Modified Beta Modified-Weibull Distribution

Noname manuscript No. (will be inserted by the editor)

Modified Beta Modified-Weibull Distribution Abdus Saboor · Muhammad Nauman Khan · Gauss M. Cordeiro · Marcelino A.R. Pascoa · Juliano Bortolini · Shahid Mubeen

Received: date / Accepted: date

Abstract We introduce a flexible modified beta modified–Weibull model, which can accommodate both monotonic and non–monotonic hazard rates such as a useful long bathtub shaped hazard rate in the middle. Several distributions can be obtained as special cases of the new model. We demonstrate that the new density function is a linear combination of modified–Weibull densities. We obtain the ordinary and central moments, generating function, conditional moments and mean deviations, residual life functions, reliability measures and mean and variance (reversed) residual life. The method of maximum likelihood and a Bayesian procedure are used for estimating the model parameters. We compare the fits of the new distribution and other competitive models to two real data sets. We prove empirically that the new distribution gives the best fit among these distributions based on several goodness–of–fit statistics. Keywords Bayesian analysis · Modified beta distribution · Moments · Simulation study First and Second Author Department of Mathematics, Kohat University of Science & Technology, Kohat, Pakistan E-mail: [email protected], and E-mail: [email protected] Third Author Departamento de Estat´ıstica, Universidade Federal de Pernambuco, 50740-540, Brazil E-mail: [email protected] Fourth and fifth Author Departamento de Estat´ıstica, Universidade Federal de Mato Grosso, 78075-850, Brazil E-mail: [email protected], and E-mail: [email protected] Sixth Author Department of Mathematics, Sargodha University, Pakistan E-mail: [email protected]

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Abdus Saboor et al.

1 Introduction The Weibull distribution is a very popular distribution for modeling lifetime data with monotone hazard rates. When modeling monotone hazard rates, the Weibull distribution may be an initial choice because of its negatively and positively skewed density shapes. However, the shortcomings of the Weibull distribution in terms of modeling phenomenon with non–monotone hazard rates such as bathtub shaped with long useful life in middle period and unimodal failure rates which are common in reliability and biological studies are very common. To cope with bathtub shaped hazard rate, several new classes of distributions were proposed based on modifications of the Weibull distribution. A good review of some of these models is presented in Pham and Lai (2007) and the references therein. Adding parameters to an existing distribution enables one to obtain classes of more flexible distributions. Nadarajah et al. (2014) introduced an interesting method for adding three new parameters to a baseline distribution. The generated distribution provides more flexibility to model many types of data in terms of its various hazard rate shapes and its further skewness. Let G(x) be the baseline cumulative distribution function (cdf). The cdf of their generated family is given by 1 F (x) = B(a, b)

Z

cG(x) (c−1)G(x)+1

0

xa−1 (1 − x)b−1 dx,

(1)

where a > 0, b > 0 and c > 0 are additional parameters, whose role is to introduce skewness, tail weight and modality of the generated distribution R1 (Nadarajah et al., 2014) and B(a, b) = 0 ta−1 (1 − t)b−1 dt is the beta function. Their family generalizes the beta–family of distributions generated by beta random variables (Jones, 2004) when c = 1. Recently, various forms of beta–compounded distributions are investigated for c = 1. For example, beta exponential (Nadarajah and Kotz, 2006), beta modified Weibull (Silva et al., 2010), beta generalized Weibull (Singla et al., 2011), beta–exponentiated Weibull (Cordeiro et al., 2013a), beta–Weibull geometric (Cordeiro et al., 2013), beta generalized normal (Cintra et al., 2014) and beta Sarhan–Zaindin modified–Weibull (Saboor et al., 2016), among others. There are only two forms of modified beta–compounded distributions: the first one is the modified beta normal pioneered by Nadarajah et al. (2014) and the second is the modified beta Weibull introduced by Khan (2015). The probability density function (pdf) corresponding to (1) is given by f (x) =

ca g(x){G(x)}a−1 {1 − G(x)}b−1 . B(a, b){1 − (1 − c)G(x)}a+b

(2)

The modified–Weibull (MW) distribution defined by Sarhan and Zaindin (2009) has cdf and pdf (for x > 0) given by k

G(x) = 1 − e−λ x−β x , λ > 0 , β > 0 , k > 0,

Modified Beta Modified-Weibull Distribution

3

Table 1 Some special cases of the MBMW distribution Parametric values in (4) a=b=c=1 λ=0 c=1 c = 1 and λ = 0 c = 1 and β = 0

Sub-model Modified Weibull (Sarhan and Zaindin, 2009) Modified Beta Weibull (Khan, 2015) Beta Sarhan–Zaindin modified–Weibull (Saboor et al., 2016) Beta Weibull (Lee et al., 2007) Beta exponential (Nadarajah and Kotz, 2006)

and k g(x) = λ + β k xk−1 e−λ x−β x ,

respectively. In this paper, we proposed a new model called the modified beta modified– Weibull (MBMW) distribution by taking the MW distribution for the baseline model. So, the cdf and pdf of the MBMW distribution are defined by F (x) =

B(1 + (−c e β x

k

+λ x

+ c − 1)−1 ; a, b) , x > 0, B(a, b)

(3)

and b a−1 k k e−β x −λ x f (x) = ca k β xk−1 + λ 1 − e−β x − λ x o−a−b n k 1 − (1 − c) 1 − e−β x −λ x , (4) × B(a, b) Rz respectively, where B(z ; a, b) = 0 ta−1 (1 − t)b−1 dt is the incomplete beta function. Henceforth, we denote by X ∼MBWM(λ, β, k, a, b, c) a random variable following (4). The proposed distribution can accommodate both decreasing and increasing hazard rates as well as bathtub shaped hazard rates. Moreover, it contains several sub–models, some of them summarized in Table 1. By using (3) and (4), the hazard rate function (hrf) of X reduces to b a−1 k k e−β x −λ x h(x) = ca k β xk−1 + λ 1 − e−β x −λ x o−a−b n k 1 − (1 − c) 1 − e−β x −λ x × , B(a, b) − B(1 + (−c eβ xk +λ x + c − 1)−1 ; a, b)

(5)

where λ > 0 , β > 0 , k > 0 , a > 0 , b > 0 , c > 0. Equations (3) to (5) can be easily evaluated numerically using computational packages such as Mathematica, Maple, MATLAB and R. Plots of the pdf (4) and hrf (5) are displayed in Figures 1-4 for selected parameter values.

4

Abdus Saboor et al. Λ=0.6, k=1.3, a=5, b=2, c=3

Β=15, k=9, a=0.4, b=0.9, c=0.2 8

3.5 Λ = 0.3, Λ = 1.3, Λ = 2.0, Λ = 3.0,

3.0 2.5

Dotted Dashed Thin Thick

Β = 1.0, Dotted Β = 2.0, Dashed Β = 5.0, Thin Β = 10, Thick

6

2.0 4

1.5 1.0

2

0.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 1 Density plots of the MBMW distribution Λ=2, Β=1.5, k=0.8, b=1, c=0.6

Λ=15, Β=13, a=1.8, b=0.1, c=1.1 1.4

a = 0.5, Dotted a = 1.2, Dashed a = 2.0, Thin a = 5.0, Thick

4

1.2 k = 5.5, Dotted k = 8.0, Dashed k = 10, Thin k = 13, Thick

1.0 0.8

3

2

0.6 0.4

1

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.5

1.0

1.5

2.0

Fig. 2 Density plots of the MBMW distribution Λ=0.5, Β=1.2, k=6, a=0.9, c=1

Λ=0.7, Β=0.6, k=0.5, a=1, b=1.2

2.5

1.4

2.0

b = 0.3, Dotted b = 0.9, Dashed b = 1.6, Thin b = 2.5, Thick

1.5

c = 0.3, c = 1.0, c = 2.5, c = 5.0,

1.2 1.0


0.8 0.6

1.0

0.4 0.5

0.2

0.0

0.5

1.0

1.5

2.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fig. 3 Density plots of the MBMW distribution

Figures 1–3 indicate how the parameters λ, β, k, a, b and c affect the MBMW density and show flexibility of the density shapes, where skewness, heavy tails and modality can be observed. These plots illustrate the versatility of the MBMW distribution. Figure 4 represents increasing, decreasing and bathtub shaped hrfs. The rest of the article is organized as follows. Some explicit expressions for statistical functions of the new distribution are provided in Section 2 including moments, generating function, conditional moments and mean deviations. We also provide alternative formulae in terms of Meijer’s G–function for some of statistical functions. In Section 3, we obtain the residual life and reversed resi-


5 Λ=2.3, Β=0.1, k=12, a=0.3, b=0.9, c=0.5

Λ=0.1, Β=0.8, a=1.3, b=1.9, c=0.9 20

2.5

2.0 15

1.5

k = 0.1, k = 0.4, k = 1.3, k = 1.6,

1.0


10

5

0.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.5

1.0

1.5

Fig. 4 Hazard plots of the MBMW distribution

dual life and other mathematical properties for the proposed distribution such as survival function, hazard rate, mean and variance. Moreover, Bonferroni and Lorenz curves and scaled total time on test are determined as reliability measures. In Section 4, we obtain further structural properties. We estimate the model parameters by maximum likelihood and a Bayesian procedure in Section 5. We prove in Section 6 the flexibility of the new distribution for modeling lifetime data by means of two real data sets. Section 7 offers some concluding remarks. 2 Statistical Functions of the MBMW Distribution By using the power series (for |z| < 1, τ > 0) (1 − z)−τ =

∞ ∞ X X τ −1 m Γ (τ + n) n z , (−1)m z and (1 − z)τ −1 = Γ (τ ) n! m m=0 n=0

the pdf (4) can be expressed as

where

∞ X k ca Wm λ + k β xk−1 e−λ(m+b)x e−β(m+b)x , f (x) = B(a, b) m=0

Wm =

(6)

∞ X (−1)m Γ (a + b + n)(1 − c)n n + a − 1 . Γ (a + b) n! m n=0

Further, we can rewrite (6) as a linear representation f (x) =

∞ X

Vm gm+b (x),

(7)

m=0

where (for m ≥ 0) Vm = ca Wm /B(a, b)(m+b) and gm+b (x) is the MW density with parameters λ⋆ = (m + b)λ, β ⋆ = (m + b)β and k. So, the MBMW density function is a linear combination of MW densities.

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Abdus Saboor et al.

In the rest of this section, we obtain some statistical functions for the MBMW distribution from equation (6). The resulting expressions can be evaluated exactly or numerically with symbolic computational packages such as Mathematica, R, Matlab or Maple. In numerical applications, the infinite sums can be truncated whenever convergence is observed.

2.1 Moments and Generating Function We obtain explicit expressions for the positive and negative moments and moment generating function (mgf) of X based of the following result: I(s; α, β, γ) =

Z

=

∞ γ

xs e−β x e−αx dx 0

1 γβ (s+1)/γ

m ∞ X α (−1)m s+m+1 , Γ m! γ β 1/γ m=0

(8)

where Γ (·) is the gamma function. Applying the standard formula for the rth ordinary moment, say µ′r = E(X r ), and using (6) and (8), we have µ′r

i ∞ X ca (−1)i (m + b)λ = Wm B(a, b) m,i=0 i! {(m + b)β}1/k " # r+k+i β Γ λ Γ r+i+1 k k . + × r +1 {(m + b)β} k k {(m + b)β}(r+1)/k

(9)

The hth order negative moment can readily be determined by replacing r with −h in (9). The mgf of X can be expressed as Z ∞ ∞ X k ca M (t) = Wm et x λ + k β xk−1 e−(m+b) λ x−(m+b) β x dx, B(a, b) m=0 0 and then using (8) becomes i ∞ X (−1)i (m + b)λ − t ca Wm M (t) = B(a, b) m,i=0 i! {(m + b)β}1/k " # k β Γ 1 + ki λ Γ i+1 k + . × k {(m + b)β} k {(m + b)β}1/k

(10)

We now provide alternative formulae for the moments and mgf of X in terms of the Meijer’s G–function defined in Appendix A, which can be coded in Mathematica. These formulae can be easily implemented in analytical programs.


7

First, we consider the following result (Saboor et al., 2012): Z ∞ k (2π)1−(q+p)/2 q1/2 p η−1/2 xη−1 e−θ x es x dx = (−s)η 0 ! p p θ q 1 − i+η , i = 0, 1, ...., p − 1 p , − × Gq,p p,q j/q , j = 0, 1, ...., q − 1 s q

(11)

where ℜ(η), ℜ(θ), ℜ(s) < 0 and k is a rational number such that k = p/q, where p and q 6= 0 are integers. By combining (6) and (11), the rth ordinary moment and the mgf of X can be expressed in terms of the Meijer’s G–functions, respectively, as E(X r ) = × Gq,p p,q

∞ ca λ (2π)1−(q+p)/2 q 1/2 p r+1/2 X Wm B(a, b) ((m + b) λ)r+1 m=0 ! p q (m + b) β p , i = 0, 1, . . . , p − 1 1 − i+r+1 p j/q , j = 0, 1, . . . , q − 1 (m + b) λ q

ca β k (2π)1−(q+p)/2 q 1/2 p r+k−1/2 X Wm B(a, b) ((m + b) λ)r+k m=0 ! p q (m + b) β p , i = 0, 1, . . . , p − 1 1 − i+r+k q,p p × Gp,q j/q , j = 0, 1, . . . , q − 1 (m + b) λ q ∞

+

and

ca λ (2π)1−(q+p)/2 q 1/2 p 1/2 X Wm B(a, b) (m + b) λ − t m=0 ! p q i+1 (m + b) β p 1 − p , i = 0, 1, . . . , p − 1 j/q , j = 0, 1, . . . , q − 1 (m + b) λ − t q ∞

M (t) = × Gq,p p,q

∞ ca β k (2π)1−(q+p)/2 q 1/2 p k−1/2 X Wm k B(a, b) ((m + b) λ − t) m=0 ! p q i+k p (m + b) β , i = 0, 1, . . . , p − 1 1 − p × Gq,p . p,q j/q , j = 0, 1, . . . , q − 1 (m + b) λ − t q

+

We shall use the following lemma for J(x; r, a, b, c, λ, β, k) = in the rest of the paper.

Rx 0

y r f (y) dy

Lemma 1 We can write J(x; r, a, b,c, λ, β, k) = ×

Z

0

x

∞ X ∞ X ca (−1)i Wm ((m + b)λ)i B(a, b) m=0 i=0 i!

k y i+r λ + β k y k−1 e−(m+b) β y dy,

r = 1, 2, . . . .

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Abdus Saboor et al.

Then, we have ∞ X (−1)i ca i Wm ((m + b)λ) B(a, b) i! m,i=0 " i+r+1 λ k × , (m + b)βx i+r+1 γ k k ((m + b)β) k # i+r β k γ , + + 1, (m + b)βx i+r +1 k ((m + b)β) k

J(x; r, a, b, c, λ, β, k) =

Rb P∞ (−1)k bα+k where γ (α, b) = 0 tα−1 e−t dt = k=0 k! α+k is the lower incomplete gamma function. In terms of Meijer’s G function, we obtain ∞ X (−1)i ca Wm ((m + b)λ)i B(a, b) m,i=0 i! ! 1−i−r 1−i−r , , . . . , , − ((m + b) β)q xp −i−r q,p p p p Gp,p+q −i−r−1 p−i−r−2 , i+r 0, qq p p ,..., p

J(x; r, a, b, c, λ, β, k) =

×

+

β xp (i+r+k) (2 π)(q−1)/2

×

Gq,p p,p+q

λ q xp (i+r+1) p (2 π)(q−1)/2

!# , 2−i−r−k , . . . , p−i−r−k ,− ((m + b) β)q xp −i−r−k+1 p p p . 0 , −i−r−k , i+r+k−1 , . . . , p−i−r−k−1 qq p p p

Proof The first part is easily established. For any arbitrary function g(x), the second part can fallow from e−g(x) = G1,0 0,1

! − . g(x) 0

Let k = p/q, where p ≥ 1 , q ≥ 1 are natural co-prime numbers. The lemma is proved by using the identity Z

0

x

y

t

G1,0 0,1

! qxp (t+1) − (m + b) βy dy = 0 p(2π)(q−1)/2 ! p−t−1 1−t ,− ((m + b) β)q xp −t q,p p , p ,..., p × Gp,p+q , −t−1 t 0 , p , p , . . . , p−t−2 qq p p/q

which comes from Equation (13) of Cordeiro et al. (2014).


9

2.2 Conditional Moments and Mean Deviations In connection with lifetime distributions, it is useful to determine the conditional moments E(X r |X > t) (for r = 1, 2, . . .), which are of interest in predictive inference. The rth conditional moment of X is given by Z t 1 r r E(X ) − x f (x) dx E(X |X > t) = S(t) 0 B(a, b) [E(X r ) − J(t; r, a, b, c, λ, β, k)] . = B(a, b) − B(1 + (−c e β tk +λ t + c − 1)−1 ; a, b) r

The mean deviations provide useful information about the characteristics of a population and they can be determined from the first incomplete moment. Indeed, the amount of dispersion in a population may be measured to some extent by all deviations from the mean and median. The mean deviations of X about the mean µ′1 = E(X) and about the median M can be expressed as δ = 2F (µ′1 ) − 2m1 (µ′1 ) and η = µ′1 − 2m1 (M ), where F (µ′1 ) follows from (3) and Z z x f (x)dx = J(z; 1, a, b, c, λ, β, k). m1 (z) = 0

3 (Reversed) Residual Life Functions with Some Reliability Measures Residual life and reversed residual life random variables are used extensively in risk analysis. Accordingly, we investigate some related statistical functions, such as survival function, mean and variance in connection with the MBMW distribution. The residual life is described by the conditional random variable R(t) := X − t|X > t, t ≥ 0, and defined as the period from time t until the time of failure. Analogously, the reversed residual life can be defined as ¯ (t) := t − X|X ≤ t which denotes the time elapsed from the failure of a R component given that its lifetime ≤ t .

3.1 Properties of Residual Lifetime Function For the MBMW distribution, the survival function of the residual lifetime R(t) (for t ≥ 0) is given by (for x > 0) k

SR(t) (x) =

B(a, b) − B(1 + (−c e β(x+t) +λ(x+t) + c − 1)−1 ; a, b) S(x + t) = . S(t) B(a, b) − B(1 + (−c e βtk +λt + c − 1)−1 ; a, b)

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Abdus Saboor et al.

Then, the pdf of R(t) reduces to a−1 k k fR(t) (x) = ca (β (x + t) + λ) e−b β (x+t) −(x+t) b λ 1 − e−β (x+t) −(x+t) λ o−a−b n k 1 − (1 − c) 1 − e−β (x+t) −(x+t) λ . × B(a, b) − B(1 + (−c e β tk +λ t + c − 1)−1 ; a, b)

Consequently, the hrf of R(t) is given by

a−1 k k hR(t) (x) = ca (β (x + t) + λ) e−b β (x+t) −(x+t) b λ 1 − e−β (x+t) −(x+t) λ o−a−b n k 1 − (1 − c) 1 − e−β(x+t) −(x+t)λ × . k B(a, b) − B(1 + (−c eβ(x+t) +λ(x+t) + c − 1)−1 ; a, b)

Its mean residual life can be expressed as Z ∞ 1 K(t) = E(R(t) ) = x f (x) dx − t 1 − F (t) t 1 [E(X) − J(t; 1, a, b, c, λ, β, k)] − t , = 1 − F (t)

t ≥ 0,

where F (x), f (x) and E(X) are given in (3), (4) and (9), respectively. The variance residual life has also been of some interest in recent years (Gupta and Kirmani, 2000). For the MBMW distribution, we obtain Z ∞ 2 V (t) = V ar(R(t) ) = xS(x)dx − 2tK(t) − K(t)2 S(t) t 1 = E(X 2 ) − J(t; 2, a, b, c, λ, β, k) − t2 − 2tK(t) − K(t)2 , S(t)

where E(X 2 ) is given by (9) and J(t; 2, a, b, c, λ, β, k) is obtained from Lemma 1 by setting r = 2. 3.2 Properties of Reversed Residual Life Function

For the MBMW distribution, the survival function of the reversed residual ¯ (t) is (for 0 ≤ x < t) lifetime R k

SR¯ (t) (x) =

F (t − x) B(1 + (−c e β (t−x) +λ (t−x) + c − 1)−1 ; a, b) , = F (t) B(1 + (−c e β tk +λ t + c − 1)−1 ; a, b)

¯ (t) becomes and then the pdf of R

ia−1 h k k fR¯ (t) (x) = ca [β (t − x) + λ] e−b β (t−x) −(t−x) b λ 1 − e−β (t−x) −(t−x) λ io−a−b n h k 1 − (1 − c) 1 − e−β (t−x) −(t−x) λ × . B(1 + (−c e β tk +λ t + c − 1)−1 ; a, b)


11

¯ (t) reduces to Consequently, the hrf of R ia−1 h k k hR¯ (t) (x) = ca [β (t − x) + λ] e−b β (t−x) −(t−x) b λ 1 − e−β (t−x) −(t−x) λ io−a−b n h k 1 − (1 − c) 1 − e−β (t−x) −(t−x) λ . × k B(1 + (−c e β (t−x) +λ (t−x) + c − 1)−1 ; a, b) ¯ (t) are given by The mean and variance of R ¯(t) ) = t − L(t) = E(R

1 F (t)

Z

t 0

J(t; 1, a, b, c, λ, β, k) , F (t)

x f (x) dx = t −

and ¯(t) ) = 2tL(t) − L(t)2 − W (t) = V ar(R

2 F (t)

= 2tL(t) − L(t)2 − t2 +

Z

t

x F (x) dx

0

J(t; 2, a, b, c, λ, β, k) , F (t)

where F (t), f (x) and J(t; 2, a, b, c, λ, β, k) can be determined from (3), (4) and Lemma 1 (for r = 2), respectively.

3.3 Reliability Curves The Bonferroni and Lorenz curves have various applications not only in economics to study income and poverty but also in other sciences like reliability, insurance and medicine. The Bonferroni curve BF [F (x)] of X is given by Z x J(x; 1, a, b, c, λ, β, k) 1 . y f (y)dy = BF [F (x)] = E (X) F (x) 0 E (X) F (x) The Lorenz curve LF [F (x)] of X reduces to LF [F (x)] =

J(x; 1, a, b, c, λ, β, k) . E (X)

The scaled total time on test transform of a distribution function F (Pundir et Rt 1 ¯ al., 2005) is defined by SF [F (t)] = E(X) 0 F (y)dy, and it is important for the ageing properties of the underlying distribution and can be applied to solve geometrically some stochastic maintenance problems. Therefore, SF [F (t)] is given by SF [F (t)] =

1 {J(t; 1, a, b, c, λ, β, k) E (X) h i k t + B(a, b) − B(1 + (−c e β t +λ t + c − 1)−1 ; a, b) . B(a, b)

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Abdus Saboor et al.

4 Alternative Properties In this section, we provide alternative properties for the MBMW distribution based on the linear representation (7) and the results by Cordeiro et al. (2014).

4.1 Moments The calculations in this sectionR involve some special functions. In particular, ∞ the gamma function Γ (r) = 0 wr−1 e−w dw (r > 0), and other functions given in Appendices A and B. In order to obtain µ′s = E(X s ), we require an integral of the type ⋆

⋆

I(s; λ , β , k) =

Z

∞ ⋆

xs e−(λ

x+β ⋆ xk )

dx.

(12)

0 ⋆

We provide four representations for (12). First, by expanding e−λ series, we obtain ⋆

⋆

I(s; λ , β , k) =

Z ∞ X (−λ⋆ )j j!

j=0

=

1 kβ ⋆ (s+1)/k

x

in Taylor

∞

xs+j e−β

⋆ k

x

dx

0

⋆ j ∞ X (−1)j λ s+1+j . Γ j! k β ⋆ 1/k j=0

The above sum can be expressed in a simple form for k > 1 using the Fox– Wright generalized hypergeometric function defined in Appendix A. We have ⋆

⋆

I(s; λ , β , k) =

1 kβ ⋆ (s+1)/k

1 Ψ0

s+1 1 k , k

−

;−

λ⋆ β ⋆ 1/k

.

(13)

Applying (13) to (7), we can write µ′s =

∞ X

m=0

Vm [λ⋆ I(s; λ⋆ , β ⋆ , k) + β ⋆ k I(s + λ⋆ − 1; λ⋆ , β ⋆ , k)] .

(14)

Second, we offer two formulae for the integral (12) provided that k = p/q, where p ≥ 1 and q ≥ 1 are relatively natural co–prime numbers. We use equation (2.3.2.13) in Prudnikov et al. (1986) to obtain formulae for I(s; λ⋆ , β ⋆ , k) when 0 < k < 1 and k > 1. We exclude the case k = 1 since the model is non-identifiable. For irrational k, an approximation of vanishingly small error can be made using increasingly accurate rational approximations for k. Let z = (pp β ⋆ q )/(q q λ⋆ p ), p Fq (a1 , . . . , ap ; b1 , . . . , bq ; x) be the well-known generalized hypergeometric function and ∆(τ, a) = (a/τ, (a+ 1)/τ, . . . , (a+ τ − 1)/τ ).


13

The generalized hypergeometric function is available in Mathematica. For 0 < k < 1, we obtain ⋆

⋆

I(s; λ , β , k) =

q−1 X (−β ⋆ )j Γ (s + 1 + jp/q) j=0

λ⋆ (s+1+jp/q) j!

× p+1 Fq (1, ∆(p, s + 1 + jp/q); ∆(q, 1 + j); (−1)q z) .

(15)

For k > 1, we have ⋆

⋆

I(s; λ , β , k) =

p−1 X (−1)j q Γ ([s + 1 + j]q/p)

p β ⋆ (s+1+j)q/p j! (−1)p × q+1 Fp 1, ∆(q, [s + 1 + j]q/p); ∆(p, 1 + j); . z j=0

(16)

A fourth representation for the integral (12) also holds when k = p/q, where p ≥ 1 and q ≥ 1 are natural co-prime numbers. It follows in terms of the Meijer Gm,n p,q –function defined in Appendix B and also available in Mathematica. Based on the first equation of the proof of Lemma 1 in Section 4.1, equation (12) can be expressed in the same form of equation (2.24.3.1) given by Prudnikov et al. (1986). Hence, we obtain ⋆ q p −s 1−s β p p , p , . . . , p−s−1 ps+1/2 q,p ⋆ ⋆ p Gp,q . I(s; λ , β , k) = 0 λ⋆ p q q (2π)(p+q)/2−1 λ⋆ s+1 (17) Further, if q = 1, using equation (9.31.2) in (Gradshteyn and Ryzhik, 2007) 1 − bs n,m −1 ar z = G z Gm,n q,p p,q 1 − ar , bs

we have, as a special case of (17), the following result (Cheng et al., 2003) ! 1 ps+1/2 λ⋆ p p,1 ⋆ ⋆ I(s; λ , β , k) = G1,p . (s+1) (s+2) (s+p) β ⋆ pp p , p , . . . , p (2π)(p−1)/2 λ⋆ s+1

Equations (14), (15), (16) and (17) are the main results of this section.

5 Estimation and a Simulation Study The parameters of the MBMW distribution can be estimated from the loglikelihood based on the sample by using the SAS (PROC NLMIXED), R (optim and MaxLik functions) or the Ox program (sub-routine MaxBFGS). Additionally, some goodness-of-fit statistics are given to compare the density estimates and for model selection.

14

Abdus Saboor et al.

5.1 Maximum Likelihood Estimation In order to estimate the model parameters of the MBMW density function defined in Equation (4), the log-likelihood of the sample is maximized with respect to the parameters. Given the data set x1 , · · · , xn , the log-likelihood function for the MBMW model is given by ℓ(λ, β, k, a, b, c) = n [a log(c) − log{B(a, b)}] + b + (a − 1) − (a + b)

n X i=1

n X i=1

log 1 − e

n X i=1

−βxk i −λxi

k log e−βxi −λxi

+

n X i=1

log kβx−1+k +λ i

o n k log 1 − (1 − c)(1 − e−βxi −λxi ) .

(18)

The log-likelihood nonlinear equations can be solved using Newton-Raphson algorithms. They can be obtained from the authors upon request. In order to determine confidence intervals for the model parameters, we require the elements of the 6 × 6 information matrix J(θ) = {Jrs (θ)} (for r, s = λ, β, k, a, b, c). They can also be obtained from the authors upon request. Under standard regularity conditions, the asymptotic distribution of (θˆ − θ) is N6 (0, K(θ)−1 ), where K(θ) = E{J(θ)} is the expected informaˆ −1 ) distribution, tion matrix. The approximate multivariate normal N6 (0, J(θ) ˆ −1 is the inverse observed information matrix evaluated at θ = θ, ˆ where J(θ) can be used to construct approximate confidence intervals for the individual parameters.

5.2 Bayesian Analysis In the Bayesian approach, the information referring to the model parameters is obtained through a posterior marginal distribution. Here, we use the simulation method of Markov Chain Monte Carlo (MCMC) such as the Metropolis– Hastings algorithm. Since we have no prior information from historical data or from previous experiment, we assign conjugate but weakly informative prior distributions to the parameters. Since we assume informative (but weakly) prior distribution, the posterior distribution is a well–defined proper distribution. Here, we assume that the elements of the parameter vector to be independent and consider that the joint prior distribution of all unknown parameters has a density function given by π(λ, β, k, a, b, c) ∝ π(λ) × π(β) × π(k) × π(a) × π(b) × π(c).

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Here, λ ∼ Γ (a1 , b1 ), β ∼ Γ (a2 , b2 ), k ∼ Γ (a3 , b3 ), a ∼ Γ (a4 , b4 ), b ∼ Γ (a5 , b5 ) and c ∼ Γ (a6 , b6 ), where Γ (ai , bi ) denotes a gamma distribution with mean ai /bi and variance ai /b2i . All hyper–parameters are specified. Combining the


15

likelihood function (18) and the prior distribution (19), the joint posterior distribution for the model parameters reduces to n n Y P P ca −βb n xk −λb n xi i i=1 i=1 kβxik−1 + λ e π(λ, β, k, a, b, c|x) ∝ B(a, b) i=1 × ×

n n o−a−b Y k 1 − (1 − c) 1 − e−βxi −xi λ

i=1 n Y

i=1

k

1 − e−βxi −xi λ

a−1

π(λ, β, k, a, b, c).

(20)

The joint posterior density above is analytically intractable because the integration of the joint posterior density is not easy to perform. In this direction, we first obtain the full conditional distributions of the unknown parameters given by π(λ|x, β, k, a, b, c) ∝ e−λb ×

n n Y

i=1

π(β|x, λ, k, a, b, c) ∝ e−βb ×

n n Y

i=1

π(k|x, λ, β, a, b, c) ∝ e−βb ×

n n Y

i=1

Pn

i=1

n Y

xi

n a−1 Y k 1 − e−βxi −xi λ kβxik−1 + λ i=1

i=1

1 − (1 − c) 1 − e Pn

i=1

xk i

n Y

−βxk i −xi λ

i=1

k

1 − (1 − c) 1 − e−βxi −xi λ i=1

xk i

n Y

n Y n i=1

π(β),

i=1

1 − (1 − c) 1 − e

ca π(a|x, λ, β, k, b, c) ∝ B(a, b)

o−a−b

n a−1 Y k kβxik−1 + λ 1 − e−βxi −xi λ

i=1

π(λ),

n a−1 Y k 1 − e−βxi −xi λ kβxik−1 + λ

i=1

Pn

o−a−b

−βxk i −xi λ

k

1 − e−βxi −xi λ

o−a−b

π(k),

a

n n o−a Y k π(a), 1 − (1 − c) 1 − e−βxi −xi λ

i=1

Pn

k

Pn

π(b|x, λ, β, k, a, c) ∝ [B(a, b)] e−βb i=1 xi −λb i=1 xi n n o−b Y k π(b) 1 − (1 − c) 1 − e−βxi −xi λ −n

i=1

and π(c|x, λ, β, k, a, b) ∝ cna

n n o−a−b Y k π(c). 1 − (1 − c) 1 − e−βxi −xi λ

i=1

16

Abdus Saboor et al.

Since the full conditional distributions for the model parameters do not have explicit expressions, we require the use of the Metropolis–Hastings algorithm. 5.3 Simulation Study We evaluate the performance of the maximum likelihood method for estimating the MBMW parameters using Monte Carlo simulation with 1, 000 replications. All results are carried out using the R statistical software. For maximizing the log-likelihood function (18), we use the BFGS method in the optim subroutine. The evaluation of point estimation is performed based on the empirical mean and mean squared error (MSE) for each sample size. Six different sample sizes, n = 50, 100, 200, 400, 800 and 1, 600, are considered for each scenario of the model parameters. The empirical results are listed in Table 2 and displayed in Figure 5. The values in Table 2 and plots of Figure 5 indicate that the empirical estimates from the simulations are usually close to the true parameter values for these sample sizes. We note from the results in Table 2 that the MSEs decrease when the sample size increases as expected. Hence, the maximum likelihood method can be used effectively for estimating the parameters of the MBMW distribution. 5.4 Goodness–of–Fit Tests We adopt some statistics to verify the goodness-of-fit of certain statistical models. They are computed using the symbolic computation package Mathematica. The MLEs are computed with the Nmaximize command. The following goodness–of–fit statistics are considered to compare the fitted models: the Anderson-Darling (A∗ ), Cram´er–von Mises (W ∗ ), Liao-Shimokawa (L–S) and Kolmogorov-Smirnov (K–S) statistics with their p–values. These statistics are used to evaluate how closely a specific distribution with cdf(·) fits the corresponding empirical distribution for a given data set. The distribution with better fit than the others will be the one having the smallest statistics and largest p–value. They are given by # " n X 0.75 1 2.25 (2i − 1) log (zi (1 − zn−i+1 )) , + + 1 −n − A∗ = n2 n n i=1 # 2 "X n 2i − 1 0.5 1 ∗ W = , zi − + +1 n 2n 12n i=1 n i 1 X Max ni − zi , zi − i−1 i−1 n p K–S = Max , L–S = √ − zi , zi − , n n n i=1 zi (1 − zi ) where zi = cdf(yi ) and the yi, s are the ordered observations.


17

Table 2 Empirical means and the MSEs in parentheses. n 50 100 200 400 800 1, 600

n 50 100 200 400 800 1, 600

n 50 100 200 400 800 1, 600

λ = 0.5 ˆ λ

β = 0.2 βˆ

k=5 ˆ k

0.6317 (0.0518) 0.5956 (0.0330) 0.5609 (0.0185) 0.5385 (0.0080) 0.5116 (0.0005) 0.5042 (< 0.0001) λ = 0.25 ˆ λ 0.2193 (0.0150) 0.2453 (0.0043) 0.2502 (0.0012) 0.2490 (0.0006) 0.2504 (0.0001) 0.2501 (< 0.0001) λ = 1.5 ˆ λ 1.5769 (0.3551) 1.5122 (0.2399) 1.4983 (0.1367) 1.4969 (0.0646) 1.5023 (0.0043) 1.5002 (0.0007)

0.1618 (0.0058) 0.1676 (0.0035) 0.1758 (0.0020) 0.1834 (0.0010) 0.1973 (0.0001) 0.1999 (< 0.0001) β = 0.5 βˆ 0.4251 (0.0388) 0.4635 (0.0266) 0.4885 (0.0160) 0.4939 (0.0072) 0.5022 (0.0005) 0.5005 (< 0.0001) β = 0.5 βˆ 0.4521 (0.0327) 0.4650 (0.0199) 0.4749 (0.0118) 0.4864 (0.0055) 0.5046 (0.0005) 0.5031 (< 0.0001)

5.6265 (1.1144) 5.3297 (0.4648) 5.1730 (0.2356) 5.0812 (0.1226) 5.0097 (0.0164) 5.0029 (0.0042) k=2 ˆ k 2.3379 (0.2571) 2.2132 (0.1099) 2.1217 (0.0452) 2.0830 (0.0232) 2.0367 (0.0044) 2.01801 (0.0010) k=8 ˆ k 8.5764 (2.7425) 8.3742 (1.3913) 8.2420 (0.7150) 8.1494 (0.3369) 7.9931 (0.0543) 8.0372 (0.0136)

a=2 a ˆ 2.3375 (0.5030) 2.3162 (0.3305) 2.2514 (0.1887) 2.1998 (0.0884) 2.0613 (0.0074) 2.0250 (0.0013) a = 0.9 a ˆ 0.9216 (0.0540) 0.9433 (0.0293) 0.9402 (0.0150) 0.9310 (0.0068) 0.9089 (0.0011) 0.9063 (0.0002) a = 0.5 a ˆ 0.5508 (0.0140) 0.5367 (0.0072) 0.5275 (0.0036) 0.5238 (0.0020) 0.5151 (0.0004) 0.5075 (< 0.0001)

b = 1.75 ˆ b 2.0487 (0.5859) 2.0326 (0.3852) 1.9785 (0.2160) 1.9206 (0.0958) 1.7983 (0.0061) 1.7728 (0.0001) b = 0.5 ˆ b 0.5089 (0.0207) 0.4884 (0.0128) 0.4841 (0.0075) 0.4942 (0.0025) 0.4962 (0.0004) 0.5010 (< 0.0001) b = 0.5 ˆ b 0.5840 (0.0309) 0.5551 (0.0188) 0.5314 (0.0097) 0.5179 (0.0042) 0.5116 (0.0004) 0.5067 (< 0.0001)

c = 0.5 cˆ 0.5176 (0.0344) 0.5139 (0.0216) 0.5082 (0.0128) 0.5070 (0.0061) 0.5037 (0.0004) 0.5020 (< 0.0001) c=3 cˆ 4.2468 (2.0908) 3.9568 (1.3059) 3.6792 (0.7234) 3.4739 (0.3278) 3.0998 (0.0193) 3.0291 (0.0031) c = 0.5 cˆ 0.5442 (0.0465) 0.5404 (0.0314) 0.5364 (0.0181) 0.5284 (0.0083) 0.5049 (0.0005) 0.5023 (< 0.0001)

6 Applications

Here, we prove the potentiality of the MBMW distribution by means of two real data sets from hydrology and medical fields using both maximum likelihood estimation and Bayesian methods.

18

Abdus Saboor et al.

n = 50

2.0

n = 50

0.8

f(x)

1.0

f(x)

1.0

0.4

0.5

0.5

0.2

0.0

0.0 0.0

0.5

1.0 x

1.5

2.0

0.0 0

1

2

3

4

5

n = 100

0.8

0.0 1.0 x

1.5

2.0

2.0

1

2

3

4

5

n = 200

0.8

0.2

0.0

0.0 1.0 x

1.5

2.0

2

3

4

5

n = 400

0.0 1.0 x

1.5

2.0

2

3

4

5

n = 800

0.0 1.0 x

1.5

2.0

n = 1600

2.0

2

3

4

5

x

1.0 x

n = 1600

n = 1600

0.0 0.0

0.5

1.0 x

1.5

2.0

0.5

2.0

1.5

2.0

f(x)

1.0

f(x)

f(x)

0.2

0.0

0.0

1.5

0.4

0.5

1.5

0.0 1

0.6

1.0

2.0

0.5

0

0.8

1.5

1.0 x

n = 800

f(x)

f(x)

f(x)

0.2

0.0 0.5

0.5

1.0

0.4

0.5 0.0

0.0

1.5

0.6

1.0

1.5

0.0 1

0.8

1.5

2.0

0.5

0

x

n = 800

2.0

1.0 x

n = 400

f(x)

f(x)

f(x)

0.2

0.0 0.5

0.5

1.0

0.4

0.5 0.0

0.0

1.5

0.6

1.0

1.5

0.0 1

0.8

1.5

2.0

0.5

0

x

n = 400

2.0

1.0 x

n = 200

f(x)

0.4

0.5 0.5

0.5

1.0

f(x)

1.0

0.0

0.0

1.5

0.6

f(x)

1.5

1.5

0.0 0

x

n = 200

2.0

0.5

0.2

0.0

1.5

f(x)

0.4

0.5

1.0 x

n = 100

1.0

f(x)

1.0

0.5

0.5

1.5

0.6

f(x)

1.5

0.0

0.0

x

n = 100

2.0

n = 50

1.5

0.6

f(x)

1.5

0.5 0.0

0

1

2

3

4

5

0.0

x

0.5

1.0 x

Fig. 5 Plots of the MBMW simulated and theoretical densities for sample sizes n = 50, 100, 200, 400, 800 and 1, 600. Left panel: parameters λ = 0.5, β = 0.2, k = 5, a = 2, b = 1.75 and c = 0.5. Center panel: parameters λ = 0.25, β = 0.5, k = 2, a = 0.9, b = 0.5 and c = 3. Right panel: parameters λ = 1.5, β = 0.5, k = 8, a = 0.5, b = 0.5 and c = 0.5.

6.1 Maximum likelihood estimation In this section, the MBMW model is compared with the following distributions by means of two data sets: – Gumbel with density function (for x, µ ∈ R, σ > 0) f (x) =

e−e

−

x−µ σ

− x−µ σ

σ

;

– Modified-Weibull (MW) (Sarhan and Zaindin, 2009) with density function (for x > 0, λ, β, k > 0) k f (x) = λ + β k xk−1 e−λ x−β x ;

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– Beta exponential distribution (BE) (Nadarajah and Kotz, 2006) with density function (for x > 0, a, b, λ > 0) λ e−bλ x 1 − e−λx f (x) = B(a, b)

a−1

;


19

– Beta modified Weibull (BMW) (Silva et al., 2010) with density function (for x > 0, a, b, α, γ > 0 , λ ≥ 0) αxγ−1 (γ + λx)eλx f (x) =

1 − e−αx B(a, b)

γ λx

e

a−1

e−bαx

γ λx

e

;

– Beta exponentiated Weibull (BEW) (Cordeiro et al., 2013a) with density function (for x > 0, a, b, c, λ, β > 0) α cλc xc−1 e−(λ x) f (x) =

c

1 − e−(λ x)

c

α a−1 c α b−1 1 − 1 − e−(λ x)

B(a, b)

.

The applications refer to the event runoff and cancer patients data sets as follow: – The first data set is reported in Kang et al., (2013) and represents the event runoff data (mm) containing 34 observations as one of hydrological characteristics of a study storms observed from a small watershed in Korea (west of city of Suwon), named Baran and abbreviated as HP#6. The data were available since 1996. – The second uncensored data set was previously studied by Lee and Wang (2003) and represents the remission times (in months) of a random sample of 128 bladder cancer patients. Bladder cancer is a disease in which abnormal cells multiply without control in the bladder. The most common type of bladder cancer recapitulates the normal histology of the urothelium and is known as transitional cell carcinoma. The maximum likelihood method is used for estimating the parameters of the above distributions and the parameter estimates with their standard errors (computed by inverting the observed information matrices) are given in Table 3 for runoff data and Table 5 for cancer data. Further, the goodness–of–fit statistics are obtained for each distribution and listed at Tables 4 and 6 for both data sets. From the figures in Tables 4 and 6, we note that the MBMW distribution has the smallest statistics with the largest p–values. Then, the MBMW distribution gives the best fit among the compared distributions. We also use the likelihood ratio (LR) statistic to compare the MBMW and MW distributions for both data sets. Under the null model, the LR statistic follows the chisquare distribution (asymptotically) with 3 degree of freedom (the number of additional parameters in the wider model). Table 7 gives the LR statistics with their p–values for the runoff and cancer data sets. Based on the LR values, we can accept the MBMW distribution to more adequate for both data sets. The 95% and 99% confidence intervals for λ, β, k, a, b and c are given in Table 8 for runoff and cancer data sets.

20

Abdus Saboor et al.

Table 3 Estimates of the parameters(Standard errors in parenthesis) for runoff data Distributions G(µ, σ) MW(λ, β, k) BE(λ, a, b) BMW(α, γ, λ, a, b) BEW(α, λ, a, b, c) MBMW(λ, β, k, a, b, c)

Estimates 12.1942 (4.4252) 1.2 × 10−9 (0.0113) 24.7684 (20.0716) 3.3051 (1.4382) 22.6965 (20.0033) 0.0444 (0.1253) 0.0001 (0.0008)

25.1866 (4.0203) 0.1731 (0.0580) 99999.9999 (1.01 × 106 ) 0.3895 (0.1311) 0.0999 (0.0707) 11.9687 (8.1504)

0.5915 (0.1114) 0.0013 (0.0011) 0.0008 (0.0015) 0.0313 (0.0374) 0.3000 (0.2717)

9.9999 (8.6587) 0.0949 (0.1028) 2.7843 (4.0134)

L-S 13.1263 2.2876 13.0247 1.9563 1.9781 1.6429

K-S 0.4772 0.1388 0.3696 0.1144 0.1211 0.0947

0.1373 (0.1243) 1.0996 (0.4612) 0.0603 (0.0806)

Table 4 Goodness–of–fit statistics for runoff data Distributions G(µ, σ) MW(λ, β, k) BE( λ , a , b) BMW(α, γ, λ, a, b) BEW(α, λ, a, b, c) MBMW(λ, β, k, a, b, c)

A∗ 21.5789 0.9039 11.1641 0.5736 0.5951 0.3473

W∗ 2.8729 0.1372 1.4737 0.0891 0.0932 0.0463

p–value 0.0000 0.5282 0.0001 0.7647 0.7006 0.9201

Table 5 Estimates of the parameters(Standard errors in parenthesis) for cancer data Distributions G(µ, σ) MW(λ, β, k) BE(λ, a, b) BMW(α, γ, λ, a, b) BEW(α, λ, a, b, c) MBMW(λ, β, k, a, b, c)

Estimates 5.6438 (0.4962) 2.9 × 10−10 (0.3142) 0.6455 (0.6103) 0.4696 (0.4787) 5.0234 (8.8718) 5.8×10−13 (0.2649) 0.0538 (0.1894)

5.4212 (0.4121) 0.0938 (0.3013) 1.4485 (0.3281) 0.6661 (0.3122) 0.2644 (0.3987) 1.8130 (2.2520)

1.0478 (0.1179) 0.1791 (0.1763) 5.8×10−13 (0.0063) 0.7859 (1.2517) 0.5238 (0.5841)

2.7347 (2.0201) 2.3528 (5.9049) 1.2999 (0.9836)

0.9082 (1.5219) 0.4799 (0.4686) 0.4182 (0.5502)

Table 6 Goodness–of–fit statistics for cancer data Distributions G(µ, σ) MW(λ, β, k) BE( λ , a , b) BMW(α, γ, λ, a, b) BEW(α, λ, a, b, c) MBMW(λ, β, k, a, b, c)

A∗ 6.9405 0.9634 0.5613 0.2719 0.2550 0.0761

W∗ 1.3621 0.1543 0.0998 0.0405 0.0378 0.0119

L-S 5.9494 2.8259 2.8391 1.6898 1.6742 1.0411

K-S 0.1708 0.0700 0.0664 0.0449 0.0448 0.0298

p–value 0.0011 0.5569 0.6238 0.9581 0.9591 0.9998


21

Table 7 LR statistics (their p-values) for the runoff and cancer data sets Hypotheses H0 : a = b = c = 1 (MW) versus H1 : a 6= b 6= c 6= 1 (MBMW)

runoff data

cancer data

12.2091 (0.0067)

9.5615 (0.0227)

Table 8 Confidence intervals C.I. 95% 99%

λ [0 0.2901] [0 0.3679]

β [0 27.9434] [0 32.9967]

C.I. 95% 99%

λ [0 0.5192] [0 0.6834]

β [0 6.2269] [0 7.6231]

Runoff data k [0 0.8326] [0 1.0011] Cancer data k [0 1.6686] [0 2.0307]

a [0 10.6505] [0 13.1389]

b [0 0.2184] [0 0.2684]

c [0 0.0017] [0 0.0022]

a [0 3.2277] [0 3.8375]

b [0 1.4965] [0 1.8377]

c [0 0.4250] [0 0.5347]

6.2 Bayesian estimation We fit the MBMW distribution under the Bayesian approach. The following independent priors are considered to perform the Metropolis-Hastings algorithm: λ ∼ Γ (0.01, 0.01), β ∼ Γ (0.01, 0.01), k ∼ Γ (0.01, 0.01), a ∼ Γ (0.01, 0.01), b ∼ Γ (0.01, 0.01) and c ∼ Γ (0.01, 0.01), so that we have a vague prior distribution. Considering these prior density functions, we generate two parallel independent runs of the Metropolis-Hastings with size 150, 000 for each parameter, disregarding the first 15, 000 iterations to eliminate the effect of the initial values and, to avoid correlation problems, we consider a spacing of size 10, obtaining a sample of size 13, 500 from each chain. To monitor the convergence of the Metropolis-Hastings, we perform the methods suggested by Cowles and Carlin (1996). To monitor the convergence of the Metropolis-Hastings, we use the between and within sequence information, following the approach developed in Gelman and Rubin (1992), to obtain the potential scale reduction, b In all cases, these values were close to one, indicating the convergence of R. the chain. The approximate posterior marginal density functions for the parameters are presented in the Figures 6 and 7. In the Tables 9 and 10, we report posterior summaries for the parameters of the MBMW model, where SD represents the standard deviation from the posterior distributions of the parameters and HPD represents the 95% highest posterior density (HPD) intervals. We note that the values for the medians a posteriori (Tables 9 and 10) are quite close (as expected) to the MLEs given in the Tables 3 and 5.

7 Conclusions We introduce a six-parameter lifetime model called the modified beta modified– Weibull (MBMW) distribution, which extends several distributions widely used in the literature. The new distribution is useful to model lifetime data with increasing, decreasing and bathtub shaped hazard functions. We provide

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Table 9 Posterior summaries for the parameters from the MBMW model for runoff data. Parameter λ β k a b c

Median 0.0461 11.9656 0.3002 2.7866 0.0600 0.0001

SD 0.0188 0.0997 0.0096 0.0998 0.0045 0.0001

HPD (95%) (0.0086; 0.0823) (11.7696; 12.1583) (0.2818; 0.3201) (2.5882; 2.9794) (0.0512; 0.0691) (0.00002; 0.0004)

ˆ R 1.0030 1.0001 1.0002 1.0003 0.9999 1.0003

Table 10 Posterior summaries for the parameters from the MBMW model for cancer data. Median 6.7 × 10−8 1.8130 0.5238 1.2999 0.4181 0.0542

HPD (95%) (2.6 × 10−12 ; 1.9 × 10−7 ) (1.7931; 1.8320) (0.5058; 0.5412) (1.2803; 1.3194) (0.3986; 0.4371) (0.0404; 0.0689)

2.8

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ˆ R 1.0007 0.9999 1.0001 1.0004 0.9998 1.0006

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SD 5.9 × 10−8 0.0099 0.0089 0.01001 0.0098 0.0073

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Parameter λ β k a b c

0.045

0.055

0.065 b

0.075

0.0000

0.0004

0.0008

0.0012

c

Fig. 6 Approximate posterior marginal densities for the parameters from the MBMW model for runoff data.

closed-form expressions for the ordinary moments and generating functions in terms of gamma and Meijer’G functions. Explicit expressions for conditional moments and mean deviations, residual lifetime and reversed residual life functions, Bonferroni and Lorenz curves are also given. Some alternative properties from the liner representation (7) are also presented. We discuss the maximum likelihood method and a Bayesian approach to make inference on the model parameters. By using some goodness-of-fit statistics we prove empirically that

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1.78

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4e+06 0e+00

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Fig. 7 Approximate posterior marginal densities for the parameters from the MBMW model for cancer data.

the new model is superior to some distributions generated from other families in terms of model fitting by means of two real data applications. Acknowledgement The research of Abdus Saboor has been supported in part by the Higher Education Commission of Pakistan. References 1. Cintra RJ, Rˆ ego LC, Cordeiro GM, Nascimento ADC (2014) Beta generalized normal distribution with an application for SAR image processing. Statistics: A Journal of Theoretical and Applied Statistics, 48: 279–294. 2. Cordeiro GM, Edwin Ortega MM, Lemonte AJ (2014) The exponential-Weibull lifetime distribution. Journal of Statistical Computation and Simulation, 84: 2592–2606. 3. Cordeiro GM, Gomes AE, Da-Silva CQ, Edwin Ortega MM (2013a) The beta exponentiated Weibull distribution. Journal of Statistical Computation and Simulation, 83: 114–138. 4. Cordeiro GM, Silva GO, Edwin Ortega MM (2013) The beta–Weibull geometric distribution. Statistics: A Journal of Theoretical and Applied Statistics, 47: 817–834. 5. Cheng J, Tellambura C, Beaulieu NC (2003) Performance analysis of digital modulations on Weibull fading channels. Vehicular Technology Conference, VTC 2003-Fall. 2003 IEEE 58th; 2003 October 6-9;Vol. 1, p. 236-240 6. Cowles MK, Carlin BP (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, 91: 133–169. 7. Erd´ elyi A, Magnus W, Oberhettinger F, Tricomi FG (1995) Higher Transcendental Functions, Vol. 1, McGraw–Hill, New York, Toronto & London.

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8. Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7: 457–472. 9. Gradshteyn IS, Ryzhik IM (2007) Table of Integrals, Series, and Products. Academic Press, New York. 10. Gupta RC, Kirmani SNUA (2000) Residual coefficient of variation and some characterization results. Journal of Statistical Planning and Inference, 91: 23-31. 11. Jones MC (2004) Families of distributions arising from distributions of order statistics. Test, 13: 1–43. 12. Khan MN (2015) The modified beta Weibull distribution, Hacettepe Journal of Mathematics and Statistics, 44: 1553–1568. 13. Kang MS, Goo JH, Song I, Chun JA, Her YG, Hwang SW, Park SW (2013) Estimating design floods based on the critical storm duration for small watersheds. Journal of HydroEnvironment Research, 7: 209–218. 14. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies Vol. 204, Amsterdam: Elsevier (North-Holland) Science Publishers. 15. Lee C, Famoye F, Olumolade O (2007) Beta-Weibull distribution: some properties and applications to censored data. Journal of Modern Applied Statistical Methods, 6: 173–86. 16. Lee ET, Wang JW (2003) Statistical Methods for Survival Data Analysis (3rd ed.). New York: Wiley. 17. Luke, YL (1969) The Special Functions and Their Approximations, San Diego: Academic Press. 18. Mathai A, Saxena R (1978) The H–function with Applications in Statistics and Other Disciplines. Wiley Halsted, New York. 19. Meijer CS (1946) On the G-function I–VIII. Proc. Kon. Ned. Akad. Wet., 49: 227–237, 344–356, 457–469, 632–641, 765–772, 936–943, 1063–1072, 1165–1175. 20. Nadarajah S, Kotz S (2006) The beta exponential distribution. Reliability Engineering and System saftey, 91: 689–697. 21. Nadarajah S, Teimouri M, Shih SH (2014) Modified Beta Distributions. Sankhy¯ a Series B, 76-B: 19-48. 22. Pham H. Lai CD (2007) On Recent Generalizations of the Weibull Distribution. IEEE Transactions on Reliability, 56: 454–458. 23. Prudnikov AP, Brychkov YA, Marichev OI (1986) Integrals and Series, vol. 1–4. Gordon and Breach Science Publishers, Amsterdam. 24. Pundir S, Arora S, Jain K (2005) Bonferroni curve and the related statistical inference. Statistics and Probability Letters, 75: 140-150. 25. Saboor A, Provost SB, Ahmad M (2012) The moment generating function of a bivariate gamma-type distribution. Applied Mathematics and Computation 218: 11911–11921. 26. Saboor A, Bakouch HS, Khan MN (2016) Beta Sarhan–Zaindin modified–Weibull distribution. Applied Mathematical modelling, 40: 6604–6621. 27. Sarhan AM, Zaindin M (2009) Modified Weibull distribution. Applied Sciences, 11: 123–136. 28. Silva GO, Edwin Ortega MM, Cordeiro GM (2010) The beta modified Weibull distribution. Lifetime Data Analysis, 16: 409-430. 29. Singla N, Jain K, Sharma SK (2012) The beta generalized Weibull distribution: properties and applications. Reliability Engineering and System Safety, 102: 5-15.

Appendix A. The unified Fox–Wright generalized hypergeometric function Here, ∗ p Ψq

∞ Qp h (a, A) i X (aj )Aj n z n p Qj=1 z = q (b, B)q j=1 (bj )Bj n n! n=0

(22)

stands for the unified variant of the Fox–Wright generalized hypergeometric function with p upper and q lower parameters; (a, A)p denotes the parameter


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p–tuple (a1 , A1 ), · · · , (ap , Ap ) and aj ∈ C, bi ∈ C \ Z− 0 , Ai , Bj > 0 for all j = 1, p, i = 1, q. The power series converges for suitably bounded values of |z| when q p X X Bj > 0 . Aj + ∆p,q = 1 − j=1

j=1

Qq B In the case ∆ = 0, the convergence holds in the open disc |z| < β = j=1 Bj j · Qp −Aj . j=1 Aj The function 1 Ψ0∗ is called confluent. The convergence condition ∆1,0 = 1 − A1 > 0 is of special interest for us. We point out that the original definition of the Fox–Wright function p Ψq [z] (consult formula collection (Erd´elyi et al., 1995) and the monographs (Kilbas et al., 2006, Mathai and Saxena, 1978)) contains gamma functions instead of the generalized Pochhammer symbols used here. However, these two functions differ only up to constant multiplying factor, that is i h (a, A) i Qp Γ (aj ) h j=1 p ∗ (a, A)p z = Qq z . p Ψq p Ψq (b, B)q (b, B)q j=1 Γ (bj )

The unification’s motivation is clear - for A1 = · · · = Ap = B1 = · · · = Bq = 1, the fucntion p Ψq∗ [z] reduces exactly to the well-known generalized hypergeometric function p Fq [z]. Appendix B. Meijer G–function The symbol Gm,n p,q (·| ·) denotes Meijer’s G−function (Meijer 1946) defined in terms of the Mellin–Barnes integral as Qn Qm I 1 a1 , · · · , ap j=1 Γ (1 − aj + s) j=1 Γ (bj − s) m,n Qq Q = z s ds, Gp,q z b1 , · · · , bq 2πi C j=m+1 Γ (1 − bj + s) pj=n+1 Γ (aj − s)

where 0 ≤ m ≤ q, 0 ≤ n ≤ p and the poles aj , bj are such that no pole of Γ (bj − s), j = 1, m coincides with any pole of Γ (1 − aj + s), j = 1, n; i.e. ak − bj 6∈ N, while z 6= 0. C is a suitable integration contour, see [p.143](Luke 1969) and (Meijer 1946) for more details. The G function’s Mathematica code reads MeijerG[{{a1 , ..., an }, {an+1 , ..., ap }}, {{b1, ..., bm }, {bm+1, ..., bq }}, z].