Methods for Estimating the Parameters of the …

1 Methods for Estimating the Parameters of theWeibull DistributionMohammad A. Al-FawzanKing Abdulaziz City for Science and Box 6086, Riyadh 11442, Saudi : +966 1 4813991 Email: 2000 Revised October 20002 AbstractIn this paper, we present some Methods for Estimating weibull Parameters ,namely, shape parameter ( ) and scale parameter ( ). The Weibul distribution is animportant distribution especially for reliability and maintainability analysis. Thepresented Methods can be classified into two categories: graphical and analyticalmethods. Computational experiments on the presented Methods are Graphical MethodsUsually, the graphical Methods are used because of their simplicity andspeed.

2 However, they involve a great probability of error. Next we discuss two maingraphical weibull Probability PlottingThe weibull distribution density function (Mann et al. (1974)) is given by: )(1)()( =xexxf,0,0,0 >> x(1)The cumulative weibull distribution function is given by: )(1)( =xexF(2)where; is the shape parameter, is the scale parameter, and is the come up with the relation between the CDF and the two Parameters ( , ), we take the double logarithmic transformation of the (2) and letting = 0, we have3xxFxxFexFexFxxlnln])(11ln[ln)(])(11 ln[)(11)(1)()( = = = = The last equation is an equation of a straight line. To plot )(xF versus x, weapply the following procedure:1.

3 Rank failure times in ascending Estimate )(ixF of the ith Plot )(ixFvs. xin the weibull probability paper. To estimate )(ixFin (2 and 3) above, we may use one of the followingmethods presented in Table 1 where n is number of data )(ixFMean Rankin+1 Median Rankin + CDFin 1. Methods for Estimating )( Hazard Plotting TechniqueThe hazard plotting technique is an estimation procedure for the Weibullparameters. This is done by plotting cumulative hazard function )(xH against failuretimes on a hazard paper or a simple log-log hazard function is given below:1)()( = xxh4 The cumulative hazard function is given below: )()()(xxhxH== (3)We can transform (3) by taking the logarithm as followsln(){lnln}Hxx= lnln()lnxHx=+1 (4)From (4), we can then plot )(lnxHversus xlnusing the following procedure: 1.

4 Rank the failure times in ascending order. 2. For each failure, calculate in=+ 111() 3. For each failure, calculate Hi=++ + 12 4. Plot lnH vs. lnx. 5. Fit a straight completing the plotting, the estimated Parameters will be as follows:slopeyx1== xHat== ,12. Analytical MethodsDue to the high probability of error in using graphical Methods , we prefer touse the analytical Methods . This is motivated by the availability of high-speedcomputers. In the following, we discuss some of the analytical Methods used inestimating weibull Maximum Likelihood Estimator (MLE)The method of maximum likelihood (Harter and Moore (1965a), Harter andMoore (1965b), and Cohen (1965)) is a commonly used procedure because it hasvery desirable properties.

5 Let 1x, 2x, .., nx be a random sample of size n drawnfrom a probability density function );( xfX where is an unknown parameter. Thelikelihood function of this random sample is the joint density of the n randomvariables and is a function of the unknown parameter. Thus ==niiXxfLi1),( (5)5is the likelihood function. The maximum likelihood estimator (MLE) of , say $ ,is the value of that maximizes Lor, equivalently, the logarithm of L. Often, butnot always, the MLE of is a solution ofdLdlog =0where solutions that are not functions of the sample values 1x, 2x, .., nx arenot admissible, nor are solutions which are not in the parameter space. Now, we aregoing to apply the MLE to estimate the weibull Parameters , namely the shape andthe scale Parameters .

6 Consider the weibull pdf given in (1), then likelihood functionwill be )(111))((),;,,(ixininexxxL = = (6)On taking the logarithms of (6), differentiating with respect to and in turn andequating to zero, we obtain the Estimating equations0ln1lnln11= += ==iniiniixxxnL (7)01ln12=+ = =niixnL (8)On eliminating between these two equations and simplifying, we have0ln11ln111= ===niiniiniiixnxxx (9)which may be solved to get the estimate of $ = . This can be accomplished by theuse of standard iterative procedures ( , Newton-Raphson method). Once isdetermined, can be estimated using equation (8) asnxnii ==1 (10) Method of Moments (MOM)The method of moments is another technique commonly used in the field ofparameter estimation.

7 If the numbers 1x, 2x, .., nx represent a set of data, then anunbiased estimator for the kth origin moment is ==nikikxnm11 (11)where; km stands for the estimate of km. In weibull distribution , the kth momentreadily follows from (1) as6)1()1( kkk+ = (12)where signifies the gamma function)0(,)(01>= sdxexsxsThen from (12), we can find the first and the second moment as follows)11()1( 11 + ==km(13) + + =+=222211211 2 kkm(14)When we divide 2m by the square of 1m, we get an expression which is a function of only)11()11()21( 2222 + + + =kk(15)On taking the square roots of (15), we have the coefficient of variation)11()11()21(2 + + + =CV(16)Now, we can form a table for various CV by using (16) for different values.

8 In order to estimate and , we need to calculate the coefficient ofvariation (CV)d of the data on hand. Having done this, we compare (CV)d withCV using the table. The corresponding is the estimated one ($ ). The scaleparameter ( ) can then be estimated using the following[]{} 1) 1( + =xwhere xis the mean of the Least Squares Method (LSM)The third estimation technique we shall discuss is known as the Least SquaresMethod. It is so commonly applied in engineering and mathematics problems that isoften not thought of as an estimation problem. We assume that a linear relationbetween the two variables (see section 1). For the estimation of weibull Parameters ,we use the method of least squares and we apply it to the results of section 1.

9 Recallthat lnln])(11ln[ln = xxF(17)7 Equation (17) is a linear equation. Now, we can write = + =nininx1)11(1lnln1(18) ==niixnY1ln1(19)211211111)(ln)(lnln]))1( 1ln(ln[]}))1(1(ln{ln[)(ln = ====+=+niiniiniininininiixxnxxn (20)) ( xYe =(21)From equations (18)-(21), we can calculate the estimate of and .3. Numerical ExampleIn order to illustrate and compare the Methods described earlier, we havecoded the three analytical Methods MLE, MOM and LSM in BASIC Language and wehave used a Pentium PC with 133 MHz . To compare the three Methods , we used theMSE (Mean Squared Error) test. MSE can be calculated as below = =niiixFxFMSE12)}()( {(22)where ) (1)( ixiexF = and )(+ =nixFiExample 1:Consider the following example where ix represents the ith failure 2.

10 Data for example 3 below shows the complete results of the estimates of the shape and scaleparameters and the mean squared error (MSE) for each 3. Results of example Since MOM has the minimum MSE, then = and = comparing our results with the well-known software calledSTATGRAPHICS, we found that STATGRAPHICS is using the MLE and our MLEestimates are exactly the same as the estimates of STATGRAPHICSs , we have the advantage of having different estimates for different methodsand the ability of choosing one of Computational ResultsThe objective of our experiments is to compare the three Methods namely,LSM, MLE and MOM.