Alpha power Weibull distribution: Properties and

Communications in Statistics - Theory and Methods

ISSN: 0361-0926 (Print) 1532-415X (Online) Journal homepage: http://www.tandfonline.com/loi/lsta20

Alpha power Weibull distribution: Properties and applications M. Nassar, A. Alzaatreh, M. Mead & O. Abo-Kasem To cite this article: M. Nassar, A. Alzaatreh, M. Mead & O. Abo-Kasem (2017) Alpha power Weibull distribution: Properties and applications, Communications in Statistics - Theory and Methods, 46:20, 10236-10252, DOI: 10.1080/03610926.2016.1231816 To link to this article: http://dx.doi.org/10.1080/03610926.2016.1231816

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Date: 31 August 2017, At: 07:56

COMMUNICATIONS IN STATISTICS—THEORY AND METHODS , VOL. , NO. , – https://doi.org/./..

Alpha power Weibull distribution: Properties and applications M. Nassara , A. Alzaatrehb , M. Meada , and O. Abo-Kasema

Downloaded by [156.195.134.39] at 07:56 31 August 2017

a Department of Statistics, Faculty of Commerce, Zagazig University, Zagazig, Egypt; b Department of Mathematics, Nazarbayev University, Astana, Kazakhstan

ABSTRACT

ARTICLE HISTORY

In this paper, a new lifetime distribution is defined and studied. We refer to the new distribution as alpha power Weibull distribution. The importance of the new distribution comes from its ability to model monotone and non monotone failure rate functions, which are quite common in reliability studies. Various properties of the proposed distribution are obtained including moments, quantiles, entropy, order statistics, mean residual life function, and stress-strength parameter. The maximum likelihood estimation method is used to estimate the parameters. Two real data sets are used to illustrate the importance of the proposed distribution.

Received  February  Accepted  August  KEYWORDS

Alpha power transformation; Weibull distribution; maximum likelihood estimation; stress-strength parameter. MATHEMATICS SUBJECT CLASSIFICATION

E; E

1. Introduction The Weibull distribution is a very popular lifetime distribution in reliability theory. It is commonly used for analyzing biological, medical, and hydrological data sets. It does not provide an acceptable fit for some applications, especially, when the hazard rates are bathtub, upside down bathtub, or bimodal shapes. To overcome such weakness, in the recent years several authors have developed various generalizations and extensions of the Weibull distribution to model various types of data. Among these, Mudholkar et al. (1995, 1996) introduced and studied the exponentiated Weibull (EW) distribution by adding an extra shape parameter to the Weibull distribution to allowing bathtub-shaped hazard rate function. Also, Xie and Lai (1995) introduced the additive Weibull distribution, the Weibull extension distribution proposed by Xie et al. (2002), generalized modified Weibull (MW) distribution by Jalmar et al. (2008), the exponential-Weibull distribution by Cordeiro et al. (2014), and beta Sarhan–Zaindin MW distribution by Saboor et al. (2016). For recent methods of generalized continuous and discrete distributions, we refer the reader to Lee et al. (2013) and Alzaatreh et al. (2014). Recently, Mahdavi and Kundu (2015) proposed a new method to introduce an extra parameter to a family of distributions for more flexibility. The proposed method is called alpha power transformation (APT) and it is useful to incorporate skewness to a family of distributions. Let F (x) be the cumulative density function (CDF) of a continuous random variable X, then they define the APT of F (x) for x ∈ R as follows: F (x) α −1 if α > 0, α = 1 α−1 (1) FAPT (x) = F (x) if α = 1 CONTACT M. Nassar mezo@gmail.com Department of Statistics, Faculty of Commerce, Zagazig University, Zagazig, Egypt. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lsta. ©  Taylor & Francis Group, LLC

COMMUNICATIONS IN STATISTICS—THEORY AND METHODS


and the corresponding probability density function (PDF) as log α f (x)α F (x) if α > 0, α = 1 fAPT (x) = α−1 f (x) if α = 1

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(2)

They applied the proposed method to a one-parameter exponential distribution and generated a two-parameter alpha power exponential (APE) distribution. They studied the various properties of the proposed distribution such as explicit expressions for the moments, quantiles, moment generating function, and order statistics. Also, they analyzed real data set to show how the proposed model works in practice. The main aim of this paper is to propose and study a new lifetime model called alpha power Weibull (APW) distribution based on the method of APT. The main purpose of the new model is that the additional parameter can give several desirable properties and more flexibility in the form of the hazard and density functions. The rest of this paper is organized as follows: In Sec. 2, we introduce the APW distribution and some special cases are presented. In Sec. 3, we study some of its structural properties including quantile function, moments, moment generating function, entropy, order statistics, mean residual life function, and stressstrength parameter. In Sec. 4, we discuss the maximum likelihood estimates (MLEs) of the model parameter. In Sec. 5, the analysis of two real data sets has been presented to illustrate the potentiality of the new model. The paper is concluded in Sec. 6.

2. APW distribution Let ϕ = (α, λ, β )T . From (2), the random variable X is said to have a three-parameter APW distribution with the scale parameter λ > 0 and shape parameters α > 0, β > 0, if the CDF of x > 0 is −λxβ 1 (1 − α 1−e ) if α > 0, α = 1 1−α (3) FAPW (x, ϕ) = −λxβ if α = 1 1−e and the corresponding PDF is fAPW (x, ϕ) =

−λxβ β−1 −λxβ log α λβα 1−e x e α−1 β−1 −λxβ

λβx

e

if α > 0, α = 1 if α = 1

(4)

The survival function (S(x, ϕ)) and the hazard rate function (h(x, ϕ)) for x > 0 are, respectively, given by −λxβ α (1 − α −e ) if α > 0, α = 1 SAPW (x, ϕ) = α−1 −λxβ (5) if α = 1 1−e hAPW (x, ϕ) =

β

log(α) λβxβ−1 e−λx (α e λβxβ−1

−λxβ

− 1)−1

if α > 0, α = 1 if α = 1

(6)

Table 1 lists seven important special models of the new distribution. Figure 1 displays some plots of the APW density for selected parameter values. Plots of the h(x, ϕ) of the APW distribution for selected parameter values are given in Figure 2.

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Table . Sub-models of the APW distribution. α

λ

β

Reduced model

—   —  — 

 —  — — — —

— — —    

Alpha power one-parameter Weibull distribution Two-parameter Weibull distribution One-parameter Weibull distribution Alpha power Rayleigh distribution Rayleigh distribution Alpha power exponential distribution Exponential distribution

3. Properties of APW distribution

1.0

β = 0.2 β = 1.5 β = 0.85 β= 2 β= 2.5

0.6

=0.2 =0.5 =25 =3 =5

0.0

0.2

0.4

f(x)

0.8

α α α α α

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

0.9

h(x)

0.7

2

0.8

4

h(x)

1.0

6

1.1

8

1.2

Figure . Plots of the APW density for λ = 1 and various values of α and β.

α =1.5 β = 1.05 α =0.6 β = 1.02 α =7 β= 0.85

0.6

α =0.5 β= 0.2 α =0.005 β = 1.5 α =0.005 β = 2.75

0


In this section, we study some properties of the APW distribution including shape behavior, quantile function, moments and moment generating function, entropy, order statistics, mean residual life and mean waiting time, and stress-strength parameter.

0.0

0.5

1.0 x

1.5

2.0

0.0

0.5

1.0

1.5

2.0

x

Figure . Plots of the APW hazard rate distribution for λ = 1 and various values of α and β.

2.5

3.0


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3.1. Shape behavior In this subsection, we discuss density and hazard rate function shapes for the APW distribution. Since λ is a scale parameter, we assume, without loss of generality, that λ = 1. When α = 1, the APW distribution reduces to the Weibull distribution. In this case the shapes for the density and the hazard rate functions are well known in the literature. Now assume for the rest of this subsection that α = 1. From (4) we have f (x) = ψ (x)k(x), where ψ (x) > 0 and β k(x) = −ex (1 − β + βxβ ) + βxβ log α, x > 0. Case I: α < 1 andβ ≤ 1. Then it is easy to show that k(x) < 0 for all x > 0. Therefore, f (x) has a reversed J-shape with mode at x = 0. β Case II: α < 1 and β > 1. Consider k (x) = −βxβ−1 ξ (x) where ξ (x) = ex (1 + βxβ ) − β log α. Since α < 1 and β > 1, we have k (x) < 0. Therefore, k(x) is decreasing. Now since limx→0 k(x) = β − 1 > 0 and limx→∞ k(x) = −∞, k(x) must have one root. Furthermore,limx→0 f (x) = limx→∞ f (x) = 0. Hence, f (x) is concave and has unique mode at x > 0. Case III: α > 1 and β > 1. In this case one can see that ξ (x) > 0 for all x > 0. Also, limx→0 ξ (x) = 1 − β log α and limx→∞ ξ (x) = ∞. If α ≤ e1/β then k (x) < 0. By following the same argument as in Case II, f (x) is concave and has unique mode at x > 0. Now if α > e1/β , then ξ (x) has unique root say x0 . This implies k (x) ≥ 0 for all x ≤ x0 and k (x) ≤ 0 for all x ≥ x0 . Since limx→0 k(x) = β − 1 > 0, f (x) is concave and has unique mode at x > 0. Case IV: α > 1 and β < 1. Then it is not difficult to show that if α ≤ e1/β , k (x) → −∞ as x → 0 or x → ∞ and k (x) < 0 for all x > 0. Also the fact that limx→0 k(x) = β − 1 < 0, limx→∞ k(x) = −∞, limx→0 f (x) = ∞ and limx→∞ f (x) = 0 implies that f (x) has a reverse J-shaped. If α > e1/β , then k (x) → ∞ as x → 0 and k (x) → −∞ as x → ∞. The fact that limx→0 k(x) = β − 1 < 0 and limx→∞ k(x) = −∞ implies that k(x) has two roots say at x1 < x2 . Since limx→0 f (x) = ∞ and limx→∞ f (x) = 0, this implies that f (x) is convex on (0, x2 ) and decreases on (x2 , ∞) (see Figure 1). For the hazard rate function, we again assume λ = 1 and α = 1. On using similar approach as in the previous paragraph, one can get the following results. If α < 1 and β ≤ 1, then h(x) has a decreasing failure rate (DFR) shape. If α < 1 and β > 1, then h(x) → ∞ as x → ∞ and h(x) can have an increasing failure rate (IFR) or concave–convex shape. If α > 1 and β ≥ 1, then h(x) has an IFR shape. If α > 1 and β < 1, h(x) can have a DFR or convex–concave shape. 3.2. Simulation and quantiles Using Equation (4), the APW distribution can be easily simulated by log(α/(1 + (α − 1)U )) 1/β 1 X = − log λ log α

(7)

where U follows uniform (0, 1) distribution. The pth quantile function of APW distribution is given by 1 log(α/(1 + (α − 1)p)) 1/β (8) x p = − log λ log α

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The median can be obtained as log(2α/(1 + α)) 1/β 1 x0.5 = − log λ log α 3.3. Moment and moment generating function In this subsection, we derive the rth moments and the moment generating function of the APW distribution. Using the series representation, α −z =

∞ (− log α)k zk k! k=0

(9)


The rth moment of X can be obtained as μr = E(X r ) =

α (r/β + 1) (− log α)k (1 − α)λr/β k=1 k! kr/β ∞

(10)

and the moment generating function is α log α (− log α)kt j ( j/β + 1) Mx (t ) = (1 − α) k=0 j=0 (k + 1)! j!(k + 1) j/β λ j/β ∞

∞

(11)

Theorem 1. The rth moments of the APW distribution exist for all r ∈ N. Furthermore, α log α log α r r (i) if α < 1, (α−1)λ r/β ( β + 1) ≤ μr ≤ (α−1)λr/β ( β + 1). log α r (ii) If α > 1, then (α−1)λ r/β ( β + 1) ≤ μr ≤ r −r/β ( β + 1). (iii) If α = 1, μr = λ

α log α (α−1)λr/β

( βr + 1).

−λxβ

Proof. It is not difficult to show that for α > 1, 1 < α 1−e < α, and for α < 1, −λxβ α < α 1−e < 1. Therefore, the results in (i)–(iii) follow from (4) and the fact that ∞ −λxβ β−1 −λxβ λβα 1−e x e dx =λ−r/β ( βr + 1). Note that (i)–(iii) implies that μr exists for all 0 r. Remark: From previous theorem, it is evident that μr exists for all values of r > −β, r ∈ R. 3.4. Rényi and Shannon entropies The entropy of a random variable X measures the variation of the uncertainty. The Rényi entropy, say REX (ν ), is defined as ∞ 1 ν REX (ν ) = f (x) dx , ν > 0, ν = 1 log 1−ν −∞ Using the PDF in (4), the Rényi entropy of X can be obtained as follows: ν ∞ β log α 1 −νe−λx ν(β−1) −νλxβ log αλβ α x e dx REX (ν ) = 1−ν α−1 0

(12)


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Using the series representation in (9) and the transformation y = λxβ (k + ν ), REX (ν ) reduces to α log α log λ ν log − log β − REX (ν ) = 1−ν α−1 β ∞ k k 1 (− log α) ν (ν − (ν − 1)/β ) + log (13) 1−ν k! (k + ν )ν−(ν−1)/β k=0 The Shannon entropy is defined as SEX = E[− log f (x)]. Using the PDF in (4), SEX is ∞ α log α (− log α)k log α 1 α−1 + − SEX = log + (β − 1)λβ I αλβ log α 1 − α k=0 k! k + 2 (k + 1)2 (14) Downloaded by [156.195.134.39] at 07:56 31 August 2017

where

∞

I=

β

xβ−1 e−λ(k+1)x log x dx

0

using the transformation y = xβ and ∞ 1 e−μx log x dx = − (C + log μ) μ 0 where C is the Euler constant. Then SEX can be obtained as ∞ α log α (− log α)k=1 α−1 SEX = log + αλβ log α 1 − α k=0 k! 1 (1 − β )(C + log(λ(k + 1))) log α − + × k + 2 (k + 1)2 β (k + 1)

(15)

3.5. Order statistics Let X1 , X2 , . . . , Xn be a random sample of size n, and let Xi:n denote the ith order statistic, then, the PDF of Xi:n , say fi:n (x) is given by fi:n (x) =

n! F (x)i−1 f (x) (1 − F (x))n−i (i − 1)! (n − i)!

(16)

Substituting Equations (3) and (4) in Equation (16), we can write fi:n (x) as fi:n (x) =

β β α n−i (−1)i−1 1−e−λx i−1 −e−λx n−i f (x) (1 − α ) (1 − α ) B(i, n − i + 1)(α − 1)n−1

(17)

where B(a, b) is the beta function. Using the binomial expansion, fi:n (x) is given by i−1 n−i i+k+ j−1 λβ log α n−i i − 1 (−1) xβ−1 fi:n (x) = k j B(i, n − i + 1)(α − 1)n k=0 j=0 α −(n+ j−i+1) β

−λxβ

× e−λx α −(k+ j+1)e

(18)

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The sth moment of Xi:n can be expressed as s )= E(Xi:n

n−i i−1 ∞ n! (s/β )! log α (−1)i+k+ j−1 (− log α)m (k + j + 1)m α n+ j−i+1 λs/β (α − 1)n k=0 j=0 m=0 k! j!m!(n − i − k)!(i − j − 1))!(m + 1)s/β+1

(19)

Next, we discuss the asymptotic distributions of Xn:n and X1:n . Consider the following theorem from Von Mises (1936). Theorem 2: Let F be an absolutely continuous CDF and suppose h f (x) is non zero and differd entiable function. If limx→F −1 (1) dx ( h 1(x) ) = 0, then F ∈ D(F1 ) where F1 (x) = exp(− exp(x)). f


Consider α = 1. From (6), we have −λxβ β 1 d (α e − 1)eλx (1 + β(λxβ − 1) = lim −1 lim x→∞ dx x→∞ hAPW (x) λβ log(α)xβ −λxβ

β

Since limx→∞ (α e − 1)eλx = log α, the above limit equals to 0. Therefore, the large sample distribution of Xn:n is of extreme value type. For the case α = 1, the APW reduces to Weibull distribution. In this case, the asymptotic distribution of the sample maximum is of extreme type. Now consider the asymptotic distribution of X1:n . According to Arnold et al. (2008), the asymptotic distribution of X1:n is of Weibull type with parameter c > 0 if = xc for all x > 0. From (2), we have limε→0 FF(εx) (ε) F (εx) f (εx) = x lim = xβ ε→0 F (ε) ε→0 f (ε) lim

Hence the asymptotic distribution of sample minimum is of Weibull type with shape parameter β. 3.6. Mean residual life and mean waiting time Assuming that X is a continuous random variable with survival function (5), the mean residual life is the expected additional lifetime that a component has survived after a fixed time point t. The mean residual life function, say μ(t ), is given by t 1 E(t ) − x f (x) dx − t (20) μ(t ) = S(t ) 0 where

t

x f (x) dx =

0

∞ α log α (− log α)k γ (λ(k + 1)t β ; 1 + 1/β ) α − 1 k=0 (k + 1)! (λ(k + 1)) 1/β

(21)

Substituting Equations (5), (10), and (21) in Equation (20), we can write μ(t ) as μ(t ) =

(1 + 1/β )λ−1/β

∞ k=1

(− log α)k k! k1/β

+ log α

∞

(− log α)k k=0 (k+1)! (λ(k+1)) 1/β γ (λ(k

β

α −e−λx − 1

+ 1)t β , 1 + 1/β )

−t

a where γ (a, b) = 0 xb−1 e−x dx is the lower incomplete gamma function. The mean waiting time represents the waiting time elapsed since the failure of an object on condition that this failure had occurred in the interval [0, t]. The mean waiting time of


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X, say μ(t ¯ ), is defined by μ(t ¯ )=t−

1 F (t )

t

x f (x) dx

(22)

0

Substituting Equations (4) and (21) in Equation (22), we can write μ(t ¯ ) as μ(t ¯ )=t−

α β

α 1−e−λt − 1

log α

∞ k=0

(− log α)k γ (λ(k + 1)t β , 1 + 1/β ) (k + 1)! (λ(k + 1)) 1/β

(23)


3.7. Stress-strength parameter Suppose X1 and X2 be two continuous and independent random variables, where X1 ∼ APW (α1 , λ1 , β ) and X2 ∼ APW (α2 , λ2 , β ), then the stress-strength parameter, say R, is defined as ∞ f1 (x)F2 (x) dx (24) R= −∞

Using Equations (3) and (4), the stress-strength parameter R, can be obtained as ∞ α1 λ1 β log α1 1 −λ xβ β −λ xβ R= α1 −e 1 xβ−1 e−λ1 x α2 1−e 2 dx − (α1 − 1)(α2 − 1) 0 (α2 − 1)

(25)

Using Equations (9) and (25), R can be written as α1 α2 λ1 β log α1 (− log α1 )k (− log α2 )m (α1 − 1)(α2 − 1) k=0 m=0 k!m! ∞ 1 β × xβ−1 e−(λ1 (k+1)+mλ2 )x dx − (α2 − 1) 0 ∞

R=

∞

By applying the transformation y = xβ (λ1 (k + 1) + mλ2 ), R reduces to ∞ ∞ 1 α1 α2 λ1 log α1 (− log α1 )k (− log α2 )m R= −1 (α2 − 1) (α1 − 1) k=0 m=0 k!m! (λ1 (k + 1) + mλ2 )

(26)

4. Parameter estimation 4.1. Maximum likelihood estimation Let x1 , x2 , . . . , xn be a random sample from APW distribution; then the logarithm of the likelihood function (l), becomes = n log

log α α−1

+ n log(αβλ) + (β − 1)

n i=1

log xi − log α

n i=1

β

e−λxi − λ

n

xiβ

i=1

(27) Therefore, to obtain the MLEs of α, λ, and β, we find the first derivatives of the natural logarithm of the likelihood function (27) with respect to α, λ, and β and equating them to zero, we get the following three equations:

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M. NASSAR ET AL. n n n(α − 1 − α log α) 1 −λxβ ∂ = + − e i =0 ∂α α α (α − 1) log α α i=1

(28)

β β β n ∂ = + log α xi e−λxi − xi = 0 ∂λ λ i=1 i=1

(29)

n

n

and n n n β n ∂ −λxi β log xi + λ log α e xi log xi − λ xiβ log xi = 0 = + ∂β β i=1 i=1 i=1

(30)


Then the MLEs of the parameters α, λ, and β can be obtained by solving system of Equations (28)–30). Next, we discuss the existence of the MLEs for each parameter when the other two parameters are given. We also discuss, in some cases, the uniqueness of the MLEs. Proposition 1. If the parameters β and λ are known, then the MLE of α exists and is unique. ∂ = α1 ψ (α) where Proof. Assume that β and λ are known, then from Equation (28) we have ∂α

β 1 ψ (α) = n( log1 α − α−1 ) − C and C = ni=1 e−λxi . Now since C < n, one can easily see that limα→∞ ψ (α) = −C < 0 and limα→0 ψ (α) = n − C > 0. Therefore, ψ (α) = 0 has at least one solution in the interval (0, ∞). To show the uniqueness, it is known that log x ≤ x − 1 for 1 all x > 0. This implies that log1 α > α−1 , α = 1. Now, if α1 < α2 then log1α1 > log1α2 > α21−1 > 1 1 1 1 . Therefore, log α1 − α1 −1 > log α2 − α21−1 . This implies that ψ (α) is strictly a decreasing α1 −1 ∂ = 0 has a unique solution. function. Thus ψ (α) = 0 has a unique solution and hence, ∂α

Proposition 2. If the parameters α and β are known, then the MLE of λ exists. Moreover, the ∂ has at least one root in the interval MLE of λ is unique whenever α ≥ 1. If 0 < α < 1 then ∂λ n n [

n β , n β ]. (1−log α)

i=1 xi

i=1 xi

Proof. Consider the case α ≥ 1. Differentiating Equation (29) with respect to λ, we get

β ∂2 ∂ = −n − λβ log α ni=1 xi2β−1 e−λxi < 0. Thus, the function ∂λ is strictly decreasing. ∂λ2 λ2

n β ∂ ∂ ∂ Since limλ→0 ∂λ = − i=1 xi < 0 and limλ→∞ ∂λ = ∞, it follows that ∂λ has a unique root. Now consider the case 0 < α < 1. Then we have that β n ∂ < − xi ∂λ λ i=1 n

and β ∂ n > − (1 − log α) xi ∂λ λ i=1 n

Thus,

∂ ∂λ

has at least one root in the interval [

n

β (1−log α) ni=1 xi

, n n

β i=1 xi

].

Proposition 3. If the parameters α and λ are known, then the MLE of β exists. Proof. The proof follows from the fact that limβ→0 log xi , 0 < xi < 1 ∂ . In all cases limβ→∞ ∂β ≤ 0. 0, xi = 1 −∞, xi > 1

∂ ∂β

= ∞ and limβ→∞

∂ ∂β

=


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Table . Average values of MLEs and the corresponding MSEs (n = 50). Parameters

MSE

λ

α

β

ˆ λ

αˆ

βˆ

ˆ λ

αˆ



.

. .   . .  

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .



. .  

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .



. .  

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

.

. .   . .   . .   . .  

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

.


MLE



.





βˆ

No explicit form for the MLEs, a numerical technique may be used to solve these non linear equations. From Equation (29) and for fixed λ and β, we can obtain α(λ, ˆ β ) as follows: n n β n β β (31) xi − xi e−λxi α(λ, ˆ β ) = exp λ i=1 i=1 ˆ and βˆ can be obtained by solving the following two Then the MLE of λ and β denoted by λ non linear equations: 1 −λxβ (α(λ, ˆ β ) − 1 − α(λ, ˆ β ) log(α(λ, ˆ β )) − e i =0 1+ (α(λ, ˆ β ) − 1) log(α(λ, ˆ β )) n i=1 n

and n n n β n + log xi + λ log(α(λ, ˆ β )) e−λxi xiβ log xi − λ xiβ log xi = 0 β i=1 i=1 i=1

(32)

Numerical techniques such as Newton–Raphson method may be used to solve the non ˆ and βˆ are obtained, then α(λ, linear equations in Equation (32). Once λ ˆ β ) can be obtained

10246

M. NASSAR ET AL.

Table . Average values of MLEs and the corresponding MSEs (n = 100). Parameters

MLE

λ

α

β

ˆ λ

αˆ

βˆ

ˆ λ

αˆ

βˆ



.

. .   . .  

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .



. .  

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .



. .  

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

.

. .   . .   . .   . .  

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

.


MSE



.





ˆ βˆ ). For interval estimation and hypothesis tests on the model from Equation (31) as αˆ = α( ˆ λ, parameters, the observed information matrix is required as follows: ⎤ ⎡ Jαα Jαλ Jαβ J(ω) = ⎣ Jλα Jλλ Jλβ ⎦ Jβα Jβλ Jββ ∂ ˆ These derivatives are given in the Appendix. ˆ b). where Jab = − ∂a∂b (a, 2

4.2. Simulation study In this subsection a simulation study has been performed using Mathcad 2007 to illustrate the behavior of the MLEs in terms of the sample size n. The number of Monte Carlo replications is 1000. Two different sample sizes n = 50 and 100 were generated from APW distribution using Equation (7). We consider the values λ = (1, 2), α = (0.5, 1.5, 3, 5), and β = (0.5, 1.5, 3, 5). The simulation results are obtained for a total of 32 parameter combinations. In each setting, we obtain the average values of MLEs and the corresponding empirical mean squared errors (MSEs). The average values of the MLEs and MSEs are reported in Tables 2 and 3, respectively. Based on the results shown in Tables 2 and 3, it can be seen that


(b) 1.0

0.004

(a)

0.6 0.0

0.000

0.2

0.4

Survival Function

0.002 0.001

Density

0.003

0.8

Empirical Survival Function Fitted Survival Function

0

500

1000

1500

0

500

x

1000

1500

x

Figure . (a) The relative histogram and the fitted APW distribution. (b) The fitted APW survival function and empirical survival function for first data set. (b) 1.0

(a)

0.8 0.6 0.4

Survival.Function

1.0

0.0

0.2

0.5

Density

1.5

Empirical Survival Function Fitted Survival Function

0.0


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0.5

1.0

1.5 x

2.0

0.5

1.0

1.5

2.0

x

Figure . (a) The relative histogram and the fitted APW distribution. (b) The fitted APW survival function and empirical survival function for second data set.

the estimates are stable and quite close to the true parameter values for these sample sizes. As the sample size increases the MSE decreases in all cases. It can be due to the fact that the sample size n plays an important role in determining the efficiency of the parameters because when n increases some additional information is gathered.

5. Applications In this section, we provide two applications to two real data sets to prove the importance and flexibility of the APW distribution. The first data set corresponding to intervals in days between 109 successive coal-mining disasters in Great Britain, for the period 1875–1951, published by Maguire et al. (1952). The sorted data are given as follows: 1, 4, 4, 7, 11, 13, 15, 15, 17, 18, 19, 19, 20, 20, 22, 23, 28, 29, 31, 32, 36, 37, 47, 48, 49, 50, 54, 54, 55, 59, 59, 61, 61, 66, 72, 72, 75, 78, 78, 81, 93, 96, 99, 108, 113, 114, 120, 120, 120, 123, 124, 129, 131, 137, 145, 151, 156, 171, 176, 182, 188, 189, 195, 203, 208, 215, 217, 217, 217, 224, 228, 233, 255, 271, 275,

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(b) Q−Q Plot for APW distribution

2.0 0.5

0

1.0

1.5

Sample Quantile

1000 500

Sample Quantile

1500

(a) Q−Q Plot for APW distribution

0

500

1000

1500

1.0

1.5

2.0

Theoretical Quantile

Figure . Q-Q plot for the APW distribution for data  and data , respectively.

(b) P−P Plot for APW distribution

0.6 0.2

0.4

Empirical Cumulative Distribution

0.6 0.4 0.2

Empirical Cumulative Distribution

0.8

0.8

1.0

1.0

(a) P−P Plot for APW distribution

0.0

0.0


Theoretical Quantile

0.0

0.2

0.4

0.6

Theoretical Cumulative Distribution

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Theoretical Cumulative Distribution

Figure . P-P plot for the APW distribution for data  and data , respectively.

275, 275, 286, 291, 312, 312, 312, 315, 326, 326, 329, 330, 336, 338, 345, 348, 354, 361, 364, 369, 378, 390, 457, 467, 498, 517, 566, 644, 745, 871, 1312, 1357, 1613, 1630. The second data set is obtained from Smith and Naylor (1987). The data are the strengths of 1.5 cm glass fibers, measured at the National Physical Laboratory, England. The data are as follows: 0.55, 0.93,1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 2, 0.74, 1.04, 1.27, 1.39, 1.49, 1.53,1.59, 1.61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.5, 1.54, 1.6, 1.62, 1.66, 1.69,1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62, 1.66, 1.7, 1.77, 1.84, 0.84, 1.24, 1.3, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67, 1.7, 1.78, 1.89. We shall compare the fit of the proposed APW (and its sub-models, namely APE and Weibull (We) distributions) with several other competitive models, namely McDonald Weibull (Mc-W) (Corderio et al. 2014), beta Weibull (BW) (Lee et al. 2007), MW (Ammar and Mazen 2009), the EW (Mudholkar and Srivastava 1993), transmuted Weibull (TW) (Aryall and Tsokos 2011), gamma Lomax (GL)


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(Cordeiro et al. 2015), and Zografos–Balakrishnan log-logistic (ZBLL) (Zografos and Balakrishnan 2009) models with corresponding densities (for x > 0): kβ λ β−1 −λxβ β β k x e (1 − e−λx )αk−1 [1 − (1 − e−λx ) ]b−1 B(α, b) βλβ β−1 −b (λx)β β x e BW f (x) = (1 − e− (λx) )α−1 B(α, b)

Mc − W f (x) =

Table . MLEs (standard errors in parentheses), and the statistics −2(θˆ ),K–S, and p-values for the first data set.


Estimates

Statistics

Model

αˆ

βˆ

ˆ λ

bˆ

kˆ

−2(θˆ )

K–S

p-Value

Mc-W

. (.) . (.) . (.) . (.) . (.) . (.) . (.) . .)) . (.) — (.)

. (.) . (.) . (.) . (.) . (.) . (.) . (.) . (.) — (.) . (.)

. (.) . (.) . (.) . (.) . (.) . (.) . (.) . (.) .

. (.) . (.) —

. (.) —

.

.

.

.

.

.

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

.

—

—

.

.

.

BW APW MW EW TW GL ZBLL APE We

Table . MLEs (standard errors in parentheses), and the statistics −2(θˆ ), K–S, and p-values for the second data set. Estimates

Statistics

Model

αˆ

βˆ

ˆ λ

bˆ

kˆ

−2(θˆ )

K–S

p-Value

Mc-W

. (.) . (.) . (.) . (.) . (.) . (.) . (.) . (.) . (.) — (.)

. (.) . (.) . (.) . (.) . (.) . (.) . (.) . (.) — (.) . (.)

. (.) . (.) . (.) . (.) . (.) . (.) . (.) . (.) .

. (.) . (.) —

. (.) —

.

.

.

.

.

.

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

—

—

.

.

.

.

BW APW MW EW TW GL ZBLL APE We

.

. .

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MW f (x) = (α + λβxβ−1 )e−αx−λx


β

β β

EW f (x) = αβλβ xβ−1 e−(λx) (1 − e−(λx) )α−1 β x β−1 −( x )β x β TW f (x) = e λ (1 − α + 2αe−( λ ) ) λ λ α−1 βλβ λ GL f (x) = −β log λ+x (α)(λ + x)β+1 −2 x β α−1 x β β xβ−1 1 + log 1 + ZBLL f (x) = β λ (α) λ λ 1 a−1 α, β, λ, b, k > 0, B(a, b) = 0 w (1 − w)b−1 dw is the beta function, and (α) = where ∞ α−1 −y y e dy is the gamma function. 0 Tables 4 and 5 list the MLEs (and the corresponding standard errors in parentheses) of the parameters of all the models and the statistics: −2(θˆ ) (where (θˆ ) denotes the log-likelihood function evaluated at the MLEs), Kolmogorov–Smirnov (K–S) and p-values for the first and the second data sets, respectively. Since the APW distribution has the lowest −2(θˆ )and K–S values and largest p-values among all the other models, it could be chosen as the best model. The relative histogram and the fitted APW distribution of the first and second data sets are displayed in Figures 3a and 4a, respectively. Also, the plots of the fitted APW survival function and empirical survival function of the first and second data sets are displayed in Figures 3b and 4b, respectively. Figure 5a and b and Figure 6a and b present the Q-Q and P-P plots for data 1 and data 2, respectively, which allows us to compare the empirical distribution of the data with the APW distribution. These graphical goodness-of-fit methods also support the results in Tables 4 and 5.

6. Conclusions In this paper, we propose a new three-parameter model, called the APW distribution, which extends the Weibull distribution. In fact, the APW distribution is motivated by the wide use of the Weibull distribution in reliability theory and also for the fact that the generalization provides more flexibility to analyze lifetime data. We derive explicit expressions for the ordinary moments and moment generating function, order statistics, entropies, mean residual life, and mean waiting time. The estimation of the model parameters is approached by maximum likelihood and the observed information matrix is derived. Two applications illustrate that the proposed model provides consistently better fit than the other competitive models. We hope that the proposed model may attract wider applications in statistics.

Acknowledgments The authors are grateful for the comments and suggestions by the referee and the editor. Their comments and suggestions have greatly improved the paper.

Appendix −

∂ 2 n n[[α(α − 1) log α(− log α] − [(α − 1 − (α log α)(α + 2α log α − (1 + log α)]] = 2 − ∂α 2 α [α(α − 1) log α]2


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2β β ∂ 2 n = 2 + log α xi e−λxi 2 ∂λ λ i=1

n −λxβ e i − i=1 2 α n n n β ∂ 2 n 2β −λxiβ −λxiβ − 2 = 2 − λ log α 2xi e log xi − 2λ xi e log xi xiβ log xi + 2λ ∂β β i=1 i=1 i=1 n

−

−

n ∂ 2 1 β −λxiβ x e =− ∂α∂λ α i=1 i

−

n ∂ 2 λ β −λxiβ x e log xi =− ∂α∂β α i=1 i


and n n n 2β β ∂ 2 β = log α λ xi e−λx log xi − xiβ e−λxi log xi + xiβ log xi − ∂λ∂β i=1 i=1 i=1

References Alzaatreh, A., Famoye, F., Lee, C. (2014). T-normal family of distribution: A new approach to generalize the normal distribution. J. Stat. Distrib. Appl. 1:1–16. Ammar, M., Mazen, Z. (2009). Modified Weibull distribution. Appl. Sci. 11:123–136. Arnold, B.C., Balakrishnan, N., Nagarajah, H.N. (2008). A First Course in Order Statistics. New York: John Wiley. Aryall, G.R., Tsokos, C.P. (2011). Transmuted Weibull distribution: A generalization of the Weibull probability distribution. Eur. J. Pure Appl. Math. 2:89–102. Cordeiro, G.M., Ortega, E.M.M., Popovic, B.V. (2015). The gamma-Lomax distribution. J. Stat. Comput. Simul. 85:305–319. Cordeiro, G.M., Edwin, M.M., Lemonte, A.J. (2014). The exponential-Weibull lifetime distribution. J. Stat. Comput. Simul. 84:2592–2606. Corderio, G.M., Hashimoto, E.H., Ortega, E.M.M. (2014). The McDonald Weibull model. Statistics 48:256–278. Jalmar, M.F., Edwin, M.M., Cordeiro, G.M. (2008). A generalized modified Weibull distribution for lifetime modeling. Comput. Stat. Data Anal. 53:450–462. Lee, C., Famoye, F., Alzaatreh, A. (2013). Methods for generating families of continuous distribution in the recent decades. Wiley Interdiscip. Rev.: Comput. Stat. 5(3):219–238. Lee, C., Famoye, F., Olumolade, O. (2007). Beta Weibull distribution: Some properties and applications to censored data. J. Mod. Appl. Stat. Methods 6:173–186. Maguire, B.A., Pearson, E., Wynn, A. (1952). The time intervals between industrial accidents. Biometrika 168–180:1952. Mahdavi, A., Kundu, D. (2015). A new method for generating distributions with an application to exponential distribution. Commun. Stat. – Theory Methods. 46(13). doi.10.1080/03610926.2015.1130839 Mudholkar, G.S., Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failurereal data. IEEE T. Reliab. 42:299–302. Mudholkar, G. S., Srivastava, D.K., Friemer, M. (1995). The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics 37:436–445. Mudholkar, G.S., Srivastava, D.K., Kollia, G.D. (1996). A generalization of the Weibull distribution with application to the analysis of survival data. J. Am. Stat. Assoc. 91:1575–1583. Saboor, A., Bakouch, H.S., Khan, M.N. (2016). Beta Sarhan–Zaindin modified Weibull distribution. Appl. Math. Modell. 40(14):6604–6621.

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Smith, R.L., Naylor, J.C. (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Appl. Stat. 36:358–369. Von Mises, R. (1936). La distribution de la plus grande de n valeurs. Rev. Math. Union Interbalcanique 1:141–160. Xie, M., Lai, C.D. (1995). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab. Eng. Syst. Saf. 52:87–93. Xie, M., Tang, Y., Goh, T.N. (2002). A modified Weibull extension with bathtub failure rate function. Reliab. Eng. Syst. Saf. 76:279–285. Zografos, K., Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions and associated inference. Stat. Methodol. 6:344–362.