Estimation The Shape, Location and Scale Parameters of The Weibull Distribution
Estimation The Shape, Location and Scale Parameters of The Weibull Distribution
Estimation The Shape, Location and Scale Parameters of The Weibull Distribution
Chikr elMezouar ESTIMATION THE SHAPE, LOCATION AND SCALE PARAMETERS OF THE WEIBULL DISTRIBUTION
RT&A # 04 (19)
(Vol.1) 2010, December
ABSTRACT
In this paper we propose a new estimators of the shape, location and scale parameters of
the weibull distribution.
Keyword: Weibull Distribution, Crans method and method of moments.
1.INTRODUCTION
The shape and scale parameter estimation of weibull distribution within the traditional methods
5
and standard Bayes from work has been studied by Tummala 1980 , Ellis and Tummala
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 Crans 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:
t a c
F t 1 exp
b
, ta
, a0
2 1
The parameter a is called the location parameter. And the k th moment is defined by :
k a
1
b1
c
k
2 2
1
c
In particular, when k 1 ,
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Zouaoui Chikr elMezouar ESTIMATION THE SHAPE, LOCATION AND SCALE PARAMETERS OF THE WEIBULL DISTRIBUTION
1
c
RT&A # 04 (19)
(Vol.1) 2010, December
2 3
1 a b1
2 a
1
b1
c
2
2 4 ,
1
c
2 5
3 1
, r 1,2, , n 1
m k 1 S n t dt
k
n 1
r
1 t r 1 t r , t 0 0
n
r 0
3 2
In particular,
m1 t , the sample mean.
Cran 1988
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Zouaoui Chikr elMezouar ESTIMATION THE SHAPE, LOCATION AND SCALE PARAMETERS OF THE WEIBULL DISTRIBUTION
1 4 2
,
1 4 2 2
2
and
c
1 a
1
1
c
ln 2
ln 1 2 ln 2 4
3 3
a t 1
1
b 1
c
n
3 4
We propose, the coefficient of variation, to get an expression which is a function of c only , i.e,
C.V .
2 12
1 t 1
2 1
1 1
c c
1
1
1 1 1
c
nc
3 5
Now, we can form a table for various C.V . by using 3 5 for different c values.
In order to estimate c and b , we calculate the coefficient of variation C.V . of the data and
comparing with C.V . using the table to estimate the shape parameter c , i.e
C.V .
2 1
1 1
c c
1
1
1 1 1
c
nc
3 6
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RT&A # 04 (19)
(Vol.1) 2010, December
Zouaoui Chikr elMezouar ESTIMATION THE SHAPE, LOCATION AND SCALE PARAMETERS OF THE WEIBULL DISTRIBUTION
4- simulation results:
The objective of our experiments is to compare the proposed estimators with Crans 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 Crans method (R=1000)
10
25
50
100
proposed
Cran
MSE
MSE
a2
15.9134
333.2995
proposed
b4
17.4373
333.1223
Proposed
c2
10.3747
237.5663
Proposed
a2
1.3287
10.8445
Proposed
b4
1.8016
11.3442
Proposed
c2
1.0492
8.5543
Proposed
a2
0.2506
0.3278
Proposed
b4
0.4193
0.5162
Proposed
c2
0.2041
0.3486
Proposed
a2
0.0806
0.0914
Proposed
b4
0.1543
0.1678
Proposed
c2
0.0706
0.1173
proposed
parameters
The Best
5-Conclusion:
In this paper, we have presented both Crans 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
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TRANSACTIONS ON RELIABILITY , vol. 37, N .4.
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Zouaoui Chikr elMezouar ESTIMATION THE SHAPE, LOCATION AND SCALE PARAMETERS OF THE WEIBULL DISTRIBUTION
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(Vol.1) 2010, December
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
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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.
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