Exponentiated modified Weibull extension distribution

Reliability Engineering and System Safety 112 (2013) 137–144

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Exponentiated modified Weibull extension distribution Ammar M. Sarhan n,1, Joseph Apaloo Department of Mathematics, Statistics & Computer Science St. Francis Xavier University, Antigonish NS, Canada B2G 2W5

a r t i c l e i n f o

abstract

Article history: Received 17 February 2012 Received in revised form 6 October 2012 Accepted 8 October 2012 Available online 20 December 2012

A new modified Weibull extension distribution is proposed by Xie et al. [20]. Recently, El-Gohary et al. [9] proposed a new distribution referred to as the generalized Gompertz distribution. In this paper, we propose a new model of a life time distribution that mainly generalizes these two distributions. We refer to this new distribution as the exponentiated modified Weibull extension distribution. This distribution generalizes, in addition to the above two mentioned distributions, the exponentiated Weibull distribution, the generalized exponential and the generalized Rayleigh distributions. Parameter estimation of the four parameters of this distribution is studied. Two real data sets are analyzed using the new distribution, which show that the exponentiated modified Weibull extension distribution can be used quite effectively in fitting and analyzing real lifetime data. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.

Keywords: Generalized Gompertz Exponential Generalized Rayleigh Maximum likelihood method Reliability data analysis

1. Introduction Weibull distribution is one of the most commonly used lifetime distributions in reliability and lifetime data analysis. It is flexible in modeling failure time data, as the corresponding hazard rate function can be increasing, constant or decreasing. But in many applications in reliability and survival analysis, the hazard rate function can be of bathtub shape. The hazard rate function plays a central role to the work of reliability engineers, see Lai and Xie [13] and Bebbington et al. [3,4] and references therein. Models with a bathtub hazard rate function are needed in reliability analysis and decision making when the life time of the system is to be modeled. Many parametric probability distributions have been introduced to analyze sets of real data with bathtub-shaped hazard rates. The bathtub-shaped hazard function provides an appropriate conceptual model for some electronic and mechanical products as well as the lifetime of humans. Some work on parametric probability distributions with bathtub-shaped hazard rate functions have been considered by different authors. The exponential power distribution was suggested by Smith and Bain [19], and it was studied by Leemis [15]. A four parameter distribution was proposed by Gaver and Acar [10]. A similar distribution with increasing, decreasing, or bathtub-shaped hazard rate has been considered by Hjorth [12]. An exponentiated Weibull distribution with three parameters was suggested by Mudholkar and Srivastava [17].

n

Corresponding author. Tel.: þ1 902 266 6423; fax: þ1 902 4945130. E-mail addresses: [email protected], [email protected] (A.M. Sarhan), [email protected] (J. Apaloo). 1 Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

Chen [8] discussed an interesting two-parameter model that can be used to model bathtub-shaped hazard rate function. Though this distribution has only two parameters, it shows a bathtub shaped hazard rate. However, it is not flexible because it does not include a scale parameter. Xie et al. [20] proposed a new modified extension of the Weibull distribution with a bathtubshaped hazard rate function. We refer to this extension as the new modified Weibull extension (MWE) distribution. The MWE generalizes the two-parameter model discussed by Chen [8] and it includes one scale parameter and two shape parameters. The cumulative distribution function of the MWE distribution [20] is b

F MWE ðx; l, a, bÞ ¼ 1expfla½1eðx=aÞ g,

x Z 0, l, a, b 4 0:

ð1Þ

Setting a ¼ 1 in (1), we get the cdf of the two-parameter distribution discussed by Chen [8] as a sub-model of the MWE distribution. Since 1995, exponentiated distributions have been widely studied in statistics and numerous authors have developed various classes of these distributions. A good review of some of these models is presented by Pham and Lai [18]. Mudholkar and Srivastava [17] proposed the exponentiated Weibull distribution (EW or EW ðs, b, gÞ), with the following cdf: b g x , x Z 0, s, b, g 4 0: ð2Þ F EW ðx; s, b, gÞ ¼ 1exp

s

El-Gohary et al. [9] proposed the exponentiated Gopertz distribution, and referred to it as the generalized Gompertz (GG) distribution, whose cdf is of the form g l ð1ecx Þ F GG ðx; l,c, gÞ ¼ 1exp , l, c, g 4 0: ð3Þ c

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138

A.M. Sarhan, J. Apaloo / Reliability Engineering and System Safety 112 (2013) 137–144

Though some distributions such as the exponentiated Weibull distribution, modified Weibull extension and generalized Gompertz distribution are known to have bathtub-shaped hazard rate, they may not be able to give a good bathtub shape of the hazard rate (see [21]). However, there are fewer models whose hazard rate curves are similar to the actual bathtub shape. Zhang et al. [21] discussed the parametric analysis of some models which exhibit a good bathtub shaped hazard rate. In this paper, we propose a new distribution which generalizes the above three distributions, with the hope that it will attract many applications in different fields such as reliability, lifetime data analysis, and others. This new distribution exhibits a good bathtub shaped hazard rate which is very similar to the actual bathtub curve. Mainly, we consider our new distribution as a generalization of the MWE distribution. On the other hand, it can be considered as a generalization of the GG distribution as well as the EW distribution. We will refer to this new distribution as the exponentiated modified Weibull extension (EMWE) distribution. The EMWE distribution contains as special sub-models, in addition to the above three mentioned distributions, many distributions such as exponential, generalized exponential, Weibull, Rayleigh, generalized Rayleigh, among others. Due to the flexibility of the EMWE in accommodating different forms of the hazard rate functions, especially the ones which have wide flat portions, it seems to be a suitable distribution that can be used in a variety of problems for fitting reliability data. The EMWE distribution is not only convenient for fitting bathtub-shaped hazard rate data but it is also suitable for testing goodness-of-fit of some special sub-models such as the MWE, EW, and GG distributions. The rest of the paper is organized as follows. In Section 2, we introduce the EMWE distribution, discuss some special submodels and provide its cumulative distribution function (cdf), the probability density function (pdf) and the hazard function. A formula for generating random samples from the EMWE distribution is also given in Section 2. Section 3 discusses some important statistical properties of the EMWE distribution such as the ordinary moments and measures of skewness and kurtosis. Section 4 discusses the parameter estimation process using maximum likelihood estimates. Two applications to real data are provided in Section 5. Section 6 concludes the paper. The paper also contains an Appendix giving technical details.

is very similar to the actual bathtub shaped curve. The EMWE distribution has two scale (a, l) and two shape (b, g) parameters and generalizes several well known distributions. The following is a list of well known sub-models of the EMWE distribution. When the scale parameter a becomes very large or tends to infinity while ab1 =l remains constant, the EMWE distribution reduces to the exponentiated Weibull with scale parameter ab1 =l and shape parameters b and s ¼ ab1 =l, say EWðs, b, gÞ. When b ¼ 1 and a becomes very large or tends to infinity while ab1 =l remains constant, the EMWE distribution reduces to the generalized exponential distribution with a scale parameter 1=l and shape parameters g, say GE ð1=l, gÞ, see Gupta and Kundu [11]. When b ¼ 1, the EMWE distribution reduces to the generalized Gompertz distribution with scale parameters l and c ¼ 1=a and shape parameter g, say GG ðl,c, gÞ, see El-Gohary et al. [9]. When g ¼ 1 and a ¼ 1, the EMWE reduces to the distribution in Chen [8] with shape parameters l and b. The pdf of the EMWE distribution (5) can be written as a linear combination of the pdf of MWE distribution. For g 40, a series expansion for ð1wÞg1 , for 9w9o 1, is 1 X ð1Þj GðgÞ

ð1wÞg1 ¼

j¼0

GðgjÞj!

wj ,

ð7Þ

where Gð:Þ is the gamma function. Since for x 4 0, b SMWE ðxÞ ¼ SMWE ðx; l, a, bÞ ¼ expfla½1eðx=aÞ go 1, then using the series expansion (7) in (5), we obtain f ðx; yÞ ¼

1 X ð1Þj Gðg þ 1Þ j¼0

GðgjÞðj þ 1Þ!

f MWE ðx; ðj þ 1Þl, a, bÞ:

ð8Þ

When g is a positive integer, the index j in (8) stops at g1. The linear combination (8) enables us to obtain some mathematical properties of EMWE directly from those of the MWE distribution such as, the moments, the moment generating function, characteristic function. There are many softwares such as MATLAB, MAPLE and MATHEMATICA that can be used to compute (8) numerically. Advantage 1: One of the advantages of the EMWE distribution is that it has a closed form cdf, which can be used to generate random numbers from it by using the following simple formula: a=b 1 X ¼ a ln 1 lnð1U 1=g Þ , ð9Þ

al

2. The EMWE distribution The cdf of the exponentiated modified Weibull extension distribution with four parameters y ¼ ða, b, l, gÞ, abbreviated as EMWE distribution, is ðx=aÞb Þ

Fðx; yÞ ¼ ½1elað1e

g ,

l, a, b, g 40, x Z0:

ð4Þ

The probability density function of the EMWEðyÞ distribution is x b1 b ðx=aÞb ðx=aÞb g1 Þ Þ eðx=aÞ þ lað1e ½1elað1e , f ðx; yÞ ¼ lbg

a

l, a, b, g 4 0, x Z 0:

where U is a uniformly distributed random variable on (0, 1) interval. The formula (9) can be used to generate random samples from a wide set of sub-models of the EMWE distribution such as the exponential, generalized exponential, Rayleigh, generalized Rayleigh, Weibull, modified Weibull extension, Exponentiated Weibull, Gompertz and generalized Gompertz distributions. Interpretation: When g is a positive integer, the EMWEðyÞ distribution can be interpreted as the lifetime distribution of a parallel system consisting of g independent and identical units whose lifetime follows the MWE ðl, a, bÞ distribution.

ð5Þ

The hazard rate function of the EMWEðyÞ distribution is x b1 b ðx=aÞb Þ lbg eðx=aÞ þ lað1e a , l, a, b, g 40, x Z0: hðx; yÞ ¼ b 1g b ðx= a Þ ðx= a Þ Þ Þ 1 ½1elað1e þelað1e ð6Þ One can see from Fig. 1 that the hazard function: (1) takes a bathtub shape if either g o1 whatever the value of b or b o1 whatever the value of g and (2) is increasing if b Z 1 and g Z 1. The bathtub-shaped curve of Fig. 1(b) has quite a long flat part which

3. The moments, skewness and kurtosis The k-th ordinary moment of the EMWE distribution can be written as linear combination of those for the MWE distribution. Let mk ðyÞ and mk,MWE ðl, a, bÞ be the k-th moments of the EMWE and MWE distributions, respectively, then

mk ðyÞ ¼

1 X ð1Þj Gðg þ1Þ j¼0

GðgjÞðj þ1Þ!

mk,MWE ððj þ 1Þl, a, bÞ:

ð10Þ


0.14 θ = (49.046, 3.148, 7.181× 10−5, 0.1) θ = (49.046, 3.148, 7.181× 10−5, 1) θ = (49.046, 3.148, 7.181× 10−5, 1.5)

0.09 0.08

θ = (49.046, 3.148, 7.181× 10−5, 0.1) −5 θ = (49.046, 3.148, 7.181× 10 , 1) θ = (49.046, 3 .148, 7.181× 10−5, 1.5)

0.12

The hazard function

The probability density function

0.1

139

0.07 0.06 0.05 0.04 0.03

0.1 0.08 0.06 0.04

0.02 0.02 0.01 0

0

20

40

60

80

0

100

0

10

20

30

40

x

60

70

80

140

160

0.2

0.03

−3 θ = (49.046, 1, 7.181× 10 , 0.1) θ = (49.046, 1, 7.181× 10−3, 1) θ = (49.046, 1, 7.181× 10−3, 1.5)

0.025

θ = (49.046, 1, 7.181× 10−3, 0.1) −3 θ = (49.046, 1, 7.181× 10 , 1) θ = (49.046, 1, 7.181× 10−3, 1.5)

0.18 0.16

The hazard function


50

x

0.02

0.015

0.01

0.14 0.12 0.1 0.08 0.06 0.04

0.005

0.02 0

0 0

20

40

60

80

100

120

140

160

0

20

40

60

80

x 0.035 θ = (49.046, 0.4, 7.181× 10−3, 0.1) θ = (49.046, 0.4, 7.181× 10−3, 1) θ = (49.046, 0.4, 7.181× 10−3, 1.5)

5 4 3 2

0.025

0.02

0.015

0.01

1 0

120

θ = (49.046, 0.4, 7.181× 10−3, 0.1) θ = (49.046, 0.4, 7.181× 10−3, 1) θ = (49.046, 0.4, 7.181× 10−3, 1.5)

0.03

The hazard function


x 10−3

6

100

x

0.005 0

100

200

300

400

500

600

700

0

100

200

300

x

400

500

600

700

x

Fig. 1. The probability density function and hazard function of the EMWE distribution at different values of the vector of unknown parameters y.

Based on the first four ordinary moments of the EMWE distribution, the measures of skewness skðyÞ and kurtosis kðyÞ of the EMWE distribution can obtained as skðyÞ ¼

m3 ðyÞ3m1 ðyÞm2 ðyÞ þ 2m31 ðyÞ ½m2 ðyÞm21 ðyÞ3=2

ð11Þ

m4 ðyÞ4m1 ðyÞm3 ðyÞ þ 6m21 ðyÞm2 ðyÞ3m41 ðyÞ : ½m2 ðyÞm21 ðyÞ2

ð12Þ

and

kðyÞ ¼

Plots of the skewness and kurtosis of the EMWE distribution as a function of g for selected values of b and a ¼ 49, l ¼ 7 105 are

given in Fig. 2. It is observed that: (i) for b 4 1, both skðyÞ and kðyÞ are convex functions for g A ð0,1Þ, while for g 4 1 both skðyÞ and kðyÞ are increasing exponentially, (ii) for b r1, skðyÞ first decreases as g increases and then start to slightly increase, (iii) for b o 1, kðyÞ first decreases as g increases and then starts to increase slightly, and (iv) for b ¼ 1, kðyÞ is a concave function for g A ð0,1, while for g 41, kðyÞ decreases slightly as g increases.

4. Parameter estimation Now, we discuss the estimation of the model parameters by using the method of maximum likelihood. Let x ¼ ðx1 , . . . ,xn Þ be a

140


35

20 α =49,β = 3.0,λ = 7× 10−5

18

α =49,β = 1.0,λ = 7× 10

α = 49,β = 3.0,λ = 7× 10−5

−5

α = 49, β = 1.0, λ = 7× 10−5

30

α =49,β = 0.4,λ = 7× 10−5

16

α = 49,β = 0.4, λ = 7× 10−5

14

25

Kurtosis

Skewness

12 10 8

20

15

6 10

4 2

5 0 0

−2 0

2

1

3

4

γ

5

6

1

0

2

3

4

γ

5

6

Fig. 2. Plots of the skewness and kurtosis of the EMWE distribution as a function of g when a ¼ 49, l ¼ 7 105 and for some values of b. Table 1 Lifetimes of 50 devices, Aarset [1]. 0.1 21 79

0.2 32 82

1 36 82

1 40 83

1 45 84

1 46 84

1 47 84

2 50 85

3 55 85

random sample of the EMWE distribution with unknown parameter vector y ¼ ða, b, l, gÞ. The log-likelihood function LðyÞ for y is L ¼ n½al þ ð1bÞ ln a þ ln b þ ln l þ ln gal

n X

b

eðxi =aÞ þ

i¼1

þðb1Þ

n X

ln xi þ ðg1Þ

i¼1

n X

n 1 X

ab i ¼ 1

b

½1expflað1eðxi =aÞ Þg:

xbi

ð13Þ

i¼1

The first partial derivatives of the log-likelihood function with respect to the four parameters are n n n X X @L 1b b X ðg1Þwi, a , ¼ n lþ ½g i þ ag i, a b þ 1 xbi l @a a ð1wi Þ a i¼1 i¼1 i¼1 ð14Þ n n b x X X @L 1 xi ¼ n ln a al g i, b þ ln i þln xi @b b a a i¼1 i¼1

n X ðg1Þwi, b i¼1

ð1wi Þ

,

6 60 85

7 63 85

11 63 85

12 67 86

18 67 86

18 67

18 67

18 72

18 75

For asymptotic interval estimation of the four parameters a, b, l and g, we obtain the observed Fisher information matrix. The elements of the 4 4 observed information matrix IðyÞ ¼ T @2 L=@y @y , are given in Appendix B. The multivariate normal N 4 ð0,Iðy^ Þ1 Þ distribution can be used to construct asymptotic confidence intervals for the parameters. The asymptotic 100ð1WÞ% confidence intervals of a, b, l and g are ða^ 7 Z W=2 SEða^ ÞÞ, ðb^ 7 Z W=2 SEðb^ ÞÞ, ðl^ 7 Z W=2 SEðl^ ÞÞ, and ðg^ 7 Z W=2 SEðg^ ÞÞ, respectively, where Z a=2 is the quantile ð1W=2Þ of the standard normal distribution and SEðÞ is the square root of the diagonal element of Iðy^ Þ1 corresponding to each parameter. Different types of goodness-of-fit can be applied here to test the superiority of the EMWE distribution in comparison to some other models. Mainly in Section 5, we use Kolmogorov–Smirnov (K–S) test as a non-parametric test and the likelihood ratio (LR) test as a parametric one to illustrate how one can compare the EMWE distribution with the MWE, GG and EW distributions to fit real data sets.

ð15Þ 5. Applications

n n X X @L 1 ðg1Þwi, l , ¼ n þ a a gi @l l ð1wi Þ i¼1 i¼1

ð16Þ

n @L n X ¼ þ lnð1wi Þ, @g g i¼1

ð17Þ b

where for i ¼ 1,2, . . . ,n, g i ða, bÞ ¼ eðxi =aÞ , wi ða, b, lÞ ¼ ela½1gi ða, bÞ , wi, a ¼ lð1g i ag i, a Þelað1g i Þ , wi, b ¼ lag i, b elað1gi Þ , wi, l ¼ að1g i Þ b

b

elað1gi Þ , g i, a ¼ ðb=aÞðxi =aÞb eðxi =aÞ , and g i, b ¼ ðxi =aÞb eðxi =aÞ lnðxi =aÞ. Setting these expressions to zero, we get a system of nonlinear equations, and solving them simultaneously gives the maximum likelihood estimate (MLE) y^ ¼ ða^ , b^ , l^ , g^ Þ of the four parameters. The system of these four non-linear equations cannot be solved analytically and mathematical or statistical software should apply to get a numerical solution via iterative techniques such as the Newton-Raphson method, Burden and Faires [7].

In this section, we analyze two real data sets to demonstrate the performance of the EMWE distribution in practice. One of the data sets is a sample of 50 components taken from Aarset [1]. The other data set is a sample of 30 devices failure and running times [16]. The two data sets possess bathtub shaped hazard rates. For the two data sets, we compare the results of the fits of the EMWE, MWE, EW and GG distributions. The main reasons to compare our new model with these three models are: (1) they are sub-models of the EMWE distribution and (2) the hazard function of all of these models can take a bathtub shape. In many applications, the total time on test transform (TTT-Transform) plot can be used to get information about the shape of the hazard rate of a given data set, which helps in selecting a particular model to fit a given data set. For details about the TTT-Transform technique, we refer the reader to Barlow and Campo [2] and Bergman and Klefsjo [6].


We perform the test of the following three null hypotheses: (i) H0 : a is large and s ¼ ðab1 =lÞ1=b is finite, the data follow the EW ðs, b, gÞ distribution, (ii) H0 : g ¼ 1, the data follow the MWE ða, l, bÞ distribution, and (iii) H0 : b ¼ 1, the data follow the GG ðc ¼ 1=a, l, gÞ distribution, in favor of the alternative hypothesis Ha : the data follow EMWEða, l, b, gÞ. We used parametric and non-parametric test statistics to test the above null hypotheses against Ha. We use Kolmogorov– Smirnov (K–S) and likelihood ratio (LR) test statistics. The K–S Table 2 The MLE of the parameter(s), K–S values and the associated p-values. MLE of the parameters

K–S

p-Value

EW

0.1841

0.0590

MWE

0.1611

0.1335

Scaled TTT−Transform

s^ ¼ 91:023, b^ ¼ 4:69, g^ ¼ 0:146 a^ ¼ 101:0909, l^ ¼ 0:0141, b^ ¼ 0:8408 GG c^ ¼ 0:044, l^ ¼ 1:43 103 , g^ ¼ 0:421 EMWE a^ ¼ 49:05, l^ ¼ 7:181 105 , b^ ¼ 3:148, g^ ¼ 0:145

0.1203

0.4307

0.1013

0.6457

test rejects the null hypothesis at level W if Dn 4 dn, W , where Dn is the observed value of the K–S test statistics.

5.1. Aarset data set The data set refers to the lifetimes of 50 devices provided by Aarset (1987). Table 1 gives the measurements of the data set. Table 2 shows the MLE of the parameters of every distribution used and the observed K–S test statistic values for the four models. Fig. 3 shows the plots of the empirical and fitted scaled TTT-Transforms, the empirical and parametric survival functions, the empirical and fitted hazard and probability density functions for the Aarset data set. We can observe from Fig. 3(a) and (c) immediately that: (1) the data set has a bathtub shaped hazard rate, (2) all distributions used to fit the data set have a bathtub shaped hazard rate function, and (3) the EMWE distribution fits the data set better than all other distributions used here, because its fitted curve is closer to the empirical curve.

1

1

0.9

0.9

0.8

0.8

0.7

0.7

Survival function

The model

0.6 0.5 0.4 0.3

Empirical EMEW GG EW MEW

0.2 0.1 0

0

0.2

0.4

0.6

0.8

Empirical EMWE MWE EW GG

0.6 0.5 0.4 0.3 0.2 0.1 0

1

0

10

20

30

40

u

60

70

80

0.03


The hazard function

0.08

50

x


0.1

141

0.06

0.04

0.02


0.025

0.02

0.015

0.01

0.005

0

0 0

10

20

30

40

50

60

70

0

80

10

20

30

40

50

60

70

80

x

x

Fig. 3. (a) The empirical and estimated scaled TTT-Transform plots of the EMWE, MWE, EW and GG models for the Aarset data. (b) The empirical and estimated survival functions of the EMWE, MWE, EW and GG models for the Aarset data. (c) Empirical and estimated hazard rate functions of the EMWE, MWE, EW and GG models for the Aarset data. (d) The empirical and estimated pdfs of the EMWE, MWE, EW and GG models for the Aarset data.

Table 3 The studied H0’s, log-likelihood function values, LH0 , df and p-values. The model

H0

GG ðc, l, gÞ

b¼1

EW ðs, b, gÞ MWE ða, l, bÞ

a is large and g¼1

s ¼ ðab1 =lÞ1=b

is finite

LH0

LH 0

df

p-Value

224.127

20.540

1

5:841 106

229.114

30.513

1

3:317 108

236.247

44.778

1

2:207 1011

142


Table 4 Meeker and Escobar data, Meeker and Escobar [16]. 2 212

10 245

13 247

23 261

23 266

28 275

30 293

65 300

80 300

88 300

106 300

143 300

147 300

173 300

181 300

Table 5 The MLE of the parameters, K–S values and p-values for data (2). The model

MLE of the parameters

K–S

p-Value

EW MWE

a^ ¼ 316:633, s^ ¼ 5:819, g^ ¼ 0:158 a^ ¼ 61:445, l^ ¼ 1:533 103 , b^ ¼ 0:715 c^ ¼ 0:017, l^ ¼ 1:234 104 , g^ ¼ 0:423 a^ ¼ 197:21, l^ ¼ 5:468 106 , b^ ¼ 4:482, g^ ¼ 0:129

0.2096 0.1590

0.1236 0.3928

GG

1

1

0.9

0.9

0.8

0.8

0.7

0.7

Survival function

Scaled TTT−Transform

EMWE

0.6 0.5 0.4 Empirical EMEW GG EW MEW

0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

0.1433

0.5226

0.1313

0.6319


0.6 0.5 0.4 0.3 0.2 0.1 0

1

0

50

100

150

200

250

300

x

u −3

x 10

0.03

0.02


0.025

The hazard function

8


0.015

0.01

0.005

0

0

50

100

150

200

250

300

Empirical ENMWE NMWE EW GG

7 6 5 4 3 2 1 0

0

50

100

x

150

200

250

300

x

Fig. 4. The empirical and estimated scaled TTT-Transform plots of the EMWE, MWE, EW and GG models for Meeker and Escobar data. (b) The empirical and estimated survival functions of the EMWE, MWE, EW and GG models for Meeker and Escobar data. (c) Empirical and estimated hazard rate functions of the EMWE, MWE, EW and GG models for Meeker and Escobar data. (d) The empirical and estimated pdfs of the EMWE, MWE, EW and GG models for Meeker and Escobar data.

We can conclude from Table 2 that: (i) none of H0’s is rejected at W r0:05, (ii) only the EW distribution is rejected at W Z 0:06, (iii) both the EW and NMW distributions are rejected at W Z0:14, (iv) both the GG and EMWE distributions are not rejected at W r 0:43, and (v) the EMWE is the best model, among those discussed here, to fit the current data set because it has the biggest p-value. As concluded above, none of the four distributions is rejected at a small level of significance W r0:05. Therefore, we also performed parametric comparisons between the EMWE distribution and the EW, MWE and GG distributions based on the parametric likelihood ratio test statistics, LH0 ¼ 2ðLHa LH0 Þ, to test H0 against Ha. The value of log-likelihood function under Ha is

Table 6 The studied H0’s, log-likelihood function values, LH0 , df and p-values. The model

H0

LH 0

GG ðc, l, gÞ

b¼1

175.507 18.306 1

1:881 105

EW(s, b, g)

a is large and

177.660 22.612 1

1:982 106

179.270 44.778 1

3:724 107

s ¼ ðab1 =lÞ1=b MWE ða, l, bÞ g ¼ 1

LH 0

df p-Value

is finite

obtained as LHa ¼ 213:858. Table 3 gives the null hypothesis H0, the value of log-likelihood function under H0, LH0 , the value of the likelihood ratio test statistics, LH0 , the degree of freedom of LH0 ,


143

Table 7 Results for the model discussed in Bebbington et al. [5]. Data

MLE of y ¼ ða,b, lÞ

L

AIC

K–S

p-Value

Aarset Meeker and Escobar

(0.0624, 0.3548, 0.0233) (0.018, 0.4536, 0.007133)

227.155 178.064

460.310 362.128

0.1275 0.1477

0.3599 0.4843

df, and the corresponding p-value. From the p-values it is clear that we reject all the three null hypotheses against the EMWE distribution at level of significance W Z5:841 106 . This means that the EMWE distribution is superior in fitting the current data set when compared with the EW, MWE or GG distribution. 5.2. Meeker and Escobar data In this section we analyze a real data set of 30 devices failure and running times [16]. Table 4 gives the measurements of the data set. Table 5 gives the MLE of the parameters of all models used here and the corresponding values of K–S test statistics and p-values. From Table 5, we immediately can conclude that: (1) none of the four models is rejected at W r0:10, (2) only the EW distribution is rejected at W Z0:13, and (3) the EMWE distribution is the best model, among those applied here, to fit the current data set in the sense of having the biggest p-value. Fig. 4 shows the plots of the empirical and fitted scaled TTTTransforms, the empirical and parametric survival functions, the empirical and fitted hazard and probability density functions for the Meeker and Escobar data set. The empirical scaled TTT-Transform plot shows a bathtubshaped hazard rate of the Meeker and Escobar data, Fig. 4(a). The change point of the hazard rate occurs at very early age. The EMWE and GG models are the fitted models which show this property, Fig. 4(a). Fig. 4(c) supports this result. Also, from Fig. 4, one can see the closeness of the fitted pdf using the EMWE model to the empirical pdf. Furthermore, the fitted survival function using the EMWE model is the closest curve to the empirical survival function, Fig. 4(b). This indicates both EMWE and GG distributions are more appropriate than MWE and EW distributions in fitting the Meeker and Escobar data. As in the Aarset data set, none of the four distributions is rejected at a small level of significance W r 0:10. Therefore, we also used the parametric likelihood ratio test statistics to perform parametric comparisons between the EMWE distribution and the EW, MWE and GG distributions. The value of log-likelihood function under Ha is obtained as LHa ¼ 166:354. Table 6 shows the null hypothesis H0, the value of log-likelihood function under H0, LH0 , the value of the likelihood ratio test statistics, LH0 , the degree of freedom of LH0 , df, and the corresponding p-value. From the p-values it is clear that we reject all the three null hypotheses against the EMWE distribution at level of significance W Z1:881 105 . This means that the EMWE distribution is superior, among those models used here, in fitting the Meeker and Escobar data set when compared with the EW, MWE or GG distribution. 5.3. Comparison with a non-sub-model In the previous two sections, we analyzed two data sets using the model introduced in this paper and we compared it with some of its three competing sub-models. In this section, we compare our model with a competing model which is not a sub-model. Bebbington et al. [5], used a modified Weibull distribution, proposed by Lai et al. [14], to analyze Aarest data. For a comparison purpose, we use this distribution as a non-submodel of our model to fit the two data sets which were discussed

in Sections 5.1 and 5.2. To compare this non-sub-model with our model, we calculated: (1) the values of the Akaike information criterion (AIC) for the fitted models and (2) the values of the Kolmogorov–Smirnov test statistic (K–S) and the corresponding p-value. Table 7 shows the MLE of the three parameters of that non-sub-model, the corresponding maximum log-likelihood function L, AIC, K–S and the corresponding p-value for both two data sets. It is worthwhile to mention that, for the Aarest data, based on our computations we obtained the MLE of the non-submodel’s parameters exactly as those provided in Bebbington et al. [5]. The values of the AIC for our model are 435.716 and 340.708 for Aarset and Meeker and Escobar, respectively. The p-values corresponding to the K–S, using our model, are 0.6457 and 0.6319 for Aarset, and Meeker and Escobar, respectively. From these results, we can conclude that our model fits the two data sets better than that non-sub-model used in Bebbington et al. [5].

6. Conclusion We have introduced a four parameter distribution, the so-called exponentiated modified Weibull extension distribution, as a simple extension of either the generalized Gompertz distribution [9], or the modified Weibull extension distribution [20] or the exponentiated Weibull distribution [17]. We discussed some statistical properties of the distribution, including moments, measures of skewness and kurtosis, probability density of the order statistics and their moments. The maximum likelihood estimates of the four parameters of the new distribution are discussed and we provided the observed Fisher information matrix. Two real data sets are analyzed using the new distribution and it is compared with three immediate sub-models mentioned above. Also, our model is compared with a non-sub-model. The results of the comparisons showed that the exponentiated modified Weibull extension distribution provides a better fit than those three mentioned distributions and the non-sub-model to the two data sets. We hope our new distribution might attract wider sets of applications in lifetime data and reliability analysis.

Acknowledgments The authors are grateful to two anonymous referees for their valuable comments.

Appendix A The kth ordinary moment of the EMWE distribution is Z 1 mk ðyÞ ¼ xk f ðx; yÞ dx: 0

Substituting (5) into (A.1), we get Z lbg 1 k þ b1 ðx=aÞb þ lað1eðx=aÞb Þ ðx=aÞb g1 Þ mk ðyÞ ¼ b1 x e ½1elað1e dx:

a

0

ðA:1Þ

144


The above integral has no analytical solution, therefore numerical methods can be used to calculate the k-th moment mk ðyÞ at a specific set of the parameters y. Alternatively, the k-th ordinary moment mk ðyÞ can be written as a linear combination of those for the modified Weibull extension (MWE) distribution mk,MWE ðl, a, bÞ. To do so, substituting (8) into (A.1), we write Z 1 1 X ð1Þj Gðg þ 1Þ mk ðyÞ ¼ xk f MWE ðx; ðj þ 1Þl, a, bÞ dx: GðgjÞðj þ 1Þ! 0 j¼0 R1 Using mk,MWE ðl, a, bÞ ¼ 0 xk f MWE ðx; ðj þ 1Þl, a, bÞ dx, mk ðyÞ can be written as given in (10).

The observed Fisher information matrix is 0 1 Laa Lab Lal Lag BL C B ba Lbb Lbl Lbg C C Iðy^ Þ ¼ B , B Lla Llb Lll Llg C @ A Lga Lgb Lgl Lgg ^ y¼y

where Lyi yj ¼ @2 L=@yi @yj , for i,j ¼ 1,2,3,4. The elements of this matrix are listed below n n b X @2 L nð1bÞ bðb þ 1Þ X xi ¼ l ½2g i, a ag i, aa þ 2 a @a2 a2 a i¼1 i¼1 2 n ð1w Þw X i i, aa þ wi, a

ð1wi Þ2

i¼1

,

n n b X X @2 L n 1 xi ¼ 2 l ½g i, b ag i, ab þ ð1b ln aÞ @a@b a a a i¼1 i¼1 n b n X X ð1wi Þwi, ab þwi, a wi, b b xi ln xi ðg1Þ , ai¼1 a ð1wi Þ2 i¼1 n n X X ð1wi Þwi, al þwi, a wi, l @2 L ¼ n ½g i þ ag i, a ðg1Þ , @a@l ð1wi Þ2 i¼1 i¼1

@2 L @b

2

¼

n

b

al 2

ðg1Þ

n X

i¼1 n X

g i, bb þ

ln

i¼1

n b X xi i¼1

a

ln

ð1wi Þwi, bb þ w2i, b ð1wi Þ2

a

2

xi

,

n n X X ð1wi Þwi, bl þ wi, b wi, l @2 L ¼ a g i, b ðg1Þ , @b@l ð1wi Þ2 i¼1 i¼1

@2 L @l 2

2

¼

n

l

2

ðg1Þ

@ L n ¼ 2, @g2 g

waa ¼ l½2g i, a ag i, aa þ lð1g i g i, a Þ2 elað1g i Þ ,

ðg i, b þ ag i, ab þ lag i, b Þð1g i ag i, a Þ elað1gi Þ ,

wab ¼ l

wal ¼ ð1g i ag i, a Þð1þ

lað1g i ÞÞ elað1gi Þ , wbb ¼ la½g i, bb laðg i, b Þ2 elað1gi Þ , wbl ¼ a ½g i þ lað1g i Þg i, b elað1gi Þ , wll ¼ a2 ð1g i Þ2 elað1gi Þ , g i, aa ¼ ðb=a2 Þ b

ðxi =aÞb ½1 þ b þ bðxi =aÞb eðxi =aÞ , ðxi =aÞb

b

aÞ þ bðxi =aÞ ln ðxi =aÞ e

,

g i, ab ¼ ð1=aÞðxi =aÞb ½1 þ b ln

ðxi =

2

½1þ

and

b

g i, bb ¼ ðxi =aÞ ðlnðxi =aÞÞ

b

ðxi =aÞb eðxi =aÞ .

References

Appendix B

ðg1Þ

where

2 n ð1w Þw X i i, ll þ wi, l

ð1wi Þ

i¼1 2

n X

2

@ L wi , ¼ @a@g ð1wi Þ i¼1

,

n X wi, l @2 L , ¼ @l@g ð1w iÞ i¼1

n X wi, b @2 L , ¼ @b@g ð1wi Þ i¼1

[1] Aarset MV. How to identify bathtub hazard rate. IEEE Transactions on Reliability 1987;36(1):106–8. [2] Barlow RE, Campo RA, Total time on test processes and application to failure data analysis. Research Report 1975; No. ORC 75-8, University of California. [3] Bebbington M, Lai CD, Zitikis R. Modeling human mortality using mixtures of bathtub shaped failure distributions. Journal of Theoretical Biology 2007;245: 528–38. [4] Bebbington M, Lai CD, Zitikis R. Bathtub-type curves in reliability and beyond. Australian and New Zealand Journal of Statistics 2007;49:251–65. [5] Bebbingtona M, Lai CD, Zitikis R. Estimating the turning point of a bathtubshaped failure distribution. Journal of Statistical Planning and Inference 2008;138:1157–66. [6] Bergman BO, Klefsjo B. The total time on test concept and its use in reliability theory. Operations Research 1984;32(3):596–606. [7] Burden L, Faires JD, Numerical analysis, 9th ed. Brooks/Cole, Cengage Learing; 2011. [8] Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics and Probability Letters 2000;49: 155–61. [9] El-Gohary A, Alshamrani A, Al-Otaibi A. The Generalized Gompertz Distribution. Applied Mathematical Modeling 2013;37:13–24. [10] Gaver DP, Acar M, Analytical hazard representations for use in reliability, mortality, and simulation studies. Communications in Statistics 1979; B(8):91–111. [11] Gupta RD, Kundu D. Generalized exponential distribution. Australian and New Zealand Journal of Statistics 1999;41(2):173–88. [12] Hjorth U. A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates. Technometrics 1980;22(1):99–107. [13] Lai CD, Xie M. Stochastic ageing and dependence for reliability. New York: Springer; 2006. [14] Lai CD, Xie M, Murthy DNP. A modified Weibull distributions. IEEE Transactions on Reliability 2003;52(1):33–7. [15] Leemis LM. Lifetime distribution identities. IEEE Transactions on Reliability 1986;35:170–4. [16] Meeker WQ, Escobar LA. Statistical methods for reliability data. New York: John Wiley; 1998. [17] Mudholkar GS, Srivastava DK. Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability 1993;42(2):299–302. [18] Pham H, Lai CD. On recent generalizations of the Weibull distribution. IEEE Transactions on Reliability 2007;56:454–8. [19] Smith RM, Bain LJ. An exponential power life-testing distribution. Communications in Statistics 1975;4:469–81. [20] Xie M, Tang Y, Goh TN. A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety 2002;76:279–85. [21] Zhang T, Xie M, Tang LC, Ng SH. Reliability and modeling of systems integrated with firmware and hardware. International Journal of Reliability, Quality and Safety Engineering 2005;12(3):227–39.