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141
ESTIMATION THE SHAPE, LOCATION AND SCALE PARAMETERS OF THE
WEIBULL DISTRIBUTION
Dr. Zouaoui Chikr el-Mezouar •
University of Bechar, Algeria E-mail: Chikrtime@yahoo.fr
ABSTRACT
In this paper we propose a new estimators of the shape, location and scale parameters of the weibull distribution.
Keyword: Weibull Distribution, Cran's method and method of moments.
1.INTRODUCTION
The shape and scale parameter estimation of weibull distribution within the traditional methods and standard Bayes from work has been studied by Tummala (l980), Ellis and Tummala (1983)[4], Cran (l988)[3], Al-Fawzan (2000), and Al-Nasir (2002)[l].
This paper considers an estimation procedure based on the coefficient of variation, C.V. The recommended use of such estimators, is to provide quick, preliminary estimators of the parameters. Computational experiments on the presented method and comparison with Cran's method are reported.
2- The three-parameter weibull:
Whenever there is a minimum life (a) such that (T > a), the three-parameter weibull may be appropriate. This distribution assumes that no failures will take place to time (a). For this distribution, the cumulative distribution function. C.D.F is given by:
F (( ) = 1 - exp
—
t - a
t > a
a > 0
(2 -1)
The parameter (a) is called the location parameter. And the [kth) moment is defined by
Mk = a+-
«K
1
kc
(2 - 2)
In particular, when k = 1,
c
a + br^1 + -j (2 - 3)
is the mean time to failure, MTTF of the distribution, and when k = 2
<;+;)
^2= a + ^ ; 'J (2-4),
2c
and so the variance of this distribution is defined by
CT2 = b 2 + 2 j-
ri;+1 "
(2 - 5)
which is the same as that in the two-parameter model.
3- Estimation of the parameters:
Given the ordered random samples : t(;) < t(2) < — < t(n), the cumulative distribution function .C.D.F can be estimated by :
Sn(() = 0, t < t(;)
= -, t(r)< t < t(r+;) , r = 1,2, —, n -1 (3 -1)
n
= ^ t(n)< t
Ten the population moment, fj,'k is estimated by:
<={{ - Sn (()}t
0
= E i1 - -] {{(r+1)- t(r)} , t(0)= 0 (3 - 2)
n
In particular,
m2 = t, the sample mean.
Cran (1988)[3j expressed the parameters in terms of the lower order moments as follows:
2
c
a =
b =
^X -M22 m1+M4- 2M2 '
M - a
and
c=
ln(2)
ln(,-M2)- ln(2 -m4)
(3 - 3)
Therefore, the moment estimators of (a), (b)and (c) can be obtained from (3 -3) by substituting m[, m2 and for M , m2 and /u'4 respectively and solving.
Since the estimator of (a ) is inadmissible by being negative or by exceeding t ^ , hence we can use the alternative estimator:
îrii+1
a =t(1)
(3 - 4)
We propose, the coefficient of variation, to get an expression which is a function of (c) only , i.e,
CV. =
I
M2 -Mi Mi -1(1)
(3 - 5)
V n J
Now, we can form a table for various (CV.) by using (3 - 5) for different (c) values.
In order to estimate (c)and(b) , we calculate the coefficient of variation (CV.) of the data and comparing with (CV.) using the table to estimate the shape parameter (c) , i.e
CV. =
(3 - 6)
V n J
substituting , the scale parameter (b) can then be estimated.
4- simulation results:
The objective of our experiments is to compare the proposed estimators with Cran's estimators. We have generated random samples with known parameters for different sample sizes. To be able to compare, we calculated the mean-squared-error (MSE) for each method , and the table 1, shows the complete results.
Table (1) : Comparison between proposed method and Cran's method (R=1000)
Sample size (n) parameters proposed Cran The Best
MSE MSE
10 a = 2 15.9134 333.2995 proposed
b = 4 17.4373 333.1223 Proposed
c = 2 10.3747 237.5663 Proposed
25 a = 2 1.3287 10.8445 Proposed
b = 4 1.8016 11.3442 Proposed
c = 2 1.0492 8.5543 Proposed
50 a = 2 0.2506 0.3278 Proposed
b = 4 0.4193 0.5162 Proposed
c = 2 0.2041 0.3486 Proposed
100 a = 2 0.0806 0.0914 Proposed
b = 4 0.1543 0.1678 Proposed
c = 2 0.0706 0.1173 proposed
5-Conclusion:
In this paper, we have presented both Cran's method and proposed method (using the coefficient of variation) for estimating the three-parameter weibull distribution. It has been shown from the computational results that the method which gives the best estimates is the proposed method.
References:
1. Al-Fawzan , M.A (2000) : "Methods for Estimating parameters of the Weibull distribution" King Abdulaziz city for science and technology Saudi Arabia.
2. Cran G.W. (1988) : "Moment Estimators for the 3-parameter Weibull Distribution" IEEE
TRANSACTIONS ON RELIABILITY , vol. 37, N 0.4.
3. Ellis , W.C. and Tummala, V.M.R (1986) : "Minimum Expected Loss Estimators of the shape and scale parameters of the Weibull Distribution" IEEE TRANSACTIONS ON
RELIABILITY , vol. R-35, N °.2.
4. Tummala, V.M.R (1980) : "Minimum Expected loss estimators of reliability and shape parameter of Weibull distibution" Industrial Mathematics , the Industrial Mathematics Sciety, vol. 30, part I pp.61-67.
