A new modified Weibull distribution - Semantic Scholar
Reliability Engineering and System Safety 111 (2013) 164–170
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
A new modified Weibull distribution Saad J. Almalki n, Jingsong Yuan School of Mathematics, University of Manchester, Manchester M13 9PL, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 February 2012 Received in revised form 29 October 2012 Accepted 30 October 2012 Available online 22 November 2012
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. & 2012 Elsevier Ltd. All rights reserved.
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 and application of the truncated Weibull distribution which has a bathtub shaped hazard function. A threeparameter 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 threeparameter 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 axb by elx , which was later generalized to exponentiated form by Carrasco et al. [6] b elx
FðxÞ ¼ ð1eax
Þy ,
x Z 0:
Recent studies of the modified Weibull include [11,26,27]. n
Corresponding author. Tel.: þ44 7590894919. E-mail address: [email protected] (S.J. Almalki).
0951-8320/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2012.10.018
ð1Þ
Among the four-parameter distributions, the additive Weibull distribution (AddW) of Xie and Lai [29] with cumulative distribution function (CDF) y bxg
FðxÞ ¼ 1eax
,
x Z0,
has a bathtub-shaped hazard function consisting of two Weibull hazards, one increasing (y 4 1) and one decreasing (0 o g o1). The modified Weibull distribution of Sarhan and Zaindin (SZMW) [21] can be derived from the additive Weibull distribution by setting y ¼ 1. A four-parameter beta Weibull distribution was proposed by Famoye et al. [10]. Cordeiro et al. [8] introduced another fourparameter called the Kumaraswamy Weibull distribution. Five-parameter modified Weibull distributions include Phani’s modified Weibull [20], 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 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.
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
165
2.3. Motivation and interpretation
2. The model
The survival function of the new distribution is given by
2.1. Definition We define a new modified Weibull distribution (NMW) by the following CDF: y bxg elx
FðxÞ ¼ 1eax
,
x Z 0,
ð2Þ
where a, b, y, g and l are non-negative, with y and g being shape parameters and a and b being scale parameters and l acceleration parameter. The probability density function (PDF) is y bxg elx
f ðxÞ ¼ ðayxy1 þ bðg þ lxÞxg1 elx Þeax
,
x 4 0:
ð3Þ
It can be rewritten as f ðxÞ ¼ ½hW ðx; a, yÞ þ hMW ðx; b, g, lÞSW ðx, a, yÞSMW ðx; b, g, lÞ,
ð4Þ
where SW, hW, SMW and hMW 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.
y bxg elx
SðxÞ ¼ eax
,
x Z 0,
ð5Þ
and, the hazard function is hðxÞ ¼ ayxy1 þ bðg þ lxÞxg1 elx ,
x 40,
ð6Þ
which can be interpreted as that of a serial system with two independent components, one of which follows the Weibull distribution with parameters a and y, and the other follows the modified Weibull distribution of Lai et al. [13] with parameters b, g and l. 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 y o 1. (It will be increasing when y 4 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. Properties of the model 3.1. The hazard function
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. 3.0
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
2.5
3.2. The moments
2.0
f(x)
The hazard function can have many different shapes, including bathtub, as shown in Fig. 2. We can deduce from (6) that it is increasing if y, g Z 1, decreasing if y, g o 1 and l ¼ 0 and bathtub shaped otherwise. 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 property, so does the NMW as shown in Fig. 3.
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 Z 1 m0r ¼ xr dFðxÞ 0 Z 1 y g lx xr deax bx e ¼
1.5
1.0
0.5
0.0
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x
8
Fig. 1. Probability density functions of the NMW. 6
h(x)
Table 1 The sub-models of the NMW. Model
a b g
y
l
S(x)
Additive Weibull
–
–
–
–
0
eax
Modified Weibull
0
-
-
0
-
ebx
4
Reference y bxg
Xie and Lai [29]
g elx
Lai et al. [13]
axbxg
S–Z modified Weibull
–
–
–
1
0
e
Linear failure rate
–
–
2
1
0
eaxbx
Bain [4]
Extreme-value
0
1
0
0
–
ee
lx
Bain [4]
Weibull
–
0
0
–
0
eax
Rayleigh
–
0
0
2
0
Exponential
–
0
0
1
0
eax eax
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
2
Sarhan and Zaindin [21]
y
Weibull [28]
2
Bain [4] Bain [4]
0 0.0
0.5
1.0
x Fig. 2. Hazard functions of the NMW.
1.5
166
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
Z
1
¼
y bxg elx
rxr1 eax
Substituting (9) and (10) into (7), we get r1 X r1 1 f r:n ðxÞ ¼ ð1Þ‘ hðxÞeðn þ ‘ þ 1rÞHðxÞ Bðr,nr þ 1Þ ‘ ¼ 0 ‘
dx
0
¼
Z 1 X 1 X ðbÞn ðlnÞm 1 r1 ng þ m axy rx x e dx n!m! 0 n¼0m¼0
r1 r1 n1 X ¼n ð1Þ‘ hðxÞeðn þ ‘ þ 1rÞHðxÞ ‘ r1 ‘ ¼ 0
1 X 1 r X ðbÞn ðlnÞm ðng þ m þ rÞ=y ng þ mþ r a G ¼ , yn¼0m¼0 n!m! y
r1 r1 n1 X ¼n ‘ r1 ‘ ¼ 0
for r ¼ 1,2, . . ., where GðÞ is the gamma function.
g lx
y
ð1Þ‘ ðayxy1 þ bðg þ lxÞxg1 elx Þeðn þ ‘ þ 1rÞðax þ bx e Þ r1 r1 n1 X ð1Þ‘ f ðx; a‘ , b‘ , y, g, lÞ, ¼n ðn þ ‘ þ 1rÞ ‘ r1 ‘ ¼ 0
3.3. Order statistics It will also be useful to derive the pdf of the rth order statistic X ðrÞ of a random sample X1, y, Xn drawn from the NMW with parameters a, b, y, g and l. From Balakrishnan and Nagaraja [3], the pdf of X ðrÞ is given by f r:n ðxÞ ¼
FðxÞr1 ð1FðxÞÞnr f ðxÞ , Bðr,nr þ 1Þ
ð7Þ
where Bð,Þ is the beta function. Using FðxÞ ¼ 1eHðxÞ ,
i
ð1Þ‘ þ i b ðilÞj ðig þ jÞ
f ðxÞ ¼ hðxÞeHðxÞ ,
ð8Þ
where h(x) is the hazard function (6) and cumulative hazard, then
HðxÞ ¼ axy þ bxg elx
ð9Þ
and r 1 X r1 FðxÞr1 ¼ ð1eHðxÞ Þr1 ¼ ð1Þ‘ e‘HðxÞ :
0.20
α= α= α= α=
0.15
ðn þ ‘ þ 1rÞðig þ j þ kÞ=y þ 1i aðig þ j þ kÞ=y
y
ð10Þ
‘
3,β = 5e−06,γ = 0.01,θ = 0.05,λ = 0.15 0.25,β = 5e−04,γ = 0.001,θ = 0.5,λ = 0.09 0.11,β = 2.5e−07,γ = 1,θ = 1,λ = 0.12 0.025,β = 5e−07,γ = 1,θ = 1,λ = 0.12
4. Parameter estimation Given a random sample x1, y, xn from the NMW with parameters ða, b, y, g, lÞ, 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 L¼
n X
g1 lxi e þ ayxy1 Þ
lnðbðg þ lxi Þxi
i
i¼1
a
n X
n X g xi elxi :
xyi b
i¼1
h(x)
ig þ j þ k :
G
is the
ð1FðxÞÞnr ¼ eðnrÞHðxÞ ,
‘¼0
where f ðx; a‘ , b‘ , y, g, lÞ is the PDF of the NMW with parameters a‘ ¼ ðn þ ‘ þ 1rÞa, b‘ ¼ ðn þ ‘ þ 1rÞb, y, g and l. Using (7), the kth non-central moment of the rth order statistic X ðrÞ is 1 X 1 X r1 r1 nk n1 X m0ðr:nÞ ¼ k y r1 i ¼ 0 j ¼ 0 ‘ ¼ 0 ‘
ð11Þ
i¼1
Setting the first partial derivatives of ‘ with respect to a, b, y, g and l to zero, the likelihood equations are
0.10
n X
n X yxiy1 xyi ¼ 0, hðx ; a , b , g , y , l Þ i i¼1 i¼1
0.05
g1 n X ðg þ lxi Þx elxi i
0.00 0
20
40
60
80
x
hðxi ; a, b, g, y, lÞ i¼1
n X axy1 ð1 þ y lnðxi ÞÞ i
Fig. 3. Hazard functions of the NMW with long useful life period.
i¼1
hðxi ; a, b, g, y, lÞ
ð12Þ
n X g xi elxi ¼ 0,
ð13Þ
i¼1
a
n X
xyi lnðxi Þ ¼ 0,
ð14Þ
i¼1
Table 2 MLEs of parameters and corresponding standard errors in brackets for the Aarset data. Model
a^
b^
g^
NMW
0.071
7:015 108
0.016
0.595
0.197
(0.031)
(1:501 107 ) 0.062 (0.027)
(3.602)
(0.128)
(0.184)
0.356) (0.113) 0.477
4.214 (1.033)
MW ða ¼ 0, y ¼ 0) AddW
1:133 10
0.086
(l ¼ 0)
(0.036)
(0.102)
SZMW
(5:183 108 ) 0.013
8:408 109
4.224
(y ¼ 1, l ¼ 0)
(2:819 103 )
(4:204 108 )
(1.140)
8
y^
l^
0.023 (4:845 103 )
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
g1 n X x elxi ððg þ lxi Þ lnðxi Þ þ1Þ i
hðxi ; a, b, g, y, lÞ
i¼1
g n X ð1 þ g þ lxi Þx elxi i
i¼1
hðxi ; a, b, g, y, lÞ
n X g xi elxi lnðxi Þ ¼ 0,
ð15Þ
i¼1
n X gþ1 xi elxi ¼ 0:
ð16Þ
i¼1
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]). Model
Log-lik
K–S
P-value
AIC
BIC
NMW MW AddW SZMW BMW
212.90 227.16 221.51 229.88 220.80
0.088 0.129 0.127 0.151 0.127
0.803 0.346 0.365 0.185 0.365
435.8 460.3 451.0 465.8 451.6
445.4 466.0 458.7 471.5 461.2
167
The maximum likelihood estimates can be obtained by solving the non-linear equations numerically for a, b, y, g and l. 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 Kolmogorov–Smirnov (K–S) statistic, as well as Akaike information criterion (AIC) [2] and Bayesian information criterion (BIC) [23] values.
1
0.8 Scaled TTT−Transform
0.15
S(x)
0.9
Nonpar NMW MW SZMW AddW BMW
0.10
0.05
0.7 0.6 0.5 0.4 0.3 0.2
0.00
0.1 0
20
40
60
80
0
X
Empirical NMW
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
X
1.0
0.030 NMW MW SZMW AddW BMW
0.025
0.8 0.6 f(x)
h(x)
0.020
Kaplan−Meier NMW MW SZMW AddW BMW
0.015
0.4 0.010 0.2
0.005 0.000
0.0 0
20
40
60
80
0
20
40
60
80
X
x
Fig. 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.
Table 4 MLEs of parameters and corresponding standard errors in brackets for the Meeker and Escobar data. Model
a^
NMW
0.024
5:991 108
0.012
0.629
0.056
(0.019)
(1.29)
(0.15)
(0.024)
MW
(8:164 108 ) 0.018
0:454
7:133 103
(a ¼ 0, y ¼ 0)
(0.018)
(0.220)
(2:113 103 )
b^
g^
y^
AddW
1:320 107
0.019
0.604
2.830
(l ¼ 0)
(7:435 107 )
(0.018)
(0.197)
(0.974)
SZMW
2:939 103
1:497 109
3.585
(y ¼ 1, l ¼ 0)
(9:290 104 )
(1:114 108 )
(1.314)
l^
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S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
5.1. Aarset data
5.2. Meeker and Escobar 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).
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 (Fig. 4a and b). Again the NMW distribution (Table 4) provides a better fit than the BMW, as can be seen from Table 5 (Fig. 5).
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 MW AddW SZMW BMW
166.18 178.06 178.11 177.90 167.55
0.148 0.182 0.191 0.186 0.161
0.482 0.242 0.197 0.221 0.378
344.4 362.1 364.2 361.8 345.1
351.4 366.3 369.8 366.0 352.1
6. Sub-model of the NMW with c ¼ 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 g ¼ 1. We test the reduced model H0 : g ¼ 1Þ against the original model Ha : g a1Þ. For each data set, Table 6 shows ML estimates of the four parameter NMW, the loglikelihood value under H0, likelihood ratio statistic (LRT) with Pvalue 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-value¼0.252) and 2.45 (P-value¼0.118), respectively, on 1 d.f. Therefore we can choose the reduced model with four 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.
1 0.9 0.015
Nonpar NMW MW SZMW AddW BMW
Scaled TTT−Transform
h(x)
0.010
0.8
0.005
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.000
0 0
50
100
150
200
250
300
Empirical NMW
0
0.1
0.2
0.3
X
0.010
0.4
0.5 X
0.6
1.0 NMW MW SZMW AddW BMW
0.008
f(x)
0.4
0.002
0.2
0.000
0.0 50
100
150 x
200
250
300
0.9
1
0.6
0.004
0
0.8
Kaplan−Meier NMW MW SZMW AddW BMW
0.8
S(x)
0.006
0.7
0
50
100
150
200
250
300
X
Fig. 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.
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
169
Table 6 Results of fitting NMW with g ¼ 1 to both data sets. Data Aarset Meeker
a^
l^
Log-lik
AIC
LRT (P-value)
0.531
0.160
213.56
435.1
1.31
0.105
(0.104)
(0.011)
(0.252)
(0.603)
3:5 108
0.675
0.039
2.45
0.153
(4:2 108 )
(0.141)
ð4:0 103 )
(0.118)
(0.440)
b^
y^
0.092
2:2 108
(0.039)
(2:1 108 )
0.017 (0.013)
167.40
1.0
0.030 0.025
Kaplan−Meier NMW 4NMW
0.8
NMW 4NMW
S(x)
S(x)
342.8
K–S (P-value)
0.020
0.6
0.015 0.4 0.010 0.2
0.005
0.0
0.000 0
20
40
60
0
80
20
40
x
60
1.0
0.010
Kaplan−Meier NMW 4NMW
NMW 4NMW
0.008
0.8
0.006
0.6
f(x)
f(x)
80
X
0.004
0.4
0.2
0.002
0.0
0.000 0
50
100
150
200
250
0
300
50
100
150
x
200
250
300
X
Fig. 6. (a and b) Fitted pdf and survival functions for Aarst data and (c and d) those for Meeker and Escobar data, five parameters (solid lines) vs four parameters (dotted lines).
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. 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.
Acknowledgments 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.
Appendix A The log-likelihood function of the NMWða, b, y, g, lÞ can be written as LðW Þ ¼
n X
g
½lnðhðxi ; W ÞÞaxyi bxi elx ,
ð17Þ
i¼1
where hðxi ; W Þ is the hazard rate function (6) of the NMW and W ¼ ða, b, y, g, lÞ is the vector of parameters. The second partial derivatives are as follows: Laa ¼
2 n X ha ðxi ; W Þ i¼1
hðxi ; W Þ
,
170
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
Lab ¼
n X ha ðxi ; W Þhb ðxi ; W Þ i¼1
i¼1
Lag ¼
Lbb ¼
ðhðxi ; W ÞÞ
ha ðxi ; W Þhl ðxi ; W Þ
i¼1
ðhðxi ; W ÞÞ2
2 n X hbg ðxi ; W Þ hðxi ; W Þ
,
,
ðhðxi ; W ÞÞ2
n X
hðxi ; W Þhbg ðxi ; W Þhb ðxi ; W Þhg ðxi ; W Þ
i¼1
ðhðxi ; W ÞÞ2
n X
hðxi ; W Þhbl ðxi ; W Þhb ðxi ; W Þhl ðxi ; f Þ
i¼1
ðhðxi ; W ÞÞ2
Lbg ¼
Lbl ¼
2 hðxi ; W Þhyy ðxi ; W Þ hy ðxi ; W Þ
i¼1
ðhðxi ; W ÞÞ2
Lyg ¼
n X hy ðxi ; W Þhg ðxi ; W Þ
ðhðxi ; W ÞÞ2
i¼1
Lyl ¼
n X hy ðxi ; W Þhl ðxi ; W Þ
ðhðxi ; W ÞÞ2
i¼1
i¼1
ðhðxi ; W ÞÞ2
g lxi
bxi e
i¼1
ðhðxi ; W ÞÞ2
d X
hðxi ; W Þhll ðxi ; W Þðhl ðxi ; W ÞÞ2
i¼1
ðhðxi ; W ÞÞ2
i
hay ðxi ; W Þ ¼ xiy1 ð1 þ y lnðxi ÞÞ, ð1 þðg þ lxi Þ lnðxi ÞÞelxi ,
g
hbl ðxi ; W Þ ¼ xi ð1 þ g þ lxi Þ lnðxi ÞÞelxi , hyy ðxi ; W Þ ¼ axiy1 ð2 þ y lnðxi ÞÞ lnðxi Þ, ð2 þ ðg þ lxi Þ lnðxi ÞÞ lnðxi Þelxi ,
g
hgl ðxi ; W Þ ¼ bxi ð1 þ ð1þ g þ lxi Þ lnðxi ÞÞelxi , gþ1
hll ðxi ; W Þ ¼ bxi
ð2þ g þ lxi Þ lnðxi ÞÞelxi ,
ha ðxi ; W Þ ¼ yxiy1 , g1
hb ðxi ; W Þ ¼ xi
ðg þ lxi Þ lnðxi Þelxi ,
hy ðxi ; W Þ ¼ ahay ðxi ; W Þ,
! 2
ln ðxi Þ ,
g þ 1 lxi e
bxi
g þ 2 lxi bx e
where
g1
2
axi ln ðxi Þ ,
hðxi ; W Þhgl ðxi ; W Þhg ðxi ; W Þhl ðxi ; W Þ
hgg ðxi ; W Þ ¼ bxi
,
! y
n X
g1
!
,
hðxi ; W Þhgg ðxi ; W Þðhg ðxi ; W ÞÞ2
hbg ðxi ; W Þ ¼ xi
g þ 1 lxi x e
,
n X
Lll ¼
!
g
xi elxi lnðxi Þ ,
i
n X
Lyy ¼
Lgl ¼
,
n X hb ðxi ; W Þhy ðxi ; W Þ i¼1
Lgg ¼
,
2
n X
i¼1
Lby ¼
References
n X ha ðxi ; W Þhg ðxi ; W Þ i¼1
Lal ¼
hl ðxi ; W Þ ¼ bhbl ðxi ; W Þ:
! hðxi ; W Þhay ðxi ; W Þha ðxi ; W Þhy ðxi ; W Þ y xi lnðxi Þ , 2 hðxi ; W Þ
n X
Lay ¼
hg ðxi ; W Þ ¼ bhbg ðxi ; W Þ,
,
ðhðxi ; W ÞÞ2
! ,
! 2
ln ðxi Þ ,
[1] Aarset MV. How to identify bathtub hazard rate. IEEE Transactions on Reliability 1987;36(1):106–8. [2] Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control 1974;AC-19:716–23. [3] Balakrishnan AN, Nagaraja HN. A first course in order statistics. New York: Wiley-Interscience; 1992. [4] Bain LJ. Analysis for the linear failure-rate life-testing distribution. Technometrics 1974;16(4):551–9. [5] Bebbington MS, Lai CD, Zitikis R. A flexible Weibull extension. Reliability Engineering & System Safety 2007;92(6):719–26. [6] Carrasco M, Ortega EM, Cordeiro GM. A generalized modified Weibull distribution for lifetime modeling. Computational Statistics and Data Analysis 2008;53(2):450–62. [7] Fisher RA. On the mathematical foundation of theoretical statistics. Philosophical Transactions of the Royal Society A 1922;222:309–68. [8] Cordeiro GM, Ortega EM, Nadarajah S. The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute 2010;347:1399–429. [9] El-Gohary A, Alshamrani A, Al-Otaibi A. The generalized Gompertz distribution. Applied Mathematical Modeling 2011;37(1):1324. [10] Famoye F, Lee C, Olumolade O. The beta-Weibull distribution. Journal of Statistical Theory and Applications 2005;4(2):121–36. [11] Jiang H, Xie M, Tang LC. On MLEs of the parameters of a modified Weibull distribution for progressively type-2 censored samples. Journal of Applied Statistical Science 2010;37(4):617–27. [12] Kuo W, Zuo MJ. Optimal reliability modeling: principles and applications. Wiley; 2002. [13] Lai CD, Xie M, Murthy DNP. A modified Weibull distribution. IEEE Transactions on Reliability 2003;52(1):33–7. [14] Lindley DV. Approximate Bayesian method. Trabajos Estadist 1980;31(1): 223–45. [15] Marshall AW, Olkin I. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 1997;84(3):641–52. [16] Meeker WQ, Escobar LA. Statistical methods for reliability data, vol. 78. New York: John Wiley; 1998. [17] Mudholkar GS, Srivastava DK. Exponentiated Weibull family for analysing bathtub failure rate data. IEEE Transactions on Reliability 1993;42(2): 299–302. [18] Murthy DNP, Xie M, Jiang R. Weibull models, vol. 358. New York: Wiley; 2003. [19] Nadarajah S, Cordeiro GM, Ortega EMM. General results for the betamodified Weibull distribution. Journal of Statistical Computation and Simulation 2011;81(10):1211–32. [20] Phani KK. A new modified Weibull distribution function. Communications of the American Ceramic Society 1987;70(8):182–4. [21] Sarhan AM, Zaindin M. Modified Weibull distribution. Applied Sciences 2009;11:123–36. [22] Sarhan AM, Ahmad, AEBA, Alasbahi IA. Exponentiated generalized linear exponential distribution. Applied Mathematical Modeling, accepted. [23] Schwarz G. Estimating the dimension of a model. Annals of Statistics 1978;6:461–4. [24] Silva GO, Ortega EM, Cordeiro GM. The beta modified Weibull distribution. Lifetime Data Analysis 2010;16:409–30. [25] Singla N, Jain K, Kumar Sharma S. The beta generalized Weibull distribution: properties and applications. Reliability Engineering & System Safety 2012;102:5–15. [26] Soliman AA, Abd-Ellah AH, Abou-Elheggag NA. Modified Weibull model: a Bayes study using MCMC approach based on progressive censoring data. Reliability Engineering & System Safety 2012;100:48–57. [27] Upadhyaya SK, Gupta A. A Bayes analysis of modified Weibull distribution via Markov chain Monte Carlo simulation. Journal of Statistical Computation and Simulation 2010;80(3):241–54. [28] Weibull WA. Statistical distribution function of wide applicability. Journal of Applied Mechanics 1951;18:293–6. [29] Xie M, Lai CD. Reliability analysis using an additive Weibull model with bathtub-shaped failure ratefunction. Reliability Engineering System Safety 1995;52:87–93. [30] Xie M, Tang Y, Goh TN. A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering System Safety 2002;76(3): 279–85. [31] Zhang T, Xie M. On the upper truncated Weibull distribution and its reliability implications. Reliability Engineering System Safety 2011;96(1): 194–200.
Contents lists available at SciVerse ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
A new modified Weibull distribution Saad J. Almalki n, Jingsong Yuan School of Mathematics, University of Manchester, Manchester M13 9PL, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 February 2012 Received in revised form 29 October 2012 Accepted 30 October 2012 Available online 22 November 2012
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. & 2012 Elsevier Ltd. All rights reserved.
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 and application of the truncated Weibull distribution which has a bathtub shaped hazard function. A threeparameter 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 threeparameter 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 axb by elx , which was later generalized to exponentiated form by Carrasco et al. [6] b elx
FðxÞ ¼ ð1eax
Þy ,
x Z 0:
Recent studies of the modified Weibull include [11,26,27]. n
Corresponding author. Tel.: þ44 7590894919. E-mail address: [email protected] (S.J. Almalki).
0951-8320/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2012.10.018
ð1Þ
Among the four-parameter distributions, the additive Weibull distribution (AddW) of Xie and Lai [29] with cumulative distribution function (CDF) y bxg
FðxÞ ¼ 1eax
,
x Z0,
has a bathtub-shaped hazard function consisting of two Weibull hazards, one increasing (y 4 1) and one decreasing (0 o g o1). The modified Weibull distribution of Sarhan and Zaindin (SZMW) [21] can be derived from the additive Weibull distribution by setting y ¼ 1. A four-parameter beta Weibull distribution was proposed by Famoye et al. [10]. Cordeiro et al. [8] introduced another fourparameter called the Kumaraswamy Weibull distribution. Five-parameter modified Weibull distributions include Phani’s modified Weibull [20], 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 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.
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
165
2.3. Motivation and interpretation
2. The model
The survival function of the new distribution is given by
2.1. Definition We define a new modified Weibull distribution (NMW) by the following CDF: y bxg elx
FðxÞ ¼ 1eax
,
x Z 0,
ð2Þ
where a, b, y, g and l are non-negative, with y and g being shape parameters and a and b being scale parameters and l acceleration parameter. The probability density function (PDF) is y bxg elx
f ðxÞ ¼ ðayxy1 þ bðg þ lxÞxg1 elx Þeax
,
x 4 0:
ð3Þ
It can be rewritten as f ðxÞ ¼ ½hW ðx; a, yÞ þ hMW ðx; b, g, lÞSW ðx, a, yÞSMW ðx; b, g, lÞ,
ð4Þ
where SW, hW, SMW and hMW 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.
y bxg elx
SðxÞ ¼ eax
,
x Z 0,
ð5Þ
and, the hazard function is hðxÞ ¼ ayxy1 þ bðg þ lxÞxg1 elx ,
x 40,
ð6Þ
which can be interpreted as that of a serial system with two independent components, one of which follows the Weibull distribution with parameters a and y, and the other follows the modified Weibull distribution of Lai et al. [13] with parameters b, g and l. 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 y o 1. (It will be increasing when y 4 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. Properties of the model 3.1. The hazard function
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. 3.0
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
2.5
3.2. The moments
2.0
f(x)
The hazard function can have many different shapes, including bathtub, as shown in Fig. 2. We can deduce from (6) that it is increasing if y, g Z 1, decreasing if y, g o 1 and l ¼ 0 and bathtub shaped otherwise. 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 property, so does the NMW as shown in Fig. 3.
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 Z 1 m0r ¼ xr dFðxÞ 0 Z 1 y g lx xr deax bx e ¼
1.5
1.0
0.5
0.0
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x
8
Fig. 1. Probability density functions of the NMW. 6
h(x)
Table 1 The sub-models of the NMW. Model
a b g
y
l
S(x)
Additive Weibull
–
–
–
–
0
eax
Modified Weibull
0
-
-
0
-
ebx
4
Reference y bxg
Xie and Lai [29]
g elx
Lai et al. [13]
axbxg
S–Z modified Weibull
–
–
–
1
0
e
Linear failure rate
–
–
2
1
0
eaxbx
Bain [4]
Extreme-value
0
1
0
0
–
ee
lx
Bain [4]
Weibull
–
0
0
–
0
eax
Rayleigh
–
0
0
2
0
Exponential
–
0
0
1
0
eax eax
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
2
Sarhan and Zaindin [21]
y
Weibull [28]
2
Bain [4] Bain [4]
0 0.0
0.5
1.0
x Fig. 2. Hazard functions of the NMW.
1.5
166
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
Z
1
¼
y bxg elx
rxr1 eax
Substituting (9) and (10) into (7), we get r1 X r1 1 f r:n ðxÞ ¼ ð1Þ‘ hðxÞeðn þ ‘ þ 1rÞHðxÞ Bðr,nr þ 1Þ ‘ ¼ 0 ‘
dx
0
¼
Z 1 X 1 X ðbÞn ðlnÞm 1 r1 ng þ m axy rx x e dx n!m! 0 n¼0m¼0
r1 r1 n1 X ¼n ð1Þ‘ hðxÞeðn þ ‘ þ 1rÞHðxÞ ‘ r1 ‘ ¼ 0
1 X 1 r X ðbÞn ðlnÞm ðng þ m þ rÞ=y ng þ mþ r a G ¼ , yn¼0m¼0 n!m! y
r1 r1 n1 X ¼n ‘ r1 ‘ ¼ 0
for r ¼ 1,2, . . ., where GðÞ is the gamma function.
g lx
y
ð1Þ‘ ðayxy1 þ bðg þ lxÞxg1 elx Þeðn þ ‘ þ 1rÞðax þ bx e Þ r1 r1 n1 X ð1Þ‘ f ðx; a‘ , b‘ , y, g, lÞ, ¼n ðn þ ‘ þ 1rÞ ‘ r1 ‘ ¼ 0
3.3. Order statistics It will also be useful to derive the pdf of the rth order statistic X ðrÞ of a random sample X1, y, Xn drawn from the NMW with parameters a, b, y, g and l. From Balakrishnan and Nagaraja [3], the pdf of X ðrÞ is given by f r:n ðxÞ ¼
FðxÞr1 ð1FðxÞÞnr f ðxÞ , Bðr,nr þ 1Þ
ð7Þ
where Bð,Þ is the beta function. Using FðxÞ ¼ 1eHðxÞ ,
i
ð1Þ‘ þ i b ðilÞj ðig þ jÞ
f ðxÞ ¼ hðxÞeHðxÞ ,
ð8Þ
where h(x) is the hazard function (6) and cumulative hazard, then
HðxÞ ¼ axy þ bxg elx
ð9Þ
and r 1 X r1 FðxÞr1 ¼ ð1eHðxÞ Þr1 ¼ ð1Þ‘ e‘HðxÞ :
0.20
α= α= α= α=
0.15
ðn þ ‘ þ 1rÞðig þ j þ kÞ=y þ 1i aðig þ j þ kÞ=y
y
ð10Þ
‘
3,β = 5e−06,γ = 0.01,θ = 0.05,λ = 0.15 0.25,β = 5e−04,γ = 0.001,θ = 0.5,λ = 0.09 0.11,β = 2.5e−07,γ = 1,θ = 1,λ = 0.12 0.025,β = 5e−07,γ = 1,θ = 1,λ = 0.12
4. Parameter estimation Given a random sample x1, y, xn from the NMW with parameters ða, b, y, g, lÞ, 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 L¼
n X
g1 lxi e þ ayxy1 Þ
lnðbðg þ lxi Þxi
i
i¼1
a
n X
n X g xi elxi :
xyi b
i¼1
h(x)
ig þ j þ k :
G
is the
ð1FðxÞÞnr ¼ eðnrÞHðxÞ ,
‘¼0
where f ðx; a‘ , b‘ , y, g, lÞ is the PDF of the NMW with parameters a‘ ¼ ðn þ ‘ þ 1rÞa, b‘ ¼ ðn þ ‘ þ 1rÞb, y, g and l. Using (7), the kth non-central moment of the rth order statistic X ðrÞ is 1 X 1 X r1 r1 nk n1 X m0ðr:nÞ ¼ k y r1 i ¼ 0 j ¼ 0 ‘ ¼ 0 ‘
ð11Þ
i¼1
Setting the first partial derivatives of ‘ with respect to a, b, y, g and l to zero, the likelihood equations are
0.10
n X
n X yxiy1 xyi ¼ 0, hðx ; a , b , g , y , l Þ i i¼1 i¼1
0.05
g1 n X ðg þ lxi Þx elxi i
0.00 0
20
40
60
80
x
hðxi ; a, b, g, y, lÞ i¼1
n X axy1 ð1 þ y lnðxi ÞÞ i
Fig. 3. Hazard functions of the NMW with long useful life period.
i¼1
hðxi ; a, b, g, y, lÞ
ð12Þ
n X g xi elxi ¼ 0,
ð13Þ
i¼1
a
n X
xyi lnðxi Þ ¼ 0,
ð14Þ
i¼1
Table 2 MLEs of parameters and corresponding standard errors in brackets for the Aarset data. Model
a^
b^
g^
NMW
0.071
7:015 108
0.016
0.595
0.197
(0.031)
(1:501 107 ) 0.062 (0.027)
(3.602)
(0.128)
(0.184)
0.356) (0.113) 0.477
4.214 (1.033)
MW ða ¼ 0, y ¼ 0) AddW
1:133 10
0.086
(l ¼ 0)
(0.036)
(0.102)
SZMW
(5:183 108 ) 0.013
8:408 109
4.224
(y ¼ 1, l ¼ 0)
(2:819 103 )
(4:204 108 )
(1.140)
8
y^
l^
0.023 (4:845 103 )
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
g1 n X x elxi ððg þ lxi Þ lnðxi Þ þ1Þ i
hðxi ; a, b, g, y, lÞ
i¼1
g n X ð1 þ g þ lxi Þx elxi i
i¼1
hðxi ; a, b, g, y, lÞ
n X g xi elxi lnðxi Þ ¼ 0,
ð15Þ
i¼1
n X gþ1 xi elxi ¼ 0:
ð16Þ
i¼1
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]). Model
Log-lik
K–S
P-value
AIC
BIC
NMW MW AddW SZMW BMW
212.90 227.16 221.51 229.88 220.80
0.088 0.129 0.127 0.151 0.127
0.803 0.346 0.365 0.185 0.365
435.8 460.3 451.0 465.8 451.6
445.4 466.0 458.7 471.5 461.2
167
The maximum likelihood estimates can be obtained by solving the non-linear equations numerically for a, b, y, g and l. 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 Kolmogorov–Smirnov (K–S) statistic, as well as Akaike information criterion (AIC) [2] and Bayesian information criterion (BIC) [23] values.
1
0.8 Scaled TTT−Transform
0.15
S(x)
0.9
Nonpar NMW MW SZMW AddW BMW
0.10
0.05
0.7 0.6 0.5 0.4 0.3 0.2
0.00
0.1 0
20
40
60
80
0
X
Empirical NMW
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
X
1.0
0.030 NMW MW SZMW AddW BMW
0.025
0.8 0.6 f(x)
h(x)
0.020
Kaplan−Meier NMW MW SZMW AddW BMW
0.015
0.4 0.010 0.2
0.005 0.000
0.0 0
20
40
60
80
0
20
40
60
80
X
x
Fig. 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.
Table 4 MLEs of parameters and corresponding standard errors in brackets for the Meeker and Escobar data. Model
a^
NMW
0.024
5:991 108
0.012
0.629
0.056
(0.019)
(1.29)
(0.15)
(0.024)
MW
(8:164 108 ) 0.018
0:454
7:133 103
(a ¼ 0, y ¼ 0)
(0.018)
(0.220)
(2:113 103 )
b^
g^
y^
AddW
1:320 107
0.019
0.604
2.830
(l ¼ 0)
(7:435 107 )
(0.018)
(0.197)
(0.974)
SZMW
2:939 103
1:497 109
3.585
(y ¼ 1, l ¼ 0)
(9:290 104 )
(1:114 108 )
(1.314)
l^
168
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
5.1. Aarset data
5.2. Meeker and Escobar 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).
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 (Fig. 4a and b). Again the NMW distribution (Table 4) provides a better fit than the BMW, as can be seen from Table 5 (Fig. 5).
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 MW AddW SZMW BMW
166.18 178.06 178.11 177.90 167.55
0.148 0.182 0.191 0.186 0.161
0.482 0.242 0.197 0.221 0.378
344.4 362.1 364.2 361.8 345.1
351.4 366.3 369.8 366.0 352.1
6. Sub-model of the NMW with c ¼ 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 g ¼ 1. We test the reduced model H0 : g ¼ 1Þ against the original model Ha : g a1Þ. For each data set, Table 6 shows ML estimates of the four parameter NMW, the loglikelihood value under H0, likelihood ratio statistic (LRT) with Pvalue 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-value¼0.252) and 2.45 (P-value¼0.118), respectively, on 1 d.f. Therefore we can choose the reduced model with four 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.
1 0.9 0.015
Nonpar NMW MW SZMW AddW BMW
Scaled TTT−Transform
h(x)
0.010
0.8
0.005
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.000
0 0
50
100
150
200
250
300
Empirical NMW
0
0.1
0.2
0.3
X
0.010
0.4
0.5 X
0.6
1.0 NMW MW SZMW AddW BMW
0.008
f(x)
0.4
0.002
0.2
0.000
0.0 50
100
150 x
200
250
300
0.9
1
0.6
0.004
0
0.8
Kaplan−Meier NMW MW SZMW AddW BMW
0.8
S(x)
0.006
0.7
0
50
100
150
200
250
300
X
Fig. 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.
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
169
Table 6 Results of fitting NMW with g ¼ 1 to both data sets. Data Aarset Meeker
a^
l^
Log-lik
AIC
LRT (P-value)
0.531
0.160
213.56
435.1
1.31
0.105
(0.104)
(0.011)
(0.252)
(0.603)
3:5 108
0.675
0.039
2.45
0.153
(4:2 108 )
(0.141)
ð4:0 103 )
(0.118)
(0.440)
b^
y^
0.092
2:2 108
(0.039)
(2:1 108 )
0.017 (0.013)
167.40
1.0
0.030 0.025
Kaplan−Meier NMW 4NMW
0.8
NMW 4NMW
S(x)
S(x)
342.8
K–S (P-value)
0.020
0.6
0.015 0.4 0.010 0.2
0.005
0.0
0.000 0
20
40
60
0
80
20
40
x
60
1.0
0.010
Kaplan−Meier NMW 4NMW
NMW 4NMW
0.008
0.8
0.006
0.6
f(x)
f(x)
80
X
0.004
0.4
0.2
0.002
0.0
0.000 0
50
100
150
200
250
0
300
50
100
150
x
200
250
300
X
Fig. 6. (a and b) Fitted pdf and survival functions for Aarst data and (c and d) those for Meeker and Escobar data, five parameters (solid lines) vs four parameters (dotted lines).
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. 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.
Acknowledgments 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.
Appendix A The log-likelihood function of the NMWða, b, y, g, lÞ can be written as LðW Þ ¼
n X
g
½lnðhðxi ; W ÞÞaxyi bxi elx ,
ð17Þ
i¼1
where hðxi ; W Þ is the hazard rate function (6) of the NMW and W ¼ ða, b, y, g, lÞ is the vector of parameters. The second partial derivatives are as follows: Laa ¼
2 n X ha ðxi ; W Þ i¼1
hðxi ; W Þ
,
170
S.J. Almalki, J. Yuan / Reliability Engineering and System Safety 111 (2013) 164–170
Lab ¼
n X ha ðxi ; W Þhb ðxi ; W Þ i¼1
i¼1
Lag ¼
Lbb ¼
ðhðxi ; W ÞÞ
ha ðxi ; W Þhl ðxi ; W Þ
i¼1
ðhðxi ; W ÞÞ2
2 n X hbg ðxi ; W Þ hðxi ; W Þ
,
,
ðhðxi ; W ÞÞ2
n X
hðxi ; W Þhbg ðxi ; W Þhb ðxi ; W Þhg ðxi ; W Þ
i¼1
ðhðxi ; W ÞÞ2
n X
hðxi ; W Þhbl ðxi ; W Þhb ðxi ; W Þhl ðxi ; f Þ
i¼1
ðhðxi ; W ÞÞ2
Lbg ¼
Lbl ¼
2 hðxi ; W Þhyy ðxi ; W Þ hy ðxi ; W Þ
i¼1
ðhðxi ; W ÞÞ2
Lyg ¼
n X hy ðxi ; W Þhg ðxi ; W Þ
ðhðxi ; W ÞÞ2
i¼1
Lyl ¼
n X hy ðxi ; W Þhl ðxi ; W Þ
ðhðxi ; W ÞÞ2
i¼1
i¼1
ðhðxi ; W ÞÞ2
g lxi
bxi e
i¼1
ðhðxi ; W ÞÞ2
d X
hðxi ; W Þhll ðxi ; W Þðhl ðxi ; W ÞÞ2
i¼1
ðhðxi ; W ÞÞ2
i
hay ðxi ; W Þ ¼ xiy1 ð1 þ y lnðxi ÞÞ, ð1 þðg þ lxi Þ lnðxi ÞÞelxi ,
g
hbl ðxi ; W Þ ¼ xi ð1 þ g þ lxi Þ lnðxi ÞÞelxi , hyy ðxi ; W Þ ¼ axiy1 ð2 þ y lnðxi ÞÞ lnðxi Þ, ð2 þ ðg þ lxi Þ lnðxi ÞÞ lnðxi Þelxi ,
g
hgl ðxi ; W Þ ¼ bxi ð1 þ ð1þ g þ lxi Þ lnðxi ÞÞelxi , gþ1
hll ðxi ; W Þ ¼ bxi
ð2þ g þ lxi Þ lnðxi ÞÞelxi ,
ha ðxi ; W Þ ¼ yxiy1 , g1
hb ðxi ; W Þ ¼ xi
ðg þ lxi Þ lnðxi Þelxi ,
hy ðxi ; W Þ ¼ ahay ðxi ; W Þ,
! 2
ln ðxi Þ ,
g þ 1 lxi e
bxi
g þ 2 lxi bx e
where
g1
2
axi ln ðxi Þ ,
hðxi ; W Þhgl ðxi ; W Þhg ðxi ; W Þhl ðxi ; W Þ
hgg ðxi ; W Þ ¼ bxi
,
! y
n X
g1
!
,
hðxi ; W Þhgg ðxi ; W Þðhg ðxi ; W ÞÞ2
hbg ðxi ; W Þ ¼ xi
g þ 1 lxi x e
,
n X
Lll ¼
!
g
xi elxi lnðxi Þ ,
i
n X
Lyy ¼
Lgl ¼
,
n X hb ðxi ; W Þhy ðxi ; W Þ i¼1
Lgg ¼
,
2
n X
i¼1
Lby ¼
References
n X ha ðxi ; W Þhg ðxi ; W Þ i¼1
Lal ¼
hl ðxi ; W Þ ¼ bhbl ðxi ; W Þ:
! hðxi ; W Þhay ðxi ; W Þha ðxi ; W Þhy ðxi ; W Þ y xi lnðxi Þ , 2 hðxi ; W Þ
n X
Lay ¼
hg ðxi ; W Þ ¼ bhbg ðxi ; W Þ,
,
ðhðxi ; W ÞÞ2
! ,
! 2
ln ðxi Þ ,
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