THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION

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Volume 22, Number 1, 2011, Pages 25-55

This paper is available online at http://pphmj.com/journals/adas.htm

: tion Classifica ject

Sub s Mathematic

2010 Kindly provide.

Keywords and phrases: generating function, information matrix, maximum likelihood, moment, Weibull distribution, Weibull negative binomial distribution.

Received March 2, 2011

THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION

CRISTIANE RODRIGUES, GAUSS M. CORDEIRO^∗, CLARICE G. B. DEMÉTRIO and EDWIN M. M. ORTEGA Departamento de Ciências Exatas

Universidade de São Paulo 13418-900, Piracicaba, SP, Brazil e-mail: crisrodrigues_27@hotmail.com clarice@esalq.usp.br

edwin@esalq.usp.br

∗Departamento de Estatística

Universidade Federal de Pernambuco e-mail: gausscordeiro@uol.com.br

Abstract

We propose the Weibull negative binomial distribution that is a quite flexible model to analyze positive data, and includes as special sub- models the Weibull, Weibull Poisson and Weibull geometric distributions.

Some of its structural properties follow from the fact that its density function can be expressed as a mixture of Weibull densities. We provide explicit expressions for moments, generating function, mean deviations, Bonferroni and Lorenz curves, quantile function, reliability and entropy.

The density of the Weibull negative binomial order statistics can be expressed in terms of an infinite linear combination of Weibull densities.

We obtain two alternative expressions for the moments of order statistics.

The method of maximum likelihood is investigated for estimating the model parameters and the observed information matrix is calculated. We propose a new regression model based on the logarithm of the new

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distribution. The usefulness of the new models is illustrated in three applications to real data.

1. Introduction

The Weibull distribution is a very popular model that has been extensively used over the past decades for analyzing data in survival analysis, reliability engineering and failure analysis, industrial engineering to represent manufacturing and delivery times, extreme value theory, weather forecasting to describe wind speed distributions, wireless communications and insurance to predict the size of reinsurance claims. In hydrology, it is applied to extreme events such as annually maximum one-day rainfalls and river discharges. The need for extended forms of the Weibull distribution arises in many applied areas but the emergence of such extensions in the statistics literature is only very recent. Following an idea due to Adamidis and Loukas [1] for a mixing procedure of distributions, we define the Weibull negative binomial (WNB) distribution and study several of its mathematical properties. The Weibull distribution represents only a special sub-model of the new distribution.

Let W₁,...,W_Z be a random sample from a Weibull density function with scale parameter a > 0 and shape parameter b>0, namely ^g_a_,_b

( )

ω = âbω^b⁻¹ê⁻â^ω^b

(

for ω> 0

)

. We assume that the random variable Z has a zero truncated negative binomial distribution with probability mass function

( )

⎟

[(

⁻^β

)

⁻ ⁻

]

⁻ ^∈ ^∈ ₊

⎠

⎜ ⎞

⎝

⎛ + − β

=

β z N s R

z z s s

z

P ^z 1 1 ^s 1 , ,

,

; ¹

and β∈

(

0,1

)

. Here Z and W are assumed independent random variables. Let X =

(

,...,

)

.

minW₁ W_z Then f

(

x|z;a,b

)

= abzx^b⁻¹e⁻^azx^b and the marginal probability density function (pdf) of X with four parameters reduces to

( )

[( ) ] (

1

)

⁽ ⁾, , 0,

(

0,1

)

,

1 1

; ¹ −β ¹ > β∈

− β

−

= β

θ abs₋ x ⁻e⁻ e⁻ ⁻ ⁺ x s

x

f ^b ^ax ^ax ^s

s

b b

(1) where θ =

(

a,b, s,β

)

. In the sequel, (1) is refereed to as the WNB density function which is customary for such names to be given to models arising via the operation of compounding (mixing) distributions.

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By integrating (1) leads to its cumulative distribution function (cdf) given by

( ) [( ) ( ) ]

[(

1

)

1

]

^, ⁰^.

1

; 1 >

− β

−

− β

= −

θ ⁻ ₋ e⁻ ⁻ x

x

F s

s ax

s ^b

(2) The hazard rate function corresponding to (2) is

( ) ( )

⁽ ⁾

[(

1

)

1

]

^.

; 1

1 1

− β

−

β

−

= β θ

τ − −

+

−

s ax

s ax ax

b

b b b

e e s e

x abx (3)

(a) (b)

Figure 1. Plots of the WNB density for some parameter values.

(a) (b)

Figure 2. Plots of the WNB hazard rate function for some parameter values.

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If X is a random variable with density function (1), then we write X ~

(

^, ^, ^,

)

^.

WNBa b s β It extends several distributions previously considered in the literature and we study some structural properties of (1). In fact, the Weibull distribution with parameters a and b is clearly a special sub-model when s=1 and

.

→0

β For s =1 and b =1 in addition to β →0, it yields the exponential distribution. The WNB distribution also contains the exponential Poisson (EP) (Kus [11]) and Weibull Poisson (WP) distributions (Bereta et al. [5]) as sub-models when

, λ s

=

β s→∞ and, in the second case, in addition to b =1. For s=1, equation (1) reduces to the Weibull geometric (WG) density function (Barreto-Souza et al.

[4]). In Figure 1, we plot the WNB density for selected parameter values. For all values of the parameters, the density function (1) tends to zero as x→∞. In Figure 2, we plot the WNB hazard rate function for selected parameter values, showing its flexibility. The WNB density can be widely applied in many areas of engineering and biology.

The rest of the article is organized as follows: In Section 2, we demonstrate that the WNB density function can be expressed as a mixture of Weibull densities. This result is important to provide some mathematical properties of the new model directly from those properties of the Weibull distribution. A range of mathematical properties is considered in Sections 3 to 6. These include moments, skewness and kurtosis, quantile function, generating function, mean deviations and Bonferroni and Lorenz curves. In Section 7, we show that the density of the WNB order statistics is a linear combination of Weibull densities. Two explicit expressions for the moments of the WNB order statistics are obtained in this section while the reliability and the Rényi entropy are derived in Sections 8 and 9, respectively. Maximum likelihood estimation and inference are discussed in Section 10. A log-Weibull negative binomial regression model is proposed in Section 11. Three applications to real data in Section 12 illustrate the importance of the new models. Concluding remarks are addressed in Section 13.

2. Expansion for the Density Function

Equations (1) and (2) are straightforward to compute using any statistical software. However, we obtain expansions for ^f( )^x and ^F( )^x in terms of an infinite weighted sum of cdf’s and pdf’s of Weibull distributions, respectively. Using the

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Lagrange expansion (Consul and Famoye [7, Section 1.2.6]) for

(

1−βe⁻^ax^b

)

⁻⁽^s⁺¹⁾,

( )

[ ]

⁽ ⁾ ^{( ⁾

[

( )

]

} ( )

∑

^∞

∑

=

∞

=

− −

−

− ⎟ β

⎠

⎜ ⎞

⎝

= ⎛ + β

−

⎟ β

⎠

⎜ ⎞

⎝

⎛ + + +

++

0 0

1

1 ,

1 1 1

1

k k

k ax k

ax

ax^b ^b e ^b

k k e s

k e k s k s

s

(4) equation (1) can be further expanded as

( )

[( ) ] ∑

^∞ ⁽ ⁾

=

+

−

− ⎟β +

⎠

⎜ ⎞

⎝

⎛ +

− β

= − θ

0

1 1

1 .

1 1

;

k

x k a b k s

b

e k x

k abs s

x f

Hence

( ) ∑

^∞ ₍ ₎

( )

= ω +

= θ

0

,

1 ,

;

k

b k a

kg x

x

f (5)

where

(

1

) [(

1

)

1

]

^,

1

− β

− +

⎟⎠

⎜ ⎞

⎝ β ⎛ +

=

ω ₋

+

s k

k k

k k s s

(6) and g_a₍_k₊₁₎_,_b( )x is the Weibull density function with scale parameter a(k+1) and

shape parameter b. Clearly,

∑

^∞_k₌₀^ω^k ⁼¹^.

Equation (5) reveals that the WNB density function is a mixture of Weibull densities that holds for any parameter values. This result is important to obtain some of its mathematical properties from those of the Weibull distribution. The formulas related with the WNB distribution turn out manageable, as it is shown in the rest of this article, and with the use of modern computer resources with analytic and numerical capabilities, may turn into adequate tools comprising the arsenal of applied statisticians. Elementary integration of (5) gives the WNB cumulative distribution

( ) ∑

^∞ ₍ ₎

( )

= ω +

= θ

0

,

1 ,

;

k

b k a

kG x

x

F (7)

where G_a₍_k₊₁₎_,_b( )x =1−e⁻^a⁽^k⁺¹⁾^x^b denotes the Weibull cumulative function with scale parameter a(k +1) and shape parameter b.

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3. Moments

From now on, let X ~ WNB

(

a,b,s,β

)

. The ordinary, central, inverse and factorial moments of the WNB distribution can be obtained from an infinite weighted linear combination of those quantities for Weibull distributions. For example, the sth moment of the Weibull distribution with parameters a and b is

(

+¹

)

^,

Γ

=

τ′_s a⁻^s^b s b where ^Γ( )^α ⁼

∫

₀^∞^w^α⁻¹^e⁻^w^dw is the gamma function. The sth generalized moment of the WNB distribution immediately comes from (5) as

( ) ( )

( )

[ ]

∑

^∞

= +

+ ω Γ

=

= μ′

0

. 1

1

k k sb

s s

k a

b X s

E (8)

Various closed form expressions can be obtained from (8) as particular cases.

The central moments ( )μ_s and the cumulants ( )κ_s of X are easily obtained from the ordinary moments by

∑ ( )

= ⎟μ′ μ′ −

⎠

⎜ ⎞

⎝

− ⎛

= μ

p

r

r p p r

p r

p

0

1 1 and

∑

⁻

= ⎟κ μ′−

⎠

⎜ ⎞

⎝

⎛

−

− − μ′

= κ

1

1 ,

p 1

r

r p r p

p r

p

respectively. Here κ₁ =μ′₁, κ₂ =μ′₂ −μ′₁², κ₃ = μ′₃−3μ′₂μ′₁+2μ′₁³, κ₄ =μ′₄ − ,

6 12

3

4μ′₃μ′₁− μ′₂² + μ′₂μ′₁²− μ′₁⁴ etc., respectively. The skewness and kurtosis measures can be obtained from the classical relationships involving cumulants:

Skewness( )X = κ₃ κ³₂² and Kurtosis( )X = κ₄ κ²₂, namely

Skewness

( ) ( ) ( ) ( ) ( )

( )

X

X E X E X E X

X E ₃₂

3 2 3

Var

2

3 +

= −

and

Kurtosis

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

^.

Var

3 6

4

2

4 2

2 3

4

X

X E X E X E X E X E X

X = E − + −

Plots of the skewness and kurtosis of X as functions of s and β are given in Figures 3 and 4. Both skewness and kurtosis increase with s for fixed β and increase with β for fixed s.

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Finally, the nth descending factorial moment of X is given by

[

^{( )}

]

=

[ (

−

)(

−

) (

− +

) ]

=

∑

ⁿ=

( )

μ′

r r

n E X X X X n sn r

X

E 1 2 " 1 0 , ,

where s

(

n,r

) ( )

= r!⁻¹

[

D^rx^{( )}ⁿ

]

_x₌₀ are the Stirling numbers of the first kind. The WNB factorial moments are obtained from (8).

(a) (b)

Figure 3. The WNB skewness

(

for a =0.3andb =1.4

)

as function of s (for fixed β) and as function of β (for fixed s).

(a) (b)

Figure 4. The WNB kurtosis

(

for a =0.3andb =1.4

)

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4. Quantile Measures

The WNB quantile function corresponds to the inverse of (2) given by

( )

u F ¹

(

u;

) {

log

[ ( {[ (

¹1 1 1

)

^s

]

u

(

1

) }

^s ¹^s

)]

¹^a

}

¹^b.

Q = ⁻ θ = β⁻ − − −β ⁻ + −β⁻ ⁻ ⁻ (9)

Then using (9), simulation from the WNB random variate is straightforward by

{

log

[ ( {[ (

¹1 1 1

)

^s

]

V

(

1

) }

^s ¹^s

)]

¹^a

}

¹^b,

X = β⁻ − − −β ⁻ + −β ⁻ ⁻ ⁻

where V is a uniformly distributed random variable over the interval (0, 1). We develop a script using the software R (R Development Core Team, 2008) to simulate X in Appendix A.

(a) (b)

Figure 5. The Bowley skewness of the WNB distribution

(

for a =0.3andb =1.4

)

We now compute quantile measures for the skewness and kurtosis. The Bowley skewness (Kenney and Keeping [9]) is based on quartiles

( ) ( ) ( )

( )

3 4 Q

( )

14 ^, Q

4 1 Q 2 1 Q 2 4 3 Q

− +

= −

B (10)

and the Moors kurtosis (Moors [13]) is based on octiles

( ) ( ) ( ) ( )

( )

6 8

( )

2 8 ^, 8 1 8 3 8 5 8 7

Q Q

Q M Q

−

− +

= − (11)

where Q( )u is calculated from (9). These measures are less sensitive to outliers and

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they exist even for distributions without moments. Plots of (10) and (11) for selected parameter values are given in Figures 5 and 6, respectively. For fixed β, when s increases, the Bowley skewness and the Moors kurtosis first increase to a maximum and then decrease. The behavior of both measures when β increases depend upon the value fixed for s.

(a) (b)

Figure 6. The Moors kurtosis of the WNB distribution

(

for a = 0.3 andb =1.4

)

5. Moment Generating Function

The moment generating function (mgf) M( )t = E

[

exp( )tX

]

follows from the power series expansion for the exponential function and (8),

( ) ( )

( )

[ ]

∑

^∞

= +

ω +

= Γ 0 ,

.

! 1

1 k

s

s b s

k t s k

a b t s

M

Now, we derive two explicit expressions for M( )t using Meijer G-function and Wright generalized hypergeometric function. Let A = absβ

[(

1−β

)

^−s −1

]

. First, we have

( )

⁼

∫

0^∞

( )

⁻

(

⁻

) [

⁻^β

(

⁻

)]

⁻⁽ ⁺⁾

1

1exp 1 exp

exptx x ax ax dx

A t

M ^b ^b ^b ^s

( ) [ ( ) ]

∑

^∞₌ ⁺ ^⎟⎠

∫

^∞ ⁻ ⁻ ⁺

⎜ ⎞

⎝ β ⎛ +

=

0 0

1

1 exp exp 1 .

k

b b

k tx x ak x dx

k k

A s (12)

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From the Meijer G-function defined by

( ) ( )

∫ ∏ ∏

∏ ∏

−

+

= = +

= =

−

− Γ +

Γ

−

− Γ +

Γ

= π

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

L

t p

n j

p

m j

j j

m

j

n

j

j j

q n p

m q

p x dt

t b t

a

t a t

b i

b b

a x a

G

1 1

1 , 1 ,

1 1 2

1 ...,

, ..., ,

and the result exp

{

−g

( )

x

}

=G¹₀^,_,⁰₁

( ( ) )

g x |⁻₀ for g( )⋅ an arbitrary function, we can write

[ ( ) ]

∫

^∞ ⁻ ⁻ ⁺

=

0

1exptx ak 1 x dx x

K ^b ^b

( ) ( ( ) )

∫

^∞ ⁻ ⁺ ^|⁻

=

0 1,0 0

1 , 0

1exptx G ak 1 x dx.

x^b ^b

If we assume that b= p q, where p ≥1 and q ≥1 are co-prime integers, then equation (2.24.1.1) in Prudnikov et al. [15, Volume 3] yields

( ) ( )

⁽ ⁾

{ ( [ ) ] }

( )

₀_, ¹_,_..., ¹ ^.

..., 2 ,

1 , 1

2

, 1 1 , 2 2 1

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

− + π

= ⁻ ₊− ⁻₋ ⁻

q q q

p b p p

b p

b q

t

p b k

G a t

K p _p _q

p q q b

p p q q

p b b

From (12) and the last two equations, we obtain

( ) ( )

[( ) ]( )

⁽ ⁾

∑

^∞

=

− + +

−

− −

⎟⎠

⎜ ⎞

⎝ β ⎛ + π

− β

−

= −

0 1 1

2 2

1

2 1

1 _k

k q

p s

b b

k k t s

t absp M

{ ( [ ) ] }

( )

₀_, ¹_,_..., ¹ ^.

..., 2 ,

1 , 1 ¹

,

, ⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

− −

−

× + ⁻

q q q

p b p p

b p

b q

t

p b k

G a _p _q

p q q b

p p

q (13)

The condition b= p q in (13) is not restrictive, since every real number can be approximated by a rational number.

A second representation for the integral M can be obtained using the Wright generalized hypergeometric function given by

( ) ( )

( )

!.

; , ..., , ,

, ..., , ,

0 1 1 1

1 1

1

∑

∏

∞

=

+ β Γ

+ α Γ

⎥=

⎥

⎦

⎤

⎢⎢

⎣

⎡

β β

α α

Ψ

n

n q

j

j j p

j

j j

q q

p p q

p n

x n B

n A x

B B

A A

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We assert that

{ [ ( ( ) ) ] }

∑ ∫

^∞

=

∞ + − − + ⁻

=

0 0

1exp 1 ¹

!

j

b b b

j j

dx x k

a t j x

K t

( ) ∑

^∞

( [ ( ) ] )

=

⎟⎠

⎜ ⎞

⎝⎛ + + Γ

= +

−

0

! 1 1 1

1

j

j b

b j j

k a l k

ba

( )

[ ]

^⎥^⎥⎦

⎤

⎢⎢

⎣

⎡

− + + Ψ

= ₋₁

1

; 1 , 1 1 1

0

1 b

k a b t k

ba (14)

provided that b>1. Combining (12) and (14), we obtain the second representation for M( )t

( ) [( ) ] ( ) ( )

( )

[ ]

∑

^∞

= +

− ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− +

⎟ Ψ

⎠

⎜ ⎞

⎝

⎛ + + β

− β

= − ₋

0

0 1 1

1

1 1 ;

, 1 1 1

1 _k ^b

k

s ak

b t k

k s k t s

M (15)

provided that b>1. Clearly, special formulas for the mgf of the Weibull, WG and WP distributions can be readily determined from equations (13) and (15) by substitution of known parameters.

6. Mean Deviations

The mean deviations about the mean (δ₁( )X = E( X −μ′₁ )) and about the median (δ₂( )X = E( X −m )) can be written as

( ) ₁ ( )₁ ( )₁

1 = 2μ′ μ′ −2 μ′

δ X F T and δ₂

( )

X =μ′₁ −2T

( )

m , (16) respectively, where μ′₁ = E( )X is given by (8), F( )^μ′₁ comes from (2), m=

( )

Q

( )

12

Median X = is computed from (9) and ^T

( )

^q ⁼

∫

₀^q^xf

( )

^x ^dx^.

From (5) and using the incomplete gamma function ^γ(^α^, ^x)⁼

∫

₀^x^w^α⁻¹^e⁻^ω^dw^,

we obtain

( ) ∑

^∞

[ ( ) ] ( ( ) )

=

− +

+ + γ

= ω

0

1

1 1 , 1 .

k 1

b b

k b ak q

k a q

T (17)

Then the mean deviations can be calculated from (17). Further, we can obtain

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Bonferroni and Lorenz curves from (17). These curves have applications in economics, reliability, demography, insurance and medicine and are defined by

( )

p =T

( ) (

q pμ′₁

)

B and L

( )

p =T

( )

q μ′₁, respectively, where q=Q

( )

p is calculated by (9) for a given probability p.

7. Order Statistics

The density function f_i:_n( )x of the ith order statistic for i =1,...,n corresponding to the random variables X₁,..., X_n following the WNB distribution, can be written as

( ) ( ) ( ) ∑

⁻

( ) { ( ) }

=

−

− +

⎟ −

⎠

⎜ ⎞

⎝

⎛ − +

= −

1

0

: 1 1 1 ,

1 ,

1 ⁱ

j

i j n j

n

i F x

j x i

i f n i x B f

where f( )x is the pdf (1), F( )x is the cdf (2) and B(a,b)= Γ(a+b) ( ) ( )

[

Γa Γb

]

is the beta function. Setting u =exp

(

−ax^b

)

, we can write from (5) and (7),

( ) ( ) ∑

^∞

( )

=

+

+ −

+ ω

= −

0

1

: 1 1

1

, m

m m b

n

i m x u

i n i B x ab f

( )

1 .

1

0 1 1

0

i j n

k k k i

j

j u

j

i ^∞ ⁺ ⁻

=

− +

= ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ω

⎟⎠

⎜ ⎞

⎝

− ⎛ −

×

∑ ∑

We use throughout an equation of Gradshteyn and Ryzhik [8, Section 0.314] for a power series raised to a positive integer j,

∑

^∞

=

∞

=

⎟ =

⎟

⎠

⎞

⎜⎜

⎝

⎛

0 , 0

,

i i i j j

i

iyi c y

a (18)

whose coefficients c_j_,_i

(

for i =1,2,...

)

are easily obtained from the recurrence equation

( ) ∑ ( )

= −

− − +

=

i

m

m i j m i

j ia jm i m a c

c

1 1 , 0

, , (19)

where c_j_,₀ = a₀^j. Hence, the coefficients c_j_,_i come directly from c_j_,₀,...,c_j_,_i₋₁

(13)

and, therefore, from a₀,...,a_i. Using (18), it follows that

( ) ( ) ∑

^∞

( )

=

+

+ −

+ ω

= −

0

1 1

: 1

1 ,

m

m b m

n

i m x u

i n i B x ab f

∑

⁻

( ) ∑

=

∞

=

−

− +

+ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

− ⎛ −

×

1

0 0

1 , .

1 1

i

j k

k k i j j n

n

j u c u

j i

Here the constants c_n₊_j₋_i_,_k are determined from equations (6) and (19) as

( ) ∑ [ ( ) ]

= + − −

− −

+ = ω + − − + ω

k

l

l k i j n l k

i j

n k l n j k l c

c

1

, 1

0

, 1 ,

where c_n₊_j₋_i_,₀ = ω₀ⁿ⁺^j⁻ⁱ. Hence, c_n₊_j₋_i_,_k can be calculated from c_n₊_j₋_i_,₀,...,

1 , −

− +j i k

cn and, therefore, from ω₀,...,ω_k₋₁. By combining terms, we obtain

( ) ( ) ( )

( ) ( )

∑ ∑

^∞

=

−

=

− +

+

− +

− + + +

⎟ω

⎠

⎜ ⎞

⎝ + ⎛ −

−

=

0 ,

1

0

,

: 1 , 1

1 1 1

m k

i

j

k i j n j m

n

i m n j i k B i n i

j c m i

x f

(

m 1 n j i k

)

x^b ¹u^m ¹ ⁿ ^j ⁱ ^k.

ab + + + − + ⁻ ⁺⁺ ⁺ ⁻⁺

×

Setting δ_k_,_m_,_j = a(m+1+n + j−i+k) in the above expression, we obtain

( ) ∑ ∑

^∞

( ) ( )

=

−

= η δ

=

0 ,

1

0

: , , ,

, , s

r i

j

b n

i x k m j g x

f km j (20)

whose coefficients η(k,m, j) are easily obtained from

( ) ( ) ( )

(

1

) (

, 1

)

^.

1 , 1 1

, ,

,

+

− +

− + + +

⎟ω

⎠

⎜ ⎞

⎝ + ⎛ −

−

=

η ⁺ ⁻

i n i B k i j n m

j c m i

j m k

k i j n m j

Equation (20) shows that the density function of the WNB order statistics can be expressed as an infinite weighted linear combination of Weibull densities. We can derive some mathematical measures of the WNB order statistics directly from those quantities of the Weibull distribution.

(14)

Using the linear combination representation (20), the moments of the WNB order statistics can be written directly in terms of the Weibull moments as

( ) ( ) ∑ ∑

^∞

( )

=

−

= δ + η

Γ

=

0 ,

1

0 , ,

: , , ,

1

m k

i

j

b s

j m k s

n i

j m b k

s X

E (21)

where δ_k_,_m_,_j and η(k,m, j) are given in Section 7.

Alternatively, we obtain another closed form expression for these moments using a result due to Barakat and Abdelkader [3] applied to the independent and identically distributed (i.i.d.) case

( ) ∑ ( ) ( )

+

−

=

− +

− ⎟

⎠

⎜ ⎞

⎝

⎟⎛

⎠

⎜ ⎞

⎝

⎛

−

− −

=

n

i n j

j i

n r j

n

i I r

j n i n r j

X E

1

: 1 ,

1 (22)

where

( )

⁼

∫

0^∞ ⁻

{

⁻

( ) }

11 F x dx. x

r

I_j ^r ^j

The integral can be written as

( ) ∫

^∞

∑ ∑ ∫

^∞

{ ( ) )

=

∞ −

∞

= +

− = − +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ω

=

0 0 0

1 ,

0 1

1 exp ,

k

b r

k j j

k k k r

j r x u dx c x a k j x dx

I

where

(

ω

)

−

∑

=

(

− +

)

ω −

= ^k

m m j k m

k

j k jm k m c

c 1 1 ,

0 ,

and c_j_,₀ =ω₀^j. Setting y = a(j+k)x^b, we obtain

( ) ( )

( )

∑

^∞

=

−

Γ +

=

0

1 , .

k

b r k b j

r

j j k

b c r a b r I

From equation (22), we have

( ) ( ) ( )

( )

∑ ∑

+

−

=

∞

=

− +

−

⎟ +

⎠

⎜ ⎞

⎝

⎟⎛

⎠

⎜ ⎞

⎝

⎛

−

− − Γ

=

n i n

j k rb

k i j

n j b

r rn

i j k

c j

n i n b j

r a rb X

E

1 0

1 ,

: 1 1 .

1 (23)

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We can compute the moments of the WNB order statistics by three different ways: (i) from equation (21) that involves two infinite sums and one finite sum but no complicated function; (ii) from equation (23) that involves only two sums, one infinite and other finite; or (iii) by direct numerical integration. The moments of the WNB order statistics listed in Table 1 are in agreement using the three methods.

Table 1. Moments of the WNB order statistics for n = 5,a = 4,b = 2,β = 0.8 and

=3 s

Order statistic X₁_:₅ X₂_:₅ X₃_:₅ X₄_:₅ X₅_:₅

=1

r 0.294022 0.481234 0.674377 0.930862 1.451229

= 2

r 0.112577 0.269695 0.513943 0.982929 2.558717

= 3

r 0.052002 0.172134 0.439308 1.182069 5.581500

= 4

r 0.027881 0.123392 0.419637 1.631341 15.08906 Variance 0.026128 0.038109 0.059160 0.116426 0.452652 Skewness 0.837768 0.762007 0.898139 1.267313 1.820482 Kurtosis 3.945583 4.059123 4.717588 6.474108 8.371872

8. Reliability

Here we derive the reliability R = Pr

(

X₂ < X₁

)

when X₁ and X₂ have independent WNB(a₁,b, s₁,β₁) and WNB(a₂,b,s₂,β₂) distributions with the same shape parameter b. The density of X₁ and the cdf of X₂ are obtained from equations (5) and (7) as

( )

∑

^∞ ( )

=

+

− + ω

=

0

1 1 1 1

1

1 1

r

r r

b r u

bx a x

f and ( )

∑

^∞

=

ω +

−

=

0

1 2 2

2 1 ,

j juj

x

F (24)

where u_i =exp

(

−a_ix^b

)

for i =1,2 and the constants ω₁_r and ω₂_j are given by (8) with the corresponding parameters of the distributions of X₁ and X₂, respectively. We have

( ) ( )

∫

^∞

=

0 f1 x F2 x dx, R

(16)

and then

( )

∫

^∞

∑ ∑

^∞

=

∞ +

=

+

−

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛ − ω

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ω +

=

0 0

1 2 2 0

1 1 1

1

1 a r 1 u 1 u dx

bx R

j

j j r

r r

b

( ) { [ ( ) ( ) ] }

∑

^∞₌ ^ω ^ω ⁺

∫

^∞ ⁻ ⁻ ⁺ ⁺ ⁺

−

=

0

, 0 1 1 2

1 2

1 1 exp 1 1 .

1

j r

b b

j

r a r bx a r a j x dx

By the application of

∫

0^∞ ⁻

(

⁻^μ

)

⁼^μ⁻

1

1exp x dx ,

bx^b ^b we obtain

( )

( ) ( )

[ ]

∑

^∞

= + + +

ω ω

− +

=

0

, 1 2

2 1

1 .

1 1

j r

j a r

a

a R r

9. Entropy

The entropy of a random variable X with density f( )x is a measure of variation of the uncertainty. Rényi entropy is defined by

( )

log

( )

,

1 1

⎭⎬

⎫

⎩⎨

ρ ⎧

= −

ρ

∫

^f ^x^ρ^dx

I_R (25)

where f( )x is the pdf of X, ρ> 0 and ρ≠1. For a random variable X with a WNB distribution, we have

( ) [( ) ]

( )

⁽ ⁾

∫

0^∞ ^ρ ⁼ ₋_β^ρ⁻^ρ ^ρ₋ ^ρ

∫

0^∞ ^ρ ⁻ ^β^ρ ⁻ ^ρ ⁻^β ⁻ ⁻^ρ ⁺ 1

1 1 .

1 1

dx e

e s x

b dx a

x

f ^b ^ax ^ax ^s

s

b b

Using the Lagrange expansion (4) in the last equation, we obtain

( ) ( )

( )

⁽ ⁾

∫

^∞

∑

^∞

= ρ − +

+ ρ ρ

−

+ −

ρ

⎟ β

⎠

⎜ ⎞

⎝

⎛ρ + + − ν

=

0 0 1

1 , 1

1 1

k b b

k

k k k dx s

x

f (26)

where

( )

[(

¹

)

¹

]

^.

1 1 1

ρ

−

− ρ

− ρ ρ

− β

−

+ ρ

− ρ

= Γ

ν s

bb b b

a

s (27)

(17)

Substituting (26) and (27) into (25), we can write

( ) ( ) ( ) [ ( ) ( )] [ ( ) ]

⎪⎩

⎪⎨

⎧

+ ρ

− ρ Γ + +

− ρ + ρ ρ

= −

ρ s b⁻ a b b b

I_R log 1 log log log 1

1

1 ₁

[( ) ] ( )

( )

^.

1 log 1

1 1

log

0 1

⎪⎭

⎪⎬

⎫

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

+ ρ

⎟ β

⎠

⎜ ⎞

⎝

⎛ρ + + − +

− β

− ρ

−

∑

^∞

= ρ−ρ +

+

− ρ

k b b

s k

k k k s

10. Estimation and Inference

Here we consider the estimation of the model parameters of the WNB distribution by the method of maximum likelihood. Suppose X₁,..., X_n is a random sample from (1) and θ = (a,b,s,β)^T is the parameter vector. The log-likelihood (LL) function, say logL = log

{

L

(

a,b,s,β

) }

for the four parameters is

( )

{ } ( ) ( ) ( ) ∑ ( )

=

− + +

= β

n

i

xi

b b n a n s

b a L

1 log 1 log

log ,

, , log

( ) ( )

∑

=

+ β +

−

n

i b

i n n s

x a

1

log log

( ) ∑ ( ) [( ) ]

=

−

− − −β −

β

− +

−

n

i

s ax_i^b

e s

1

. 1 1

log 1

log

1 (28)

It follows that the maximum likelihood estimators (MLEs) are the simultaneous solutions of the equations:

( )

∑ ∑

= = −

− =

β

− + β

+

n

i

n

i ax

ax b b i

i a

n e

e s x

x _b

i ib

1 1

, 1

1

( ) ( ) ( ) ( )

∑ ∑ ∑

= = = −

− =

β

− + β

+

−

n

i

n

i

n

i ax

i ax bi i

b i

i b

n e

e x s ax

x x

x

a _b

i b i

1 1 1

, 1

1 log log

log

( ) ( ) ( )

[( ) ]

∑

= −

− −

− β

−

β

− β

+ −

= β

−

n

i

s ax s

s e n

b i

1 1 1

1 log 1 1

log