- Email: [email protected]
On the moments of the modified Weibull distribution
Reliability Engineering and System Safety 90 (2005) 114–117 www.elsevier.com/locate/ress
Short communication
On the moments of the modified Weibull distribution Saralees Nadarajah Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA Received 3 April 2004; accepted 2 September 2004 Available online 11 November 2004
Abstract In a recent paper, Xie et al. [Reliab Eng Syst Saf 2002;76:279–85] proposed a modification of the Weibull distribution to allow it to exhibit bathtub-shaped failure rate functions. In this note, we derive explicit algebraic formulas for the kth moment of the distribution. The formulas allow one to calculate the moments for a wide range of values of k and the shape parameter of the modified Weibull. They also allow one to calculate the moments more easily with less computational time. Besides being of interest in their own, we believe that these formulas will be useful for any follow-up mathematical study of the proposed distribution. q 2004 Elsevier Ltd. All rights reserved. Keywords: Weibull distribution; MAPLE; Gamma function
1. Introduction The Weibull distribution is commonly used to model lifetimes of systems. However, it does not exhibit a bathtubshaped failure rate function and thus it cannot be used to model the complete lifetime of a system. In a recent paper Xie et al. [1] proposed a modification of the Weibull distribution which exhibits bathtub-shaped failure rate functions. The survivor function of this distribution is given by x b RðxÞ Z exp la 1 K exp (1) a for xO0, lO0, aO0 and bO0. The corresponding pdf (probability density function) is x bK1 x b x b f ðxÞ Z lb exp C la 1 K exp : (2) a a a The authors derived parameter estimation methods for this distribution and illustrated its applicability by means of two examples. However, the mathematical properties of the distribution were not studied in detail. In particular, only integral representations were provided for the mean and variance associated with (1). E-mail address: [email protected]. 0951-8320/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2004.09.002
In this note, we derive explicit algebraic forms for the kth moment associated with (1), which can be written as ðN ðN x b kK1 kK1 exp la 1 K exp dx: k x RðxÞ dx Z k x a 0 0 (3) We show that (3) can be reduced to explicit forms— involving only standard special functions—for a wide range of values of k and b (Sections 2 and 3). These formulas, apart from being of interest in their own, will be useful for any follow-up mathematical study of the proposed distribution. They could also be used to achieve substantial gain in computational time as shown in Section 2. Another modification of the Weibull distribution introduced by Mudholkar et al. [2] is the exponentiated Weibull distribution given by the survivor function RðxÞ Z 1 K ½1 K expfKðlxÞa ga for xO0, aO0, aO0 and lO0. The moments of this distribution have also been of some interest. Mudholkar and Hutson [3] provided a non-closed form integral representation for the moments while—most recently—Nassar and Eissa [4] derived a finite sum representation by restricting a to be a positive integer. Nadarajah and Gupta [5] have shown that one can derive various closed-form representations for the moments with no restrictions imposed on
S. Nadarajah / Reliability Engineering and System Safety 90 (2005) 114–117
the parameters of the distribution. These representations involve only the (standard) gamma function and its derivatives. The calculations of this note use the following functions: exponential integral, generalized hypergeometric, incomplete gamma and the zeta functions defined by ðN expðKxtÞ Eiðn; xÞ Z dt; tn 1 p Fq
p
ða1 ; .; ap ; b1 ; .; bq ; xÞ Z ðN
Gða; xÞ Z
N X ða1 Þk /ðap Þk xk ; ðb1 Þk /ðbq Þk k! kZ0
taK1 expðKtÞdt
x
N X 1 x; k kZ1
respectively, where ðcÞk Z cðcC 1Þ /ðcC kK 1Þ denotes the ascending factorial. We also need Euler’s constant gZ0.5772156649/ and the following important lemma.
,
2. Moments Theorem 1 expresses the kth moment as simple derivatives of the incomplete gamma function. Theorem 1. If X is a random variable with the pdf (2) and k/bZn is an integer then vnK1 ðlaÞKn Gðn; laÞ EðX Þ Z na expðlaÞ vnnK1 k
k
1 2 3 4 5 6 7 8 9 10
Using (3)
Using (4)
0.420 0.480 0.520 0.510 0.490 0.510 0.570 0.450 0.480 0.480
0.150 0.100 0.130 0.110 0.120 0.110 0.110 0.130 0.130 0.100
Corollary 1. If X is a random variable with the pdf (2) then EðX b Þ Z Eið1; laÞ; p2 g 2 C C g log l C g log a 12 2 ðlog lÞ2 ðlog aÞ2 C C C log l log a 2 2 K la3 F3 ð1; 1; 1; 2; 2; 2; KlaÞ; 2
2hð3Þ g3 p2 g p 3b 2 K K K C g log l EðX Þ Z K 3 6 3 6 2
p C g2 log a K 2g log l log a K 6 K gðlog lÞ2 K gðlog aÞ2 K log lðlog aÞ2 1 1 K ðlog lÞ2 log a K ðlog lÞ3 K ðlog aÞ3 3 3 C 2la4 F4 ð1; 1; 1; 1; 2; 2; 2; 2; KlaÞ;
EðX 2b Þ Z
Lemma 1. For mO0 and nO0, ðN vm mKn Gðn; mÞ xnK1 expðKmxÞðlog xÞm dx Z : vnm 1 Proof. See equation (4.358.1) in [6].
CPU time
It is evident that the CPU times for (4) are substantially smaller. Similar observations were noted for other combinations of k and b (with k/b taking an integer value). This gain in time could be crucial for potential real life applications of the modified Weibull distribution. The following corollary is the particular case of Theorem 1 for nZ1,2,3,4.
and zðxÞ Z
k
115
(4)
for nZ1,2,., where the derivative is evaluated as n/0. Proof. Apply the transformation yZexp{(x/a)b} to the right hand side of (3). One obtains ðN k k EðX Þ Z na expðlaÞ yK1 ðlog yÞk=bK1 expðKlayÞdy: 1
By Lemma 1, the above is the same as the required form in (4). , The moments of the modified Weibull distribution can be computed using either (3) or (4). In the table below, we have provided a comparison of the CPU times taken for computing E(Xk) for aZ1, lZ1, and kZbZ1,2,.,10. The computations were performed using the algebraic manipulation package, MAPLE.
3p4 p2 g 2 g 4 C2zð3Þg C C 80 4 4 p2 g C 2zð3Þ Cg3 C log l 2
p2 g log a C 2zð3Þ Cg3 C 2
2
p2 3g p2 C C 3g2 C log l log aC ðlog lÞ2 2 2 4 2
3g p2 C C ðlog aÞ2 C3g log lðlog aÞ2 2 4 C3gðlog lÞ2 log aCgðlog lÞ3 Cgðlog aÞ3 3 Clog lðlog aÞ3 C ðlog lÞ2 ðlog aÞ2 2 1 1 3 Cðlog lÞ log aC ðlog lÞ4 C ðlog aÞ4 4 4 K6la5 F5 ð1;1; 1; 1;1; 2; 2; 2;2; 2;KlaÞ:
EðX 4b Þ Z
116
S. Nadarajah / Reliability Engineering and System Safety 90 (2005) 114–117
Proof. For nZ1,2,3,4, evaluate the right hand side of (4) as n/0. , Corollary 1 can be particularly useful when b takes integer values. For example, when bZ1 the formulas give the first four moments; when bZ2 the formulas give moments of order 2, 4, 6 and 8; when bZ3 the formulas give moments of order 3, 6, 9 and 12; and, so on.
3. Special cases Theorem 1 can also be used to derive simple expressions when 1/b takes integer values. The two cases below illustrate this when 1/bZ2 and 1/bZ3. Case 1. If X is a random variable with the pdf (2) for bZ1/2 then p2 g2 1 C Cg log l Cg log a C ðlog lÞ2 2 12 2 1 2 C ðlog aÞ Clog l log aKla3 F3 ð1;1; 1; 2; 2;2;KlaÞ 2
EðXÞ Z
and
EðX 2 Þ Z
61p6 20z2 ð3Þ 10p2 zð3Þ C C 24zð5Þ g C 3 3 1008
3p4 g2 20zð3Þg3 5p2 g4 C C 8 3 12 2 10p zð3Þ 3p4 g C 24zð5Þ C C 20g2 zð3Þ C 3 4
5p2 g3 10p2 zð3Þ 5 C 24zð5Þ C g log l C C 3 3
3p4 g 5p2 g3 2 5 C C 20g zð3Þ C C g log a 4 3 4
3p 4 2 2 C 5g C 5p g C 40gzð3Þ log l log a C 4 4
3p 5p2 g2 5g4 C 20gzð3Þ C C C ðlog lÞ2 8 2 2 4
3p 5p2 g2 5g4 C 20gzð3Þ C C C ðlog aÞ2 8 2 2 C
C ð20zð3Þ C 5p2 g C 10g3 Þlog lðlog aÞ2
and p2 g2 g4 3p4 a4 C 2zð3Þg C C C 4 4 80 4
p2 g 3 C g log l C 2zð3Þ C 2
p2 g 3 C g log a C 2zð3Þ C 2 2
2
p p 3g2 2 C 3g log l log a C C C ðlog lÞ2 2 4 2 2
p 3g2 C C ðlog aÞ2 C 3glog lðlog aÞ2 4 2
EðX 2 Þ Z
C 3gðlog lÞ2 log a C gðlog lÞ3 C gðlog aÞ3 3 C log lðlog aÞ3 C ðlog lÞ2 ðlog aÞ2 2 1 3 C ðlog lÞ log a C ðlog lÞ4 4 K la4 F4 ð1; 1; 1; 1; 2; 2; 2; 2; KlaÞ: Case 2. If X is a random variable with the pdf (2) for bZ1/3 then 2
p2 g 2zð3Þ g3 p K K K EðXÞ Z K C g2 log l 6 3 3 6 2
p C g2 log a K 2g log l log a K gðlog lÞ2 K 6 K gðlog aÞ2 K log lðlog aÞ2 K ðlog lÞ2 log a 1 1 K ðlog lÞ3 K ðlog aÞ3 3 3 C 2la3 F3 ð1; 1; 1; 1; 2; 2; 2; 2; KlaÞ
C ð20zð3Þ C 5p2 g C 10g3 Þðlog lÞ2 log a
20zð3Þ 5p2 g 10g3 C C C ðlog lÞ3 3 3 3
20zð3Þ 5p2 g 10g3 C C C ðlog aÞ3 3 3 3 2
5p C 10g2 log lðlog aÞ3 C 3 2
5p 2 C 15g ðlog lÞ2 ðlog aÞ2 C 2 2
5p 2 C 10g ðlog lÞ3 log a C 3 2
2
5p 5g2 5p 5g2 4 C C C ðlog lÞ C ðlog aÞ4 12 2 12 2 C 5glog lðlog aÞ4 C 10gðlogl Þ2 ðlog aÞ3 C 10gðlog lÞ3 ðlog aÞ2 C 5gðlog lÞ4 log a C gðlog lÞ5 C gðlog aÞ5 C log lðlog aÞ5 5 10 C ðlog lÞ2 ðlog aÞ4 C ðlog lÞ3 ðlog aÞ3 2 3 5 1 C ðlog lÞ4 ðlog aÞ2 C ðlog lÞ5 log a C ðlog lÞ6 2 6 1 C ðlog aÞ6 K 120la7 F7 6 !ð1; 1; 1; 1; 1; 1; 1; 2; 2; 2; 2; 2; 2; 2; KlaÞ:
S. Nadarajah / Reliability Engineering and System Safety 90 (2005) 114–117
Note that the expressions in the two cases above are elementary except for the two hypergeometric terms, which can be evaluated by using widely available numerical routines. Similar expressions can be obtained for any integer value of 1/b.
Acknowledgements The authors would like to thank Professor M. Xie and the editor for carefully reading the paper and for their help in improving the paper.
117
References [1] Xie M, Tang Y, Goh TN. A modified Weibull extension with bathtubshaped failure rate function. Reliab Eng Syst Saf 2002;76:279–85. [2] Mudholkar GS, Srivastava DK, Freimer M. The exponentiated Weibull family. Technometrics 1995;37:436–45. [3] Mudholkar GS, Hutson AD. The exponentiated Weibull family: some properties and a flood data application. Commun Stat Theory Meth 1996;25:3059–83. [4] Nassar MM, Eissa FH. On the exponentiated Weibull distribution. Commun Stat Theory Meth 2003;32:1317–36. [5] Nadarajah S, Gupta AK. On the moments of the exponentiated Weibull distribution. Commun Stat Theory Meth 2005;34. [6] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products, 6th ed. San Diego: Academic Press; 2000.
Short communication
On the moments of the modified Weibull distribution Saralees Nadarajah Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA Received 3 April 2004; accepted 2 September 2004 Available online 11 November 2004
Abstract In a recent paper, Xie et al. [Reliab Eng Syst Saf 2002;76:279–85] proposed a modification of the Weibull distribution to allow it to exhibit bathtub-shaped failure rate functions. In this note, we derive explicit algebraic formulas for the kth moment of the distribution. The formulas allow one to calculate the moments for a wide range of values of k and the shape parameter of the modified Weibull. They also allow one to calculate the moments more easily with less computational time. Besides being of interest in their own, we believe that these formulas will be useful for any follow-up mathematical study of the proposed distribution. q 2004 Elsevier Ltd. All rights reserved. Keywords: Weibull distribution; MAPLE; Gamma function
1. Introduction The Weibull distribution is commonly used to model lifetimes of systems. However, it does not exhibit a bathtubshaped failure rate function and thus it cannot be used to model the complete lifetime of a system. In a recent paper Xie et al. [1] proposed a modification of the Weibull distribution which exhibits bathtub-shaped failure rate functions. The survivor function of this distribution is given by x b RðxÞ Z exp la 1 K exp (1) a for xO0, lO0, aO0 and bO0. The corresponding pdf (probability density function) is x bK1 x b x b f ðxÞ Z lb exp C la 1 K exp : (2) a a a The authors derived parameter estimation methods for this distribution and illustrated its applicability by means of two examples. However, the mathematical properties of the distribution were not studied in detail. In particular, only integral representations were provided for the mean and variance associated with (1). E-mail address: [email protected]. 0951-8320/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2004.09.002
In this note, we derive explicit algebraic forms for the kth moment associated with (1), which can be written as ðN ðN x b kK1 kK1 exp la 1 K exp dx: k x RðxÞ dx Z k x a 0 0 (3) We show that (3) can be reduced to explicit forms— involving only standard special functions—for a wide range of values of k and b (Sections 2 and 3). These formulas, apart from being of interest in their own, will be useful for any follow-up mathematical study of the proposed distribution. They could also be used to achieve substantial gain in computational time as shown in Section 2. Another modification of the Weibull distribution introduced by Mudholkar et al. [2] is the exponentiated Weibull distribution given by the survivor function RðxÞ Z 1 K ½1 K expfKðlxÞa ga for xO0, aO0, aO0 and lO0. The moments of this distribution have also been of some interest. Mudholkar and Hutson [3] provided a non-closed form integral representation for the moments while—most recently—Nassar and Eissa [4] derived a finite sum representation by restricting a to be a positive integer. Nadarajah and Gupta [5] have shown that one can derive various closed-form representations for the moments with no restrictions imposed on
S. Nadarajah / Reliability Engineering and System Safety 90 (2005) 114–117
the parameters of the distribution. These representations involve only the (standard) gamma function and its derivatives. The calculations of this note use the following functions: exponential integral, generalized hypergeometric, incomplete gamma and the zeta functions defined by ðN expðKxtÞ Eiðn; xÞ Z dt; tn 1 p Fq
p
ða1 ; .; ap ; b1 ; .; bq ; xÞ Z ðN
Gða; xÞ Z
N X ða1 Þk /ðap Þk xk ; ðb1 Þk /ðbq Þk k! kZ0
taK1 expðKtÞdt
x
N X 1 x; k kZ1
respectively, where ðcÞk Z cðcC 1Þ /ðcC kK 1Þ denotes the ascending factorial. We also need Euler’s constant gZ0.5772156649/ and the following important lemma.
,
2. Moments Theorem 1 expresses the kth moment as simple derivatives of the incomplete gamma function. Theorem 1. If X is a random variable with the pdf (2) and k/bZn is an integer then vnK1 ðlaÞKn Gðn; laÞ EðX Þ Z na expðlaÞ vnnK1 k
k
1 2 3 4 5 6 7 8 9 10
Using (3)
Using (4)
0.420 0.480 0.520 0.510 0.490 0.510 0.570 0.450 0.480 0.480
0.150 0.100 0.130 0.110 0.120 0.110 0.110 0.130 0.130 0.100
Corollary 1. If X is a random variable with the pdf (2) then EðX b Þ Z Eið1; laÞ; p2 g 2 C C g log l C g log a 12 2 ðlog lÞ2 ðlog aÞ2 C C C log l log a 2 2 K la3 F3 ð1; 1; 1; 2; 2; 2; KlaÞ; 2
2hð3Þ g3 p2 g p 3b 2 K K K C g log l EðX Þ Z K 3 6 3 6 2
p C g2 log a K 2g log l log a K 6 K gðlog lÞ2 K gðlog aÞ2 K log lðlog aÞ2 1 1 K ðlog lÞ2 log a K ðlog lÞ3 K ðlog aÞ3 3 3 C 2la4 F4 ð1; 1; 1; 1; 2; 2; 2; 2; KlaÞ;
EðX 2b Þ Z
Lemma 1. For mO0 and nO0, ðN vm mKn Gðn; mÞ xnK1 expðKmxÞðlog xÞm dx Z : vnm 1 Proof. See equation (4.358.1) in [6].
CPU time
It is evident that the CPU times for (4) are substantially smaller. Similar observations were noted for other combinations of k and b (with k/b taking an integer value). This gain in time could be crucial for potential real life applications of the modified Weibull distribution. The following corollary is the particular case of Theorem 1 for nZ1,2,3,4.
and zðxÞ Z
k
115
(4)
for nZ1,2,., where the derivative is evaluated as n/0. Proof. Apply the transformation yZexp{(x/a)b} to the right hand side of (3). One obtains ðN k k EðX Þ Z na expðlaÞ yK1 ðlog yÞk=bK1 expðKlayÞdy: 1
By Lemma 1, the above is the same as the required form in (4). , The moments of the modified Weibull distribution can be computed using either (3) or (4). In the table below, we have provided a comparison of the CPU times taken for computing E(Xk) for aZ1, lZ1, and kZbZ1,2,.,10. The computations were performed using the algebraic manipulation package, MAPLE.
3p4 p2 g 2 g 4 C2zð3Þg C C 80 4 4 p2 g C 2zð3Þ Cg3 C log l 2
p2 g log a C 2zð3Þ Cg3 C 2
2
p2 3g p2 C C 3g2 C log l log aC ðlog lÞ2 2 2 4 2
3g p2 C C ðlog aÞ2 C3g log lðlog aÞ2 2 4 C3gðlog lÞ2 log aCgðlog lÞ3 Cgðlog aÞ3 3 Clog lðlog aÞ3 C ðlog lÞ2 ðlog aÞ2 2 1 1 3 Cðlog lÞ log aC ðlog lÞ4 C ðlog aÞ4 4 4 K6la5 F5 ð1;1; 1; 1;1; 2; 2; 2;2; 2;KlaÞ:
EðX 4b Þ Z
116
S. Nadarajah / Reliability Engineering and System Safety 90 (2005) 114–117
Proof. For nZ1,2,3,4, evaluate the right hand side of (4) as n/0. , Corollary 1 can be particularly useful when b takes integer values. For example, when bZ1 the formulas give the first four moments; when bZ2 the formulas give moments of order 2, 4, 6 and 8; when bZ3 the formulas give moments of order 3, 6, 9 and 12; and, so on.
3. Special cases Theorem 1 can also be used to derive simple expressions when 1/b takes integer values. The two cases below illustrate this when 1/bZ2 and 1/bZ3. Case 1. If X is a random variable with the pdf (2) for bZ1/2 then p2 g2 1 C Cg log l Cg log a C ðlog lÞ2 2 12 2 1 2 C ðlog aÞ Clog l log aKla3 F3 ð1;1; 1; 2; 2;2;KlaÞ 2
EðXÞ Z
and
EðX 2 Þ Z
61p6 20z2 ð3Þ 10p2 zð3Þ C C 24zð5Þ g C 3 3 1008
3p4 g2 20zð3Þg3 5p2 g4 C C 8 3 12 2 10p zð3Þ 3p4 g C 24zð5Þ C C 20g2 zð3Þ C 3 4
5p2 g3 10p2 zð3Þ 5 C 24zð5Þ C g log l C C 3 3
3p4 g 5p2 g3 2 5 C C 20g zð3Þ C C g log a 4 3 4
3p 4 2 2 C 5g C 5p g C 40gzð3Þ log l log a C 4 4
3p 5p2 g2 5g4 C 20gzð3Þ C C C ðlog lÞ2 8 2 2 4
3p 5p2 g2 5g4 C 20gzð3Þ C C C ðlog aÞ2 8 2 2 C
C ð20zð3Þ C 5p2 g C 10g3 Þlog lðlog aÞ2
and p2 g2 g4 3p4 a4 C 2zð3Þg C C C 4 4 80 4
p2 g 3 C g log l C 2zð3Þ C 2
p2 g 3 C g log a C 2zð3Þ C 2 2
2
p p 3g2 2 C 3g log l log a C C C ðlog lÞ2 2 4 2 2
p 3g2 C C ðlog aÞ2 C 3glog lðlog aÞ2 4 2
EðX 2 Þ Z
C 3gðlog lÞ2 log a C gðlog lÞ3 C gðlog aÞ3 3 C log lðlog aÞ3 C ðlog lÞ2 ðlog aÞ2 2 1 3 C ðlog lÞ log a C ðlog lÞ4 4 K la4 F4 ð1; 1; 1; 1; 2; 2; 2; 2; KlaÞ: Case 2. If X is a random variable with the pdf (2) for bZ1/3 then 2
p2 g 2zð3Þ g3 p K K K EðXÞ Z K C g2 log l 6 3 3 6 2
p C g2 log a K 2g log l log a K gðlog lÞ2 K 6 K gðlog aÞ2 K log lðlog aÞ2 K ðlog lÞ2 log a 1 1 K ðlog lÞ3 K ðlog aÞ3 3 3 C 2la3 F3 ð1; 1; 1; 1; 2; 2; 2; 2; KlaÞ
C ð20zð3Þ C 5p2 g C 10g3 Þðlog lÞ2 log a
20zð3Þ 5p2 g 10g3 C C C ðlog lÞ3 3 3 3
20zð3Þ 5p2 g 10g3 C C C ðlog aÞ3 3 3 3 2
5p C 10g2 log lðlog aÞ3 C 3 2
5p 2 C 15g ðlog lÞ2 ðlog aÞ2 C 2 2
5p 2 C 10g ðlog lÞ3 log a C 3 2
2
5p 5g2 5p 5g2 4 C C C ðlog lÞ C ðlog aÞ4 12 2 12 2 C 5glog lðlog aÞ4 C 10gðlogl Þ2 ðlog aÞ3 C 10gðlog lÞ3 ðlog aÞ2 C 5gðlog lÞ4 log a C gðlog lÞ5 C gðlog aÞ5 C log lðlog aÞ5 5 10 C ðlog lÞ2 ðlog aÞ4 C ðlog lÞ3 ðlog aÞ3 2 3 5 1 C ðlog lÞ4 ðlog aÞ2 C ðlog lÞ5 log a C ðlog lÞ6 2 6 1 C ðlog aÞ6 K 120la7 F7 6 !ð1; 1; 1; 1; 1; 1; 1; 2; 2; 2; 2; 2; 2; 2; KlaÞ:
S. Nadarajah / Reliability Engineering and System Safety 90 (2005) 114–117
Note that the expressions in the two cases above are elementary except for the two hypergeometric terms, which can be evaluated by using widely available numerical routines. Similar expressions can be obtained for any integer value of 1/b.
Acknowledgements The authors would like to thank Professor M. Xie and the editor for carefully reading the paper and for their help in improving the paper.
117
References [1] Xie M, Tang Y, Goh TN. A modified Weibull extension with bathtubshaped failure rate function. Reliab Eng Syst Saf 2002;76:279–85. [2] Mudholkar GS, Srivastava DK, Freimer M. The exponentiated Weibull family. Technometrics 1995;37:436–45. [3] Mudholkar GS, Hutson AD. The exponentiated Weibull family: some properties and a flood data application. Commun Stat Theory Meth 1996;25:3059–83. [4] Nassar MM, Eissa FH. On the exponentiated Weibull distribution. Commun Stat Theory Meth 2003;32:1317–36. [5] Nadarajah S, Gupta AK. On the moments of the exponentiated Weibull distribution. Commun Stat Theory Meth 2005;34. [6] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products, 6th ed. San Diego: Academic Press; 2000.