takes the following form α,β, )=1 exp { αx βx ,x

(1.3) here γ> ,α, 0 such that β> 0. This distribution generalizes the following distributions: (1) ED( ) when = 0; (2) RD( ) when =0 , = 2; (3) LFRD( α, ) when = 2; and (4) WD( β, ) when =0. The main objective of this article is to estimate the three unknown param- eters of the MWD( α,β, ). We use the maximum likelihood and least squares procedures to derive such estimates. The estimators are obtained by using the data of type II censoring testing without replacement. Also the asymp- totic conﬁdence intervals of the parameters are discussed. Further,

we study whether this distribution ﬁts a set of real data better than other distributions. Two criteria are used for this purpose. These are the Kolmogorov-Smirnov test statistic and the likelihood ratio test statistic. Monte Carlo simulation technique is used to study the performance of the estimators obtained. For this purpose, we used the mathematical program Matlab7 The rest of this paper is organized as follows. Some properties of the MWD( α,β, ) are presented in Section 2. Section 3 gives the parameter es- timations using both maximum likelihood and least squares

techniques. We
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Modiﬁed Weibull distribution 543 apply the theoretical results discussed here on sets of real data in section 4. Simulation study is provided in section 4. Finally, section 5 concludes the paper. 2 The MWD The survival function of the MWD( α,β, ) takes the following form ) = exp { αx βx ,x (2.4) here γ> ,α, 0 such that β> 0. The pdf of the MWD( α,β, )is α,β, )=( βγx ) exp { αx βx ,x (2.5) Figure 1 shows some patterns of the pdf of MWD( α,β, ), which may have a single mode

or no mode at all. 0.2 0.4 0.6 0.8 1.2 1.4 1.6 0.5 1.5 2.5 The probability density function =1, =2.5, =0.5 =1, =1, =1.5 =1, =1, =1.5 Figure 1. Diﬀerent patterns of the probability density function. The hazard rate function of MWD( α,β, )is α,β, )= βγx (2.6) It is easy to verify that the hazard rate function is either constant if =1or increasing if γ> 1 or decreasing if γ< 1. Figure 2 shows diﬀerent patterns of the hazard rate function.
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544 M. Zaindin and A. M. Sarhan 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5 2.5 The hazard

function =1, =0.5, =0.5 =1, =0.5, =0.5 =1, =0.5, =0.5 Figure 2. Diﬀerent patterns of the hazard rate function. 3 Parameters Estimation In this section, we use the maximum likelihood and least squares procedures to derive point estimates of the unknown parameters of the MWD( α,β, ). Also, we will derive the asymptotic interval estimates of the parameters. Henceforth we shall consider the data of type II censoring without replace- ment. In such type of data, it is assumed that identical items are put on the life test. The testing process is terminated at the time of th item

failure. The number of observations is decided before the data are collected. Assume the times to failures are <... . Let denote the information obtained from the life testing. It means the number of all items to be tested the number of failed items , and their times to failure ,x ,..., x . That is, n, r ,x ,...,x . It is assumed also that the life time of each item follows the MWD( α,β, ) with pdf given by (2.5). 3.1 The Maximum Likelihood Estimators In this subsection, we use the maximum likelihood procedure to derive the point and interval estimates of the parameters. 3.1.1 Point

Estimators The likelihood function of is, see Lawless (1982), )= )! α,β, =1 α,β, (3.1)
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Modiﬁed Weibull distribution 545 Substituting (2.5) and (2.4) into (3.1), we get )! =1 βγx exp { αT (1) βT (3.2) where )=( The log-likelihood, denoted ), is )= =1 ln βγx αT (1) βT (3.3) where is a constant and it is given by =ln ln( )!. Calculating the ﬁrst partial derivatives of with respect to α,β, and equating each to zero, we get the likelihood equations as in the following system of nonlinear equations

of α,β, 0= =1 βγx (1) (3.4) 0= =1 γx βγx (3.5) 0= =1 (1 + ln( )) βγx βT (3.6) where )=( ln( )+ ln( ). To ﬁnd out the maximum likelihood estimators of α,β, , we have to solve the above system of nonlinear equations (3.4)-(3.6) with respect to α,β, .As it seems, this system has no closed form solution in α,β, . Then we have to use a numerical technique method, such as Newton-Raphson method, to obtain the solution. 3.1.2 Asymptotic Conﬁdence Bounds The approximate conﬁdence intervals of the

parameters based on the asymp- totic distributions of the MLE of the parameters α,β, are derived in this subsection. For the observed information matrix for α,β, , we ﬁnd the sec-
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546 M. Zaindin and A. M. Sarhan ond partial derivatives of as 11 =1 βγx 12 =1 γx βγx 13 =1 (1 + ln( )) βγx 22 =1 2( 1) βγx 23 =1 (1 + ln( )) βγx 33 =1 ln )+2 ln( βx βγi 1) βT where )=( ln )+ ln ). Then the observed information matrix is given by 11 12 13 21 22 23 31 32 33 11 12 13 12 22 23 13 23 33

so that the variance-covariance matrix may be approximated as 11 12 13 21 22 23 31 32 33 11 12 13 12 22 23 13 23 33 It is known that the asymptotic distribution of the MLE α, β, is given by, see Miller (1981), 11 12 13 21 22 23 31 32 33 (3.7) Since involves the parameters α,β, , we replace the parameters by the corresponding MLE’s in order to obtain an estimate of , which is denoted
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Modiﬁed Weibull distribution 547 by 11 12 13 12 22 23 13 23 33 where ij ij when ( α, β, ) replaces ( α,β, ). By using (3.7), approximate 100(1 )%

conﬁdence intervals for α,β, are determined, respectively, as θ/ 11 θ/ 22 and θ/ 33 (3.8) where is the upper th percentile of the standard normal distribution. 3.2 The least squares procedure In this section, we shall derive the least square estimators (LSEs) of the three parameters α,β, . Given the observed lifetimes <... in a certain censored sample from the MWD( α,β, ). Then the least square estimates of the parameters α,β, , denoted , respectively, can be obtained by minimizing the following quantity with respect to

α,β, =1 αx βx (3.9) where ln ) and ) is the empirical estimate of ) at the observation =1 ,...,m , given by )= That is, to get , , we have to solve the following system of non- linear equations with respect to α,β, 0= =1 =1 =1 +1 (3.10) 0= =1 =1 +1 =1 (3.11) 0= =1 ln( =1 +1 ln( =1 ln( (3.12) From the ﬁrst two equations (3.10) and (3.11) we get =1 =1 =1 +1 =1 =1 =1 =1 +1 (3.13) =1 =1 =1 +1 =1 =1 =1 =1 +1 (3.14)
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548 M. Zaindin and A. M. Sarhan Substituting (3.13) and (3.14) into (3.12) we get a non-linear equation in Solving the obtained

non-linear equation with respect to we get .Asit seems, such non-linear equation has no closed form solution in . So, we have to use a numerical technique, such as Newton Raphson or bisection method, to solve it. 4 Numerical study and simulation In this section we present practical applications of the theoretical results dis- cussed in the preceding sections with two examples. One example involves a large sample and the other with a small sample. 4.1 Example 1 This example is from McCool (1974) giving the fatigue life in hours of ten bearing of a certain type. These data are as follows: 152.7,

172.0, 172.5, 173.3, 193.0, 204.7, 216.5, 234.9, 262.6, 422.6 In this case, if we assume a Type-II censored sample of size =8,we obtain the maximum likelihood estimates of the parameters of the following distributions: (1) ED( ), (2) RD( ), (3) LFRD( α, ), (4) WE( α, ), and (5) MWD( α,β, ). Also, we compared these distributions to ﬁt the data. For comparison purpose, we use: (1) the likelihood ratio test statistics and the corresponding p-value, and (2) the mean square of the diﬀerence between the empirical cdf and ﬁtted cdf, say MSD, using each model.

Not that MSD is computed by the following relation MSD = =1 FE where and FE are the empirical and the estimated cdf computed at . The estimated cdf is computed by replacing the parameters of the model adopted with their the MLE. Table 1, gives the results obtained. Table 1: The results for example 1. Dist. Parameter estimates p-value MSD ED 2.0213 10 -52.129 16.376 2.780 10 0.04021 RD 3.958 10 -47.195 6.508 0.039 0.02306 LFRD 1.686 10 , 1.087 10 -52.456 17.03 3.679 10 0.02881 WD 7.921 10 , 1.739 -48.137 8.391 3.770 10 0.02698 MWD 1.000 10 , 6.535 10 , 3.494 -43.941 0.00821 The results shown in

Table 1 imply that the MWD(1 000 10 535 10 494) ﬁts the given data better than all other distributions tested above.
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Modiﬁed Weibull distribution 549 4.2 Example 2. In this example, we give a simulation using Monte Carlo method. This simu- lation is provided in order to study the properties of the estimators obtained using the methods used. Also, we need to investigate the inﬂuence of the val- ues of and on the accuracy of the estimators. Further, we will compare the MLE and LSE of the parameters. The following scheme has been followed in the study 1.

Determine the values of the parameters and 2. Specifying the tested and failed items and , respectively. 3. Generating a type II censored data from the MWD( α,β, ). 4. Using the generated data obtained in step 3, calculate the MLE and LSE of the parameters and 5. Repeat steps 3 and 4 times. 6. Compute the mean squared error associated with each estimate of the vector of parameters =( ) according to the following formula RMSE =1 ( where denotes either MLE of LSE of the vector computed using the random sample generated in the repetition number =1 ,N 7. The above steps are made when =(

α, β, )=(2 3), (2 3) and (4 3). 8. Repeat the entire process when =3 ,..., 10, at ﬁxed =10 Table 2 summarizes the root mean squared errors associated with the estima- tors of =( α, β, ). Table 2. The RMSE’s associated with the estimators of =( α, β, =(2 3) =(2 3) =(4 3) LS ML LS ML LS ML 4.212 17.221 5.121 15.831 6.610 14.534 2.860 2.338 2.918 3.598 4.720 4.243 2.581 2.096 2.498 2.860 3.972 4.060 1.792 2.023 2.301 2.763 3.334 4.007 1.111 1.946 1.588 2.618 2.893 3.976 1.089 1.930 1.791 2.580 2.151 3.953 0.983 1.900 1.076 2.520 2.064 3.945 10 0.917 1.889

1.143 2.010 1.317 3.035
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550 M. Zaindin and A. M. Sarhan Based on the results summarized on table 2, we could conclude that: the RMSE is greater than RMSE for each investigated , and both of them are monotonically decreasing with . From the above analysis, one can conclude that the maximum likelihood method provides better estimators than the least square method for the case studied in this paper. 5 Conclusion In this paper we discussed the parameter estimation of the MWD( α,β, based on Type-II censored data. The maximum likelihood and least square techniques have

been used. The MWD( α,β, ) is tested against diﬀerent dis- tributions using a set of real data. Based on the two criteria (the values of the log-likelihood function and average K-S test statistics), we found that the MWD( α,β, ) ﬁts the data better than those compared distributions. Further, simulation studies are used to investigate the properties of the estimators ob- tained and to compare the methods used. Further work can be done in this area such as, the Bayes technique can be used to estimate the parameters of MWD( α,β, ). Acknowledgement This

research is supported by the the Research Center, College of Science, King Saud University under the project number Stat/2008/54 References [1] L. J. Bain, ”Analysis for the Linear Failure-Rate Life-Testing Distribu- tion, Technometrics, Vol. 16, no. 4, pp. 551-559, 1974 [2] J. F. Lawless, ”Statistical Models and Methods for Lifetime Data, John Wiley and Sons, New York , 2003. [3] J. I. McCool, ”Inferential Techniques for Weibull Populations”, Aeropace Research Laboratories Report ARL TR74-0180, Wright-Patterson Air Force Base, Dayton, Ohio , 1974. [4] A. M. Sarhan and M. Zain-Din, ”New

Generalized Weibull Distribution”, Applied Science , In Press, 2008. Received: August, 2008