Oct 29, 2012 - Cite this article as: Saad J Almalki and Jingsong Yuan, A new modified Weibull distribution, Reliability Engineering and System Safety, ...
Author’s Accepted Manuscript
A new modified Weibull distribution Saad J Almalki, Jingsong Yuan
PII: DOI: Reference:
S0951-8320(12)00239-6 http://dx.doi.org/10.1016/j.ress.2012.10.018 RESS4728
To appear in:
Reliability Engineering and System Safety
Received date: Revised date: Accepted date:
2 February 2012 29 October 2012 30 October 2012
www.elsevier.com/locate/ress
Cite this article as: Saad J Almalki and Jingsong Yuan, A new modified Weibull distribution, Reliability Engineering and System Safety, http://dx.doi.org/10.1016/ j.ress.2012.10.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A New Modified Weibull Distribution Saad J Almalki and Jingsong Yuan School of Mathematics, University of Manchester, Manchester M13 9PL, UK
Abstract
We introduce a new lifetime distribution by considering a serial system with one component following a Weibull distribution and another following a modified Weibull distribution. We study its mathematical properties including moments and order statistics. The estimation of parameters by maximum likelihood is discussed. We demonstrate that the proposed distribution fits two well-known data sets better than other modified Weibull distributions including the latest beta modified Weibull distribution. The model can be simplified by fixing one of the parameters and it still provides a better fit than existing models. Keywords. Weibull distribution; additive Weibull; modified Weibull; maximum likelihood estimation.
1
Introduction
The Weibull distribution [28] has been used in many different fields with many applications, see for example [18]. The hazard function of the Weibull distribution can only be increasing, decreasing or constant. Thus it cannot be used to model lifetime data with a bathtub shaped hazard function, such as human mortality and machine life cycles. For many years, researchers have been developing various extensions and modified forms of the Weibull distribution, with number of parameters ranging from 2 to 5. The two-parameter flexible Weibull extension of Bebbington et al [5] has a hazard function that can be increasing, decreasing or bathtub shaped. Zhang and Xie [31] studied the characteristics 1
and application of the truncated Weibull distribution which has a bathtub shaped hazard function. A three-parameter model, called exponentiated Weibull distribution, was introduced by Mudholkar and Srivastave [17]. Another three-parameter model is by Marshall and Olkin [15] and called extended Weibull distribution. Xie et al [30] proposed a three-parameter modified Weibull extension with a bathtub shaped hazard function. The modified Weibull (MW) distribution of Lai et al [13] multiplies the Weibull cumulative hazard function αxβ by eλx , which was later generalized to exponentiated form by Carrasco et al [6] F (x) = (1 − e−αx
β eλx
)θ , x ≥ 0.
(1)
Recent studies of the modified Weibull include [11], [26] and [27].
Among the four-parameter distributions, the additive Weibull distribution (AddW) of Xie and Lai [29] with cumulative distribution function (CDF) F (x) = 1 − e−αx
θ −βxγ
, x ≥ 0,
has a bathtub-shaped hazard function consisting of two Weibull hazards, one increasing (θ > 1) and one decreasing (0 < γ < 1). The modified Weibull distribution of Sarhan and Zaindin (SZMW) [21] can be derived from the additive Weibull distribution by setting θ = 1. A four-parameter beta Weibull distribution was proposed by Famoye et al [10].
Five-parameter modified Weibull distributions include Phani’s modified Weibull [20], the Kumaraswamy Weibull by Cordeiro et al [8] and the beta modified Weibull (BMW) introduced by Silva et al [24] and further studied by Nadarajah et al [19]. The latest examples include the beta generalized Weibull distribution by Singla, et al [25], exponentiated generalized linear exponential distribution by Sarhan et al [22] and the generalized Gomprtz distribution by El-Gohary et al [9].
We propose a new lifetime distribution based on the Weibull and the modified Weibull (MW) distributions by combining them in a serial system. The hazard function of the new distribution is the sum of a Weibull hazard function and a modified Weibull hazard function. Section 2 gives definition, motivation and usefulness of this model and lists
2
its sub-models. Section 3 considers properties of the new distribution such as hazard, moments and order statistics. Section 4 discusses estimation of the parameters. Two real data sets are analyzed in section 5 and the results are compared with existing distributions. Section 6 concludes the paper.
2
The model
2.1
Definition
We define a new modified Weibull distribution (NMW) by the following CDF F (x) = 1 − e−αx
θ −βxγ eλx
, x ≥ 0,
(2)
where α, β, θ, γ and λ are non-negative, with θ and γ being shape parameters and α and β being scale parameters and λ acceleration parameter.
The probability density function (PDF) is ( ) θ γ λx f (x) = αθxθ−1 + β(γ + λx)xγ−1 eλx e−αx −βx e , x > 0.
(3)
It can be rewritten as f (x) = [hW (x; α, θ) + hM W (x; β, γ, λ)] SW (x, α, θ)SM W (x; β, γ, λ),
(4)
where SW , hW , SM W and hM W are survival and hazard functions of the Weibull and modified Weibull distributions respectively.
This function can exhibit different behavior depending on the values of the parameters when chosen to be positive, as shown in Fig. 1.
3
Table 1: The sub-models of the NMW.
Model
α
Additive Weibull
-
Modified Weibull
0
S-Z modified Weibull Linear failure rate
-
Extreme-value
0
Weibull
-
β 1 0
γ
θ
-
-
-
0
-
0
Xie and Lai [29]
−βxγ eλx
Lai et al [13]
−αx−βxγ
Sarhan and Zaindin [21]
−αx−βx2
Bain [4]
−eλx
Bain [4]
−αxθ
Weibull [28]
−αx2
Bain [4] Bain [4]
0
0
0
Reference
−αxθ −βxγ
0
1
0
S(x)
-
1
2
λ e
e e
e
-
-
e
0
e
-
0
0
2
0
e
Exponential
-
0
0
1
0
e−αx
3.0
Rayleigh
1.5 0.0
0.5
1.0
f(x)
2.0
2.5
α = 1.15,β = 0.5,γ = 5,θ = 0.75,λ = 2 α = 0.05,β = 5,γ = 1.25,θ = 5,λ = 0.05 α = 2,β = 0.75,γ = 15,θ = 1.2,λ = 0.05 α = 5,β = 4,γ = 5,θ = 2.5,λ = 0.15 α = 4,β = 0.5,γ = 1.5,θ = 0.4,λ = 0.75
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x
Figure 1: Probability density functions of the NMW.
2.2
Sub models
This distribution includes sub models that are widely used in survival analysis. Table 1 shows a list of models that can be derived from the NMW distribution.
4
2.3
Motivation and interpretation
The survival function of the new distribution is given by S(x) = e−αx
θ −βxγ eλx
, x ≥ 0,
(5)
and, the hazard function is h(x) = αθxθ−1 + β(γ + λx)xγ−1 eλx , x > 0,
(6)
which can be interpreted as that of a serial system with two independent components, one of which follows the Weibull distribution with parameters α and θ, and the other follows the modified Weibull distribution of Lai et al [13] with parameters β, γ and λ. Therefore the distribution can be used when there are two types of failure, e.g. a ‘normal’ type and a premature type.
The purpose of the Weibull component is to provide a decreasing hazard function when required, as in the additive Weibull [29], by choosing θ < 1. (It will be increasing when θ > 1.) The modified Weibull component has either an increasing or a bathtub shaped hazard function. Together they provide a bathtub shaped hazard (unless both hazards are increasing) with more flexibility than the additive Weibull. The flexibility is useful when there is a second peak in the distribution as shown in section 5.
3
3.1
Properties of the model
The hazard function
The hazard function can have many different shapes, including bathtub, as shown in Figure 2. We can deduce from (6) that it is increasing if θ, γ ≥ 1, decreasing if θ, γ < 1 and λ = 0 and bathtub shaped otherwise.
5
8 6 4
h(x)
0
2
α = 1.2,β = 1.5,γ = 0.5,θ = 3,λ = 0.75 α = 0.05,β = 5,γ = 1.25,θ = 5,λ = 0.05 α = 2,β = 0.75,γ = 15,θ = 1.2,λ = 0.05 α = 5,β = 0.7,γ = 0.05,θ = 0.15,λ = 0 α = 0.5,β = 0.7,γ = 5,θ = 2.5,λ = 0.15 α = 4,β = 0.5,γ = 1.5,θ = 0.4,λ = 0.75
0.0
0.5
1.0
1.5
x
Figure 2: Hazard functions of the NMW.
It is desirable for a bathtub shaped hazard function to have a long useful life period [12], with relatively constant failure rate in the middle. A few distributions have this
0.20
property, so does the NMW as shown in Figure 3.
0.10 0.00
0.05
h(x)
0.15
α = 3,β = 5e−06,γ = 0.01,θ = 0.05,λ = 0.15 α = 0.25,β = 5e−04,γ = 0.001,θ = 0.5,λ = 0.09 α = 0.11,β = 2.5e−07,γ = 1,θ = 1,λ = 0.12 α = 0.025,β = 5e−07,γ = 1,θ = 1,λ = 0.12
0
20
40
60
80
x
Figure 3: Hazard functions of the NMW with long useful life period.
6
3.2
The moments
It is customary to derive the moments when a new distribution is proposed. Using the Taylor expansion of ex twice, the rth non-central moment of the NMW is ∫ ∞ ′ µr = xr dF (x) 0 ∫ ∞ θ γ λx = − xr de−αx −βx e ∫ ∞0 θ γ λx = rxr−1 e−αx −βx e dx 0
∫ ∞ ∑ ∞ ∑ (−β)n (λn)m ∞ r−1 nγ+m −αxθ rx x e dx = n!m! 0 n=0 m=0 ) ( ∞ ∞ r ∑ ∑ (−β)n (λn)m − nγ+m+r nγ + m + r θ α , = Γ θ n!m! θ n=0 m=0
for r = 1, 2, . . ., where Γ(·) is the gamma function.
3.3
Order statistics
It will also be useful to derive the pdf of the rth order statistic X(r) of a random sample X1 , . . . , Xn drawn from the NMW with parameters α, β, θ, γ and λ. From Balakrishnan and Nagaraja [3], the pdf of X(r) is given by F (x)r−1 (1 − F (x))n−r f (x) , B(r, n − r + 1)
(7)
F (x) = 1 − e−H(x) , f (x) = h(x)e−H(x) ,
(8)
fr:n (x) = where B(., .) is the beta function.
Using
where h(x) is the hazard function (6) and H(x) = αxθ + βxγ eλx is the cumulative hazard, then, (1 − F (x))n−r = e−(n−r)H(x) ,
(9)
and F (x)
r−1
) r−1 ( ( )r−1 ∑ r−1 −H(x) = 1−e = (−1)ℓ e−ℓH(x) . ℓ ℓ=0
7
(10)
Substituting (9) and (10) into (7), we get fr:n (x) =
=
=
=
) r−1 ( ∑ r−1 1 (−1)ℓ h(x)e−(n+ℓ+1−r)H(x) , B (r, n − r + 1) ℓ ℓ=0 ( )∑ ) r−1 ( n−1 r−1 n (−1)ℓ h(x)e−(n+ℓ+1−r)H(x) , ℓ r−1 ℓ=0 ( )∑ ) r−1 ( ( ) n−1 r−1 θ γ λx n (−1)ℓ αθxθ−1 + β(γ + λx)xγ−1 eλx e−(n+ℓ+1−r)(αx +βx e ) , r−1 ℓ ℓ=0 ( ) r−1 ( ) n−1 ∑ r−1 (−1)ℓ f (x; αℓ , βℓ , θ, γ, λ), n (n + ℓ + 1 − r) r−1 ℓ ℓ=0
where f (x; αℓ , βℓ , θ, γ, λ) is the PDF of the NMW with parameters αℓ = (n + ℓ + 1 − r)α, βℓ = (n + ℓ + 1 − r)β, θ, γ and λ. Using (7), the kth non-central moment of the rth order statistic X(r) is ′ (r:n)
µk
) ( ) ∞ ∞ r−1 ( ) ( (−1)ℓ+i β i (iλ)j (iγ + j) nk n − 1 ∑ ∑ ∑ r − 1 iγ + j + k = . iγ+j+k iγ+j+k Γ ℓ θ r−1 θ (n + ℓ + 1 − r) θ +1−i α θ i=0 j=0 ℓ=0
4
Parameter estimation
Given a random sample x1 , . . . , xn from the NMW with parameters (α, β, θ, γ, λ), the usual method of estimation is by maximum likelihood [7]. Other possible approaches include Bayesian estimation using Lindley approximation [14] or MCMC [26] [27].
The log-likelihood function is given by L =
n ∑
( ln β(γ +
λxi )xγ−1 eλxi i
+
i=1
αθxθ−1 i
)
−α
n ∑ i=1
xθi
−β
n ∑
xγi eλxi .
(11)
i=1
Setting the first partial derivatives of ℓ with respect to α, β, θ, γ and λ to zero, the likelihood
8
equations are n ∑ i=1
∑ θxθ−1 i − xθi = 0, h(xi ; α, β, γ, θ, λ) n
i=1
n ∑ (γ + λxi )xγ−1 eλxi i
i=1
h(xi ; α, β, γ, θ, λ)
n ∑ αxθ−1 (1 + θ ln(xi )) i
i=1
h(xi ; α, β, γ, θ, λ)
n ∑ xγ−1 eλxi ((γ + λxi ) ln(xi ) + 1) i
i=1
h(xi ; α, β, γ, θ, λ)
−
−α
−
n ∑
xγi eλxi
= 0,
(13)
xθi ln(xi ) = 0,
(14)
xγi eλxi ln(xi ) = 0,
(15)
i=1 n ∑ i=1
n ∑ i=1
n ∑ (1 + γ + λxi )xγ eλxi i
i=1
(12)
h(xi ; α, β, γ, θ, λ)
−
n ∑
xγ+1 eλxi i
= 0.
(16)
i=1
The maximum likelihood estimates can be obtained by solving the nonlinear equations numerically for α, β, θ, γ and λ. This can be done using R, Matlab and Mathcad, among other packages. The relatively large number of parameters can cause problems especially when the sample size is not large. A good set of initial values is essential.
We have also obtained all the second partial derivatives of the log-likelihood function for the construction of the Fisher information matrix, so that standard errors of the parameter estimates can be obtained in the usual way. These are in the Appendix.
5
Applications
In this section we provide results of fitting the NMW to two well-known data sets and compare its goodness-of-fit with other modified Weibull distributions using KolmogorovSmirnov (K-S) statistic, as well as Akaike information criterion (AIC) [2] and Bayesian information criterion (BIC) [23] values.
9
Table 2: MLEs of parameters and corresponding standard errors in brackets for the Aarset data.
Model
α ˆ
βˆ
γˆ
θˆ
ˆ λ
NMW
0.071
7.015 × 10−8
0.016
0.595
0.197
(0.031)
(1.501 × 10−7 )
(3.602)
(0.128)
(0.184)
0.062
0.356
0.023
(0.027)
(0.113)
(4.845 × 10−3 )
MW (α = 0, θ = 0) AddW
1.133 × 10−8
0.086
0.477
4.214
(λ = 0)
(5.183 × 10−8 )
(0.036)
(0.102)
(1.033)
SZMW
0.013
8.408 × 10−9
4.224
(2.819 × 10−3 )
(4.204 × 10−8 )
(1.140)
(θ = 1, λ = 0)
5.1
Aarset data
The data represent the lifetimes of 50 devices [1]. Many authors have analysed this data set, including Mudholkar and Srivastava [17], Xie and Lai [29], Lai et al [13], Sarhan and Zaindin [21], and Silva et al [24].
It is known to have a bathtub-shaped hazard function (Fig. 3a) as indicated by the scaled TTT-Transform plot (Fig. 3b). Table 2 gives ML estimates of parameters of the NMW and sub-models with standard errors in brackets and goodness of fit statistics are in Table 3. We find that the NMW distribution with the same number of parameters provides a better fit than the beta modified Weiull distribution (BMW) which was the best in Silva et al [24]. It has the largest likelihood, and the smallest K-S, AIC and BIC values among those considered in this paper. It is clear in Fig. 3c that the NMW fits the left and right peaks in the histogram better and its survival function follows the Kaplan-Meier estimate more closely (Fig. 3d).
10
1
Scaled TTT−Transform
0.7
0.10
0.15
0.8
0.05
h(x)
0.9
Nonpar NMW MW SZMW AddW BMW
0.6 0.5 0.4 0.3
0.00
0.2 0.1
0
20
40
60
80
0
Empirical NMW 0
0.1
0.2
0.3
X
0.6
0.7
1.0
0.025
0.8
0.9
1
0.6
0.8
Kaplan−Meier NMW MW SZMW AddW BMW
0.4
S(x)
0.015
0.020
NMW MW SZMW AddW BMW
0.0
0.000
0.005
0.2
0.010
f(x)
0.5 X
(b)
0.030
(a)
0.4
0
20
40
60
80
0
20
40
60
x
X
(c)
(d)
80
Figure 4: For Aarst data: (a) hazard function (b) TTT-Transform plot (c) pdf and (d) survival function using NMW plus sub models, and beta modified Weibull.
11
Table 3: Log-likelihood, K-S statistic, the corresponding P -values, AIC and BIC values of models fitted to Aarst data for comparison with beta modified Weibull (Silva et al [24]).
5.2
Model
Log-lik
K-S
P -value
AIC
BIC
NMW
-212.90
0.088
0.803
435.8
445.4
MW
-227.16
0.129
0.346
460.3
466.0
AddW
-221.51
0.127
0.365
451.0
458.7
SZMW
-229.88
0.151
0.185
465.8
471.5
BMW
-220.80
0.127
0.365
451.6
461.2
Meeker and Escobar data
The data are failure and running times of a sample of 30 devices (Meeker and Escobar [16], p. 383). Two types of failures were observed for this data. It was shown by Nadarajah et al [19] to be best fit by the beta modified Weibull distribution.
The data have a bathtub shaped hazard function (Figs 4a and 4b). Again the NMW distribution (Table 4) provides a better fit than the BMW, as can be seen from Table 5.
12
Table 4: MLEs of parameters and corresponding standard errors in brackets for the Meeker and Escobar data.
Model
α ˆ
βˆ
γˆ
θˆ
ˆ λ
NMW
0.024
5.991 × 10−8
0.012
0.629
0.056
(0.019)
(8.164 × 10−8 )
(1.29)
(0.15)
(0.024)
0.018
0.454
7.133 × 10−3
(0.018)
(0.220)
(2.113 × 10−3 )
MW (α = 0, θ = 0) AddW
1.320 × 10−7
0.019
0.604
2.830
(λ = 0)
(7.435 × 10−7 )
(0.018)
(0.197)
(0.974)
SZMW
2.939 × 10−3
1.497 × 10−9
3.585
(9.290 × 10−4 )
(1.114 × 10−8 )
(1.314)
(θ = 1, λ = 0)
Table 5: Log-likelihood, K-S statistic, the corresponding P -values, AIC and BIC values of models fitted to Meeker and Escobar data
Model
Log-lik
K-S
P -value
AIC
BIC
NMW
-166.18
0.148
0.482
344.4
351.4
MW
-178.06
0.182
0.242
362.1
366.3
AddW
-178.11
0.191
0.197
364.2
369.8
SZMW
-177.90
0.186
0.221
361.8
366.0
BMW
-167.55
0.161
0.378
345.1
352.1
13
0.015
1 0.9 0.8
Scaled TTT−Transform
0.7
0.005
h(x)
0.010
Nonpar NMW MW SZMW AddW BMW
0.6 0.5 0.4 0.3
0.000
0.2 0.1
0
50
100
150
200
250
300
0
Empirical NMW 0
0.1
0.2
0.3
0.4
X
0.6
0.7
0.9
1
1.0 0.8
Kaplan−Meier NMW MW SZMW AddW BMW
0.0
0.000
0.2
0.002
0.4
0.004
f(x)
S(x)
0.6
0.008
NMW MW SZMW AddW BMW
0.006
0.8
(b)
0.010
(a)
0.5 X
0
50
100
150
200
250
300
0
50
100
150
x
X
(c)
(d)
200
250
300
Figure 5: For Meeker and Escobar data: (a) hazard function (b) TTT-Transform plot (c) pdf and (d) survival function using NMW plus sub models, and beta modified Weibull.
14
6
Sub-model of the NMW with γ = 1
To simplify the statistical inference, it is always a good idea to reduce the number of parameters of any distribution and investigate how that affects the ability of the reduced model to fit the data. In this section we reduce the number of parameters from five to four, by setting γ = 1. We test the reduced model (H0 :γ = 1) against the original model (Ha :γ ̸= 1). For each data set, Table 6 shows ML estimates of the four parameter NMW, the log-likelihood value under H0 , likelihood ratio statistic (LRT) with P -value in brackets, AIC, K-S statistic with P -value in brackets .
The likelihood ratio statistics against the full model with five parameters are 1.31 (P value = 0.252) and 2.45 (P -value=0.118) respectively on 1 d.f.. Therefore we can choose the reduced model with 4 parameters. The likelihood and AIC value also points to this model when the modified beta distribution is included in the comparison. Fig. 6 shows the reduced model is nearly as good as the full model for both data sets.
Table 6: Results of fitting NMW with γ = 1 to both data sets.
Data
Aarset
Meeker
α ˆ
βˆ
θˆ
ˆ λ
0.092
2.2 × 10−8
0.531
0.160
(0.039)
(2.1 × 10−8 )
(0.104)
(0.011)
0.017
3.5 × 10−8
0.675
0.039
(0.013)
(4.2 × 10−8 )
(0.141)
(4.0 × 10−3 )
15
Log-lik
-213.56
-167.40
AIC
435.1
342.8
LRT
K-S
(P -value)
(P -value)
1.31
0.105
(0.252)
(0.603)
2.45
0.153
(0.118)
(0.440)
7
Conclusions
A new distribution, based on Weibull and modified Weibull distributions, has been proposed and its properties studied. The idea is to combine two components in a serial system, so that the hazard function is either increasing or more importantly, bathtub shaped. By using a modified Weibull component, the distribution has flexibility to model the second peak in a distribution. We have shown that the new modified Weibull distribution fits certain well-known data sets better than existing modifications of the Weibull distribution. Reducing the number of parameters to four by fixing one of the parameters still provides a better fit than existing models.
Future work includes MCMC methods with censored data, regression problems with covariates and parameter reduction.
Acknowledgements
We would like to thank the referees for their comments and suggestions which improved the presentation of the paper. The first author wishes to thank the Saudi Arabia Culture Bureau in the UK and the Taif University for their financial support.
References [1] Aarset, M. V. (1987). How to identify bathtub hazard rate, IEEE Trans. Rel., vol. R-36, no. 1, pp.106-108. [2] Akaike, H. (1974). A new look at the statistical model identification, IEEE Trans. Automat. Control,. AC-19, 716-723. [3] Balakrishnan, A. N. and Nagaraja, H. N. (1992). A First Course in Order Statistics, New York: Wiley-Interscience.
16
1.0
0.030
0.8
0.025
0.4
S(x)
0.015
0.6
0.020
NMW 4NMW
0.0
0.000
0.005
0.2
0.010
f(x)
Kaplan−Meier NMW 4NMW
20
40
60
80
0
20
40
x
X
(a)
(b)
60
1.0
0.010
0
0.8
Kaplan−Meier NMW 4NMW
0.0
0.000
0.2
0.002
0.4
0.004
f(x)
S(x)
0.6
0.006
0.008
NMW 4NMW
80
0
50
100
150
200
250
300
0
50
100
150
x
X
(c)
(d)
200
250
300
Figure 6: (a) and (b): fitted pdf and survival functions for Aarst data, (c) and (d): those for Meeker and Escobar data, 5 parameters (solid lines) vs 4 parameters (dotted lines).
17
[4] Bain, L. J. (1974). Analysis for the Linear Failure-Rate Life-Testing Distribution, Technometrics, Vol. 16, no. 4, pp. 551-559. [5] Bebbington MS, Lai CD, Zitikis R, (2007) A flexible Weibull extension, Reliability Engineering & System Safety, 92(6), pp. 719-726 [6] Carrasco, M., Ortega, E. M. and Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling, Comput. Statist. Data Anal. 53 (2), pp. 450– 462. [7] Fisher, R. A. (1922). On the mathematical foundation of theoretical statistics, Phil. Trans. A, 222, pp 309–368. [8] Cordeiro,G. M., Ortega, E. M. and Nadarajah,S. (2010). The Kumaraswamy Weibull distribution with application to failure data, Journal of the Franklin Institute, 347, 1399–1429. [9] El-Gohary A, Alshamrani A. and Al-Otaibi A. (2012). The Generalized Gompertz Distribution, Applied Mathematical Modeling, In Press. [10] Famoye,F., Lee, C. and Olumolade, O. (2005). The beta-Weibull distribution, J. Statist. Theory Appl. 4, pp. 121–136. [11] Jiang, H., Xie, M. and Tang, L. C. (2010). On MLEs of the parameters of a modified Weibull distribution for progressively type-2 censored samples, J. Appl. Statist., 37, pp. 617–627. [12] Kuo, W. and Zuo, M. J. (2001). Optimal Reliability Modeling: Principles and Applications, Wiley. [13] Lai, C. D., Xie, M. and Murthy, D. N. P. (2003). A modified Weibull distribution, IEEE Transactions on Reliability, 52, pp. 33–37. [14] Lindley, D.V. (1980). Approximate Bayesian method, Trabajos Estadist, 31, pp. 223– 237.
18
[15] Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika , 84(3) , pp. 641–652. [16] Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data, JohnWiley, New York. [17] Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analysing bathtub failure rate data, IEEE Trans. Rel., vol. 42, no. 2, pp. 299–302. [18] Murthy, D. N. P., Xie, M. and Jiang, R. (2003). Weibull Models, New York: Wiley. [19] Nadarajah S., Cordeiro, G. M. and Ortega, E. M. M. (2011). General results for the beta-modified Weibull distribution, J. Stat. Comput. and Simu., vol. 81, no. 10, pp. 1211–1232. [20] Phani, K. K. (1987).A new modified Weibull distribution function, Communications of the American Ceramic Society, vol. 70, pp. 182–184. [21] Sarhan, A. M. and Zaindin, M. (2009). Modified Weibull distribution. Applied Sciences, vol.11, pp. 123-136. [22] Sarhan, A.M., Abd EL-Baset A. A. and Alasbahi, I. A. (2012). Exponentiated Generalized Linear Exponential Distribution, Applied Mathematical Modeling, Accepted. [23] Schwarz, G. (1978) Estimating the dimension of a model, Ann. Statist., 6. 461-464. [24] Silva, G. O., Ortega, E. M. and Cordeiro, G. M. (2010). The beta modified Weibull distribution, Lifetime Data Anal., vol. 16, pp. 409-430. [25] Singla, N., Jain, K. and Kumar Sharma, S. (2012). The Beta Generalized Weibull distribution: Properties and applications, Reliability Engineering & System Safety, 102, pp. 5-15. [26] Soliman, AA, Abd-Ellah, AH. and Abou-Elheggag, NA. (2012). Modified Weibull model: A Bayes study using MCMC approach based on progressive censoring data, Reliability Engineering & System Safety, 100, pp. 48–57.
19
[27] Upadhyaya, S. K. and Gupta, A. (2010). A Bayes analysis of modified Weibull distribution via Markov chain Monte Carlo simulation, Journal of Statistical Computation and Simulation, 80 ( 3 ), pp. 241–254. [28] Weibull, W. A. (1951). Statistical distribution function of wide applicability, J Appl Mech, vol. 18, pp.293–296. [29] Xie, M. and Lai, C. D. (1995). Reliability analysis using an additive Weibull model with bathtub-shaped failure ratefunction, Reliability Engineering & System Safety, vol. 52, pp. 87-93. [30] Xie,M., Tang,Y., Goh,T.N.(2002). A modified Weibull extension with bathtub-shaped failure rate function, Reliability Engineering & System Safety, 76(3), pp. 279–285. [31] Zhang, T., and Xie, M. (2011). On the Upper Truncated Weibull Distribution and Its Reliability Implications, Reliability Engineering & System Safety, 96(1), pp. 194–200.
20
A
Appendix
The log-likelihood function of the N M W (α, β, θ, γ, λ) can be written as
L(ϑ) =
n [ ] ∑ ln (h(xi ; ϑ)) − αxθi − βxγi eλx ,
(17)
i=1
where h(xi ; ϑ) is the hazard rate function (6) of the NMW and ϑ = (α, β, θ, γ, λ) is the vector of parameters.
The second partial derivatives are as follows ) n ( ∑ hα (xi ; ϑ) 2 Lαα = − , h(xi ; ϑ) Lαβ = − Lαθ =
i=1 n ∑
hα (xi ; ϑ)hβ (xi ; ϑ)
i=1
(h(xi ; ϑ))2
n ( ∑ h(xi ; ϑ)hαθ (xi ; ϑ) − hα (xi ; ϑ)hθ (xi ; ϑ) i=1
Lαγ
= −
Lαλ = −
(h(xi ; ϑ))2
n ∑ hα (xi ; ϑ)hγ (xi ; ϑ) i=1
Lββ
,
(h(xi ; ϑ))2
n ∑ hα (xi ; ϑ)hλ (xi ; ϑ)
(h(xi ; ϑ))2 i=1 ) n ( ∑ hβγ (xi ; ϑ) 2 = − , h(xi ; ϑ)
) ln(xi ) ,
−
xθi
−
xγi eλxi
−
xγ+1 eλxi i
, ,
i=1
Lβθ = −
n ∑ hβ (xi ; ϑ)hθ (xi ; ϑ) i=1
Lβγ
=
Lβλ = Lθθ =
,
n ( ∑ h(xi ; ϑ)hβγ (xi ; ϑ) − hβ (xi ; ϑ)hγ (xi ; ϑ) i=1 n ( ∑ i=1 ( n ∑ i=1
Lθγ
(h(xi ; ϑ))2
= −
Lθλ = −
(h(xi ; ϑ))2 h(xi ; ϑ)hβλ (xi ; ϑ) − hβ (xi ; ϑ)hλ (xi ; ϕ) (h(xi ; ϑ))2
i=1
)
) h(xi ; ϑ)hθθ (xi ; ϑ) − (hθ (xi ; ϑ))2 − αxθi ln2 (xi ) , (h(xi ; ϑ))2
n ∑ hθ (xi ; ϑ)hγ (xi ; ϑ) i=1 n ∑
) ln(xi ) ,
(h(xi ; ϑ))2
,
hθ (xi ; ϑ)hλ (xi ; ϑ) , (h(xi ; ϑ))2 21
,
Lγγ
=
n ∑
(
i=1
Lγλ = Lλλ =
h(xi ; ϑ)hγγ (xi ; ϑ) − (hγ (xi ; ϑ))2 (h(xi ; ϑ))2
) −
βxγi eλxi
n ( ∑ h(xi ; ϑ)hγλ (xi ; ϑ) − hγ (xi ; ϑ)hλ (xi ; ϑ) i=1 ( d ∑ i=1
(h(xi ; ϑ))2
2
ln (xi ) ,
) λxi 2 − βxγ+1 e ln (x ) , i i
h(xi ; ϑ)hλλ (xi ; ϑ) − (hλ (xi ; ϑ))2 − βxγ+2 eλxi i (h(xi ; ϑ))2
) ,
where hαθ (xi ; ϑ) = xθ−1 (1 + θ ln(xi )), i hβγ (xi ; ϑ) = xγ−1 (1 + (γ + λxi ) ln(xi )) eλxi , i hβλ (xi ; ϑ) = xγi (1 + γ + λxi ) ln(xi )) eλxi , (2 + θ ln(xi )) ln(xi ), hθθ (xi ; ϑ) = αxθ−1 i hγγ (xi ; ϑ) = βxγ−1 (2 + (γ + λxi ) ln(xi )) ln(xi )eλxi , i hγλ (xi ; ϑ) = βxγi (1 + (1 + γ + λxi ) ln(xi )) eλxi , (2 + γ + λxi ) ln(xi )) eλxi , hλλ (xi ; ϑ) = βxγ+1 i , hα (xi ; ϑ) = θxθ−1 i hβ (xi ; ϑ) = xγ−1 (γ + λxi ) ln(xi )eλxi , i hθ (xi ; ϑ) = αhαθ (xi ; ϑ), hγ (xi ; ϑ) = βhβγ (xi ; ϑ), hλ (xi ; ϑ) = βhβλ (xi ; ϑ).
22
