Estimation of the Bivariate Generalized Linear Failure Rate - CiteSeerX

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 17, 821 - 837

Estimation of the Bivariate Generalized Linear Failure Rate Distribution Parameters under Random Censoring Hanan M. Aly Department of Statistics, Faculty of Economics & Political Science Cairo University, Egypt [email protected]

Hiba Z. Muhammed Department of Mathematical Statistics Institute of Statistical Studies and Research Cairo University, Egypt [email protected]

Abstract Recently a new distribution, named a bivariate generalized linear failure rate distribution has been introduced by Sarhan et al. (2011). In this paper, we obtained the moment generating function for the bivariate generalized linear failure rate distribution and the maximum likelihood estimates for the unknown parameters of this distribution and their approximate variance- covariance matrix in case of left random censoring. A numerical example is carried out to discuss the properties of the estimators.

Keywords: Maximum Likelihood estimators, random censoring, generalized linear failure rate distribution, bivariate generalized linear failure rate distribution, moment generating function

822

H. M. Aly and H. Z. Muhammed

1. Introduction In analyzing lifetime data, one often uses the exponential, Rayleigh, linear failure rate or generalized exponential distributions. It is well known that the exponential distribution can has constant hazard function whereas Rayleigh, linear failure rate and the generalized exponential distribution can have monotone (increasing in the case of Rayleigh or linear failure rate and increasing/decreasing in case of the generalized exponential distribution) hazard functions. Unfortunately, in practice often one needs to consider non-monotonic functions such as bathtub shaped hazard function. Recently Sarhan and Kundu (2009) presented a new distribution; called the generalized linear failure rate distribution (GLFR) which may has bathtub shaped hazard function. The GLFR has a cdf in the form

FGLFR ( x; α , β , γ ) = [1 − e Its corresponding pdf has the form

γ

− ( βx + x 2 ) α 2

] , x > 0, α > 0 , β > 0, γ > 0 . γ

− ( βx + x 2 )

γ

− ( βx + x 2 )

2 2 f GLFR ( x; α , β , γ ) = α ( β + γx)e [1 − e ]α −1 , x > 0 . Several properties of this distribution were established by the authors. They observed that several known distributions like the exponential, the Rayleigh and the linear failure rate distribution distributions can be obtained as special cases of the GLFR distribution. Recently, Sarhan et al. (2011) introduced a new bivariate generalized linear failure rate (BGLFR) distribution, whose marginals were GLFR distribution. This new five parameters BGLFR distribution was obtained using a method similar to that used to obtain the Marshall – Olkin bivariate exponential model [Marshall and Olkin (1967)]. The proposed BGLFR was constructed from three independent GLFR distributions using a maximization process. This new distribution is a singular distribution, and it can be used quit conveniently if there are ties in the data. They provided the following two interpretations for the BGLFR model

Competing risks model: Assume a system has two components, labeled 1 and 2, and the survival time of component i is denoted by X i , i = 1, 2. It is considered that there are three independent causes of failures, which may affect the system. Only component 1 can fail due to cause 1, and similarly only component 2 can fail due to cause 2, while both the components fail at the same time due to cause 3. Let U i be the lifetime of cause i, i = 1, 2, 3 . If U 1 ,U 2 and U 3 follow a GLFR distribution, then ( X 1 , X 2 ) follow the BGLFR model. Shock model: Suppose there are three independent sources of shocks; say 1, 2 and 3. Suppose these shocks are affecting a system with two components, say 1 and 2. It is assumed that the shock from source 1 reaches the system and destroys component 1 immediately, the shock from source 2 reaches the system and

Bivariate generalized linear failure rate distribution parameters

823

destroys component 1 immediately, while if the shock from source 3 hits the system it destroys both components immediately. Let U i denote the inter-interval times, between the shocks in source i, i = 1, 2, 3 , which follow the distribution GLFR. If X 1 , X 2 denote the survival times of the components, then ( X 1 , X 2 ) follows the BGLFR model. Sarhan et al. (2011) discussed several properties of this distribution, and used the EM algorithm to compute the maximum likelihood estimators of the unknown parameters in the case of complete sampling. Random censored data arises often in clinical trails Patients may enter the study in a more or less random fashion, according to their time of diagnosis. If the study is terminated some prearranged date, then censoring times, that is the lengths of time from an individual's entry into the study until the termination of the study, are random. This type of censoring will be the main censoring mechanism. It occurs when the censoring time varies from individual to individual and is unknown in advance. The data are left censored when some job durations are incomplete at the beginning of the job spells and are observed. For example, in the study of epidemics, such as AIDS, the time of onset of infection is typically unknown. What is known is the time at which the patient reports to the doctor. Hence the time from infection to the development of the disease is left censored. This Paper is concerned with the moment generating function for the BGLFR distribution that provided in section 3. The maximum likelihood estimation, estimated variance-covariance matrix and asymptotic confidence intervals for BGLFR distribution under random left censoring are provided in Section 4. In Section 5, we introduce a numerical example to illustrate the results. Finally we conclude the paper in Section 6.

2. The Bivariate Generalized Linear Failure Rate Distribution Sarhan et al. (2011) defined the BGLFR distribution as follows: Suppose U 1 ,U 2 and U 3 are three independent random variables such that U i ~ GLFR ( α i , β , γ ) for i = 1, 2 and3. Define X 1 = max(U 1 ,U 3 ) and X 2 = max(U 2 ,U 3 ) , then it is said that the bivariate vector ( X 1 , X 2 ) has BGLFR distribution with parameters (α 1 , α 2 , α 3 , β , γ ), denoted by BGLFR (α 1 , α 2 , α 3 , β , γ ) . Then, the joint cdf of ( X 1 , X 2 ) is given as follows 3

FBGLFR ( x1 , x 2 ) = ∏ FGLFR ( xi ; α i , β , γ ) i =1

where x3 = min( x1 , x 2 ) , α 13 = α 1 + α 3 , α 23 = α 2 + α 3 and α 123 = α 1 + α 2 + α 3 . The joint cdf of ( X 1 , X 2 ) can be written as

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H. M. Aly and H. Z. Muhammed

⎧ F1 ( x1 , x 2 ) if 0 < x1 < x 2 < ∞ ⎪ FBGLFR ( x1 , x 2 ) = ⎨ F2 ( x1 , x 2 ) if 0 < x 2 < x1 < ∞ ⎪ F ( x) if 0 < x1 = x 2 = x < ∞ ⎩ 3

where

(2.1)

F1 ( x1 , x 2 ) = FGLFR ( x1 ; α 13 , β , γ ) FGLFR ( x 2 ; α 2 , β , γ ) γ

γ

− ( βx2 + x22 )

− ( βx1 + x12 )

2 2 = [1 − e ]α13 [1 − e ]α 2 , F2 ( x1 , x 2 ) = FGLFR ( x1 ; α 1 , β , γ ) FGLFR ( x 2 ; α 23 , β , γ )

γ

γ

− ( βx1 + x12 ) α1 2

= [1 − e ] [1 − e F3 ( x) = FGLFR ( x; α 123 , β , γ )

− ( β x2 + x22 ) α 23 2

]

,

γ

− ( βx + x 2 )

2 = [1 − e ]α123 . Then, joint pdf of ( X 1 , X 2 ) is given as if 0 < x1 < x 2 < ∞ ⎧ f1 ( x1 , x 2 ) ⎪ f BGLFR ( x1 , x 2 ) = ⎨ f 2 ( x1 , x 2 ) if 0 < x 2 < x1 < ∞ ⎪ f ( x) if 0 < x1 = x 2 = x < ∞. ⎩ 3 where f1 ( x1 , x 2 ) = f GLFR ( x1 ; α 13 , β , γ ) f GLFR ( x 2 ; α 2 , β , γ )

= α 13α 2 ( β + γx1 )( β + γx 2 )e

γ

− ( βx1 + x12 ) 2

γ

e

− ( β x2 + x22 ) 2

γ

γ

− ( βx1 + x12 ) α13 −1 2

[1 − e ] [1 − e f 2 ( x1 , x 2 ) = f GLFR ( x1 ; α 1 , β , γ ) f GLFR ( x 2 ; α 23 , β , γ ) = α 1α 23 ( β + γx1 )( β + γx 2 )e

γ

− ( β x1 + x12 ) 2

γ

−( βx + x 2 )

]

,

γ

e

− ( βx1 + x12 ) α1 −1 2

]

− ( βx2 + x22 ) α 2 −1 2

− ( βx2 + x22 ) 2

γ

[1 − e

(2.2)

γ

[1 − e

− ( β x2 + x22 ) α 23 −1 2

]

,

γ

− ( βx + x 2 )

2 2 and f 3 ( x) = α 3 ( β + γx)e [1 − e ]α123 −1 . The marginals of both X 1 and X 2 have GLFR ( α 13 , β , γ ) and GLFR ( α 23 , β , γ ), respectively.

3. The Moment Generating Function for BGLFR Distribution In this section, we derive the joint moment generating function for the BGLFR distribution. We then obtain the joint moment of ( X 1 , X 2 ) and the marginal moments of X 1 and X 2 respectively. The moment generating function for the BGLFR distribution is given by:

Bivariate generalized linear failure rate distribution parameters

∞

∞

M (t1 , t 2 ) = α 2α 13 ∑∑ k ij ([ i =0 j =0

γ

e γ

2

t γ (i + 1) − 2 * 1 1 {−1 − 1 [ ] Γs 2 ( i ) ( ) 1 (i + 1)( j + 1) 2γ 2 2

+[

[( i +1) s12 ( i ) + ( j +1) s 22 ( j )]

+[

t1 e 2

[( i +1) s12 ( i ) + ( j +1) s 22 ( j )]

∞

∞

i =0 j =0

γ

⋅e γ

+[

e

2

2

γ (i + 1) 2

t2 e

−

1 2

1

t 2 − t1 γ ( j − i ) − 2 1 [ ] Γs 2 (i , j ) ( )}] 3 2 2 2 1

+

τ (m, k , r ; t 2 ; s1 (i), s 2 ( j ))]) 1

[( i +1) s 22 ( j ) + ( j +1) s12 ( i )]

⋅ {e

[( i +1) s 22 ( j ) + ( j +1) s12 ( i )]

2 (i + 1)γ γ

]

t γ ( j + 1) − 2 1 + 2[ ] Γs 2 ( j ) ( )}] 2 2 2 2

t γ (i + 1) − 2 * 1 1 {−1 − 2 [ ] Γs 2 ( j ) ( ) 2 (i + 1)( j + 1) 2γ 2 2

(i + 1)( j − i ) 2

} ⋅ {e

− ( j −i ) s32 ( i , j ) 2

s32 ( i , j )] [ 2 ( i +1) s 22 ( j ) + ( j − i ) ~

γ

+[

[

2 γ (i + 1)

+ α 2α 13 ∑∑ k ij′ ([

γ

− ( j +1) s 22 ( j ) 2

γ

⋅ {e

(i + 1)( j − i ) γ

( i +1) s 42 ( j )

[ γ

γ

} ⋅ {e

− ( j +1) s12 ( i ) 2

γ s32 ( i , j ) − ( j −i ) ~ 2

γ (i + 1) 2

]

−

1 2

t γ ( j + 1) − 2 1 + 1[ ] ⋅ Γs 2 (i ) ( )}] 1 2 2 2 1

t1 − t 2 γ ( j − i ) − 2 1 [ ] Γ~s 2 (i , j ) ( )}] 3 2 2 2 1

+

τ (m, k , r; t1 ; s 2 ( j ), s1 (i ))])

t + t γ (i + 1) − 2 1 + α 3 ∑ ki + 1 2 [ ] Γs 2 ( j ) ( )] ; i < j [e 4 i +1 2 2 2 i =0 (3.1) i+ j i+ j k ij = (−1) α 13 (i ) α 2 ( j ), k ij′ = (−1) α 23 (i ) α 1 ( j ), k i = (−1) i α123 (i ), ∞

where

1

[ 2 ( i +1) s12 ( i ) + ( j − i ) s32 ( i , j )]

e2

825

e2

− ( i +1) s 42 ( j ) 2

1

t2 β (α − 1)(α − 2) K (α − i) t1 β , , s1 (i ) = − , s 2 ( j) = − γ γ (i + 1) γ γ ( j + 1) i! β t −t β t −t β t +t s 3 (i, j) = − 2 1 , ~ s3 (i, j ) = − 1 2 , s 4 ( j) = − 1 2 , γ γ( j − i) γ γ ( j − i) γ γ ( j + 1)

α (i) =

826

H. M. Aly and H. Z. Muhammed

x

Γx* (a ) = ∫ t a −1 e −t dt ,

and

0

∞

τ (m, k , r; t 2 ; s1 (i ), s 2 ( j )) = ∑

1 [ m+ ] 2 k

∑∑

m =0 k =0 r =0

(−1) m + k − r ( s1 (i )) 2 ( m − k ) +1 ( s 2 ( j )) k − r 1 2 (m + ) m! 2

1⎞ ⎛ r +2 γ ( j + 1) −( 2 ) r+2 ⎜m + ⎟ ⎛k ⎞ ] ) Γs 2 ( j ) ( 2 ⎟ ⎜⎜ ⎟⎟ ⋅ {γ [ ⎜ 2 2 2 ⎜ k ⎟ ⎝r ⎠ ⎝ ⎠ r +1

t 2 γ ( j + 1) −( 2 ) r +1 [ ] ) }. Γs 2 ( j ) ( 2 2 2 j +1 It easily to obtain different moments of X 1 and X 2 as follows: - The joint moment of X 1 and X 2 is given as: ∞ ∞ α ′ β 13 α 2 c ij + α 23 α 1 c ij ] E ( X 1 , X 2 ) = ( ) 2 Γ β (1) [∑∑ [ ( ) (i + 1)( j + 1) γ i =0 j =0 γ +

2

.[[

β

γ

+ ( )2 ( ) γ 2

1 − 2

γ (i + 1) 2

]

−

1 2

3 Γ *β 2 ( ) − Γ *β 2 (1)] + α 3 ( ) ( ) 2 γ γ

(

γ

+ α3 ( ) 2

β

γ

ci

i =0

α 13 α 2 cij + α 23 α 1 c ij′ 3 Γ β 2 ( ) [∑∑ [ ] 3 ( ) 2 i =0 j =0 γ 2 (i + 1) ( j + 1) ∞

∞

. [Γ *β 2 (1) − (i + 1) 1 − 2

∞

∑ (i + 1) ]

∞

Γ β 2 (2)∑ ( )

i =0

γ

∞

γ

−

1 2

)

3 Γ *β 2 ( )] + 2α 3 ( ) 2 γ

∞

∑ i =0

ci (i + 1)

3 2

]

ci (i + 1) 2

∞

+ ( ) ( ) [∑∑ [α 13 α 2 cij + α 23 α 1 cij′ ] . γ 2 i =0 j =0

γ

.[ ( ) 2 where

c ij = (−1) α 13 (i) α 2 ( j) e i+ j

γ β 2 ( ) ( i +1) γ

ci = (−1) i α 123 (i ) e 2

and

−

1 2

η (u , l ,1, j ,1) η (u , l ,1, j ,0) (i + 1)

γ β 2 ( ) ( i + j+1) 2 γ

3 2

−

(i + 1)

].

, c′ij = (−1) α 23 (i) α 1 ( j)e i+ j

γ β 2 ( ) ( i + j+1) 2 γ

,

Bivariate generalized linear failure rate distribution parameters

827

β ⎛s⎞ (−1) s +u −l ( ) s −l ⎜⎜ ⎟⎟ k +u +l + 4 γ ⎝ l ⎠ [ γ ( j + 1)]− 2 η (u , l , s, j , k ) = ∑∑ k 2 u =0 l =0 ( + u + 1) u! 2 k +u+l +4 Γβ 2( ), k , s ∈ Ν * . ( ) 2 γ ∞

s

- The first moment of X 1 is given as:

β γ

∞

∞

E ( X 1 ) = ( ) Γ β 2 (1) { Γ *β 2 (1)∑∑ [ ( )

(

γ

γ

)

α 13 α 2 cij + α 23 α 1 cij′ (i + 1)( j + 1)

i =0 j =0

γ − 3 ∞ ∞ − α 13 α 2 ( ) 2 Γ *β 2 ( ) ∑∑ ( ) 2 2 i =0 j = 0 γ

] −α 3

1

β

γ

+( )( ) γ 2

β

γ

+( )( ) γ 2

−

1 2

1 − 2

∞

3 Γ β 2 ( ){ α 3 ( ) 2 γ ∞

∑ i =0

∞

α 13 α 2 { ∑∑ i =0 j =0

(i + 1) cij

(i + 1)

[

3 2

i =0

}

3 2

(i + 1) ( j + 1)

( )

i =0 j =0

γ

(i + 1)

1 2

γ

−( ) 2

1 − 2

cij′

∞

− α 23 α 1Γ *β 2 (1) ∑∑

η (u,0,0, j ,1)

ci

∑ (i + 1)

cij

∞

ci

∞

(i + 1)( j + 1)

η (u,0,0, j ,0) ] }

∞ ∞ cij′ β + ( ) α 23 α 1 ∑∑ η (u, l ,1, j ,0) . 2 i =0 j =0 (i + 1)

- The second moment of X 1 is given as: ∞ ∞ cij β E ( X 12 ) = ( ) Γ β 2 (1){ α 13α 2 ∑ ∑ [ Γ *β 2 (1) ( ) γ (γ ) ( 1 )( 1 ) i + j + i =0 j =0 γ 3 Γ *β 2 ( ) ( ) 2 γ

+2 [ ∞

.∑ i =0

γ (i + 1)

∞

2

]

Γ *β 2 (2) 1 2

cij′

( )

γ

+ [

∑ (i + 1)( j + 1) j =0

γ (i + 1) 2

β γ

] + α 23 α 1 ( ) Γ *β 2 (1) . ]

β γ

∞

+ α3 ( ) ∑ i =0

ci } (i + 1)

( )

γ

3 2

}

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H. M. Aly and H. Z. Muhammed

γ

+β ( ) 2

−

3 2

3 Γ β 2 ( ){ α 3 ( ) 2 γ

∞

∑ i =0

β

ci (i + 1)

3 2

γ

+ ( ) −1 Γ β 2 (2){ α 3 ( ) 2 γ

∞

ci

∑ (i + 1) i =0

)

γ

2

∞

∞

i =0

j =0

∞

∞

∑∑

Γ *β 2 (1) (

− α 23

γ

β

γ

∞

β

γ

∞

+ ( ) ( ) α 13 α 2 ∑ γ 2 i =0 + ( ) ( ) α 23 α 1 ∑ γ 2 i =0

cij′

(i + 1)( j + 1) β γ α 1 ( ) ( ) −1 γ 2

Γ *β 2 (1) ∑ ( )

γ

− α 23 α 1 ( ) ( ) −1 2 2

i =0

∑ j =0

3 2

cij′ 3 2

}

}

(i + 1)( j + 1) η (u ,0,0, j ,0) η (u ,0,0, j ,1) η (u ,0,0, j ,2) } c ij { −2 + ∑ 1 (i + 1) (i + 1) j =0 2 (i + 1) ∞ c ij′ η (u , l ,2, j ,0). ∑ j = 0 (i + 1) ∞

Similarly, we can obtain E ( X 2 ) and E ( X 22 ) respectively as follows. Hence the covariance between X 1 and X 2 will be obtained by substituting in the following equation Cov( X 1 , X 2 ) = E ( X 1 , X 2 ) − E ( X 1 ) E ( X 2 ) .

4. Maximum Likelihood Estimation under Left Random Censoring Suppose {( x11 , x 21 ),K , ( x1n , x 2 n )} is a random sample from BGLFR distribution is subject to random left censoring by an independent vector of any random variables {( y11 , y 21 ),K , ( y1n , y 2 n )} . We observe (t1i , δ1i , t 2i , δ 2i ) , i = 1K n Where ⎧1; x1i > y1i t1i = max( x1i , y1i ) , δ 1i = ⎨ , ⎩0; otherwise ⎧1; x 2i > y 2i t 2i = max( x 2i , y 2i ) , δ 2i = ⎨ ⎩0; otherwise Therefore, if x1i < y1i , x1i is left censored and if x1i > y1i , x1i is observed. In order to write down the likelihood, we note the following four cases: i. When δ1i = δ 2i = 1 , both failure times are observed and the contribution to the likelihood is f (t1i , t 2i ) .

Bivariate generalized linear failure rate distribution parameters

ii.

829

When δ1i = 1 and δ 2i = 0 , the first component fails at t1i and the second component is censored (fails before t 2i ). The contribution to the likelihood t2 i

is ∫ f (t1i , y ) dy . 0

iii.

When δ1i = 0 and δ 2i = 1 , the first component is censored (fails before t1i ) and t1i

the second one fails at t 2i . The contribution to the likelihood is ∫ f ( x, t 2i ) dx . 0

When δ1i = δ 2i = 0 , both x1i and x 2i are censored and the contribution to the likelihood is F (t1i , t 2i ) . According to Dewan and Nandi (2010) the likelihood function of the sample of size n based on (t1i , δ 1i , t 2i , δ 2i ) , i = 1,2,K n is given by:

iv.

n

L(α) = ∏ L(t1i , δ1i ; t 2i , δ 2i ) i =1 n

= ∏ [ f (t1i , t 2i )]δ1i δ 2 i [ϕ(t1i , t 2i )]δ1i (1−δ 2 i ) [ψ (t1i , t 2i )]δ 2 i (1−δ1i ) [ F (t1i , t 2i )](1−δ1i )(1−δ 2 i ) . i =1

Where f (t1i , t 2i ) is the joint pdf of ( x1i , x 2i ) , ϕ(t1i , t 2i ) =

t2 i

∫ f (t

1i

, y ) dy ,

0

ψ (t1i , t 2i ) =

t1i

∫ f ( x, t

2i

) dx , and F (t1i , t 2i ) is the joint cdf of ( x1i , x 2i ) .

0

Let α be the vector of the parameters, and let I 1 , I 2 , I 3 denote the following sets I 1 = {i | t1i < t 2i } , I 2 = {i | t1i > t 2i } and I 3 = {i | t1i = t 2i = t i } , then the likelihood function can be written as 3

L(α ) = ∏∏ L(t1i , δ 1i ; t2i , δ 2i )

(4.1)

j =1 i∈I j

Let n1 , n 2 and n3 denote the number of elements in the sets I 1 , I 2 , I 3 respectively, In the case of BGLFR distribution we can evaluate ϕ(t1i , t 2i ) =

t2 i

∫ f (t

1i

, y ) dy and

0

ψ (t1i , t 2i ) =

t1i

∫ f ( x, t

2i

) dx as follows:

0

ϕ (t1i , t 2i ) =

t2 i

∫ 0

Where

⎧ϕ1 (t1i , t 2i ) if t1i < t 2i ⎪ f (t1i , y ) dy = ⎨ϕ 2 (t1i , t 2i ) if t1i > t 2i ⎪ϕ (t ) if t1i = t 2i = t i ⎩ 3 i

(4.2)

830

H. M. Aly and H. Z. Muhammed

ϕ1 (t1i , t 2i ) = f GLFR (t1i ; α 13 ) [ FGLFR (t 2i ; α 2 ) − FGLFR (t1i ; α 2 )], ϕ 2 (t1i , t 2i ) = f GLFR (t1i ; α 1 ) FGLFR (t 2i ; α 23 ), ϕ 3 (t i ) = f GLFR (t i ; α 1 ) FGLFR (t i ; α 23 ). and

ψ (t1i , t 2i ) =

t1i

∫ 0

where

⎧ψ 1 (t1i , t 2i ) if t1i < t 2i ⎪ f ( x, t 2i ) dx = ⎨ψ 2 (t1i , t 2i ) if t1i > t 2i ⎪ψ (t ) if t1i = t 2i = t i ⎩ 3 i

(4.3)

ψ 1 (t1i , t 2i ) = f GLFR (t 2i ; α 2 ) FGLFR (t1i ; α 13 ), ψ 2 (t1i , t 2i ) = f GLFR (t 2i ; α 23 ) [ FGLFR (t1i ; α 1 ) − FGLFR (t 2i ; α 1 )], ψ 3 (t i ) = f GLFR (t i ; α 2 ) FGLFR (t i ; α 13 ).

The Likelihood on the set I 1 is: n

L1 (α ) = ∏ ([ f 1 (t1i , t 2i )]δ1iδ 2 i ⋅ [ϕ1 (t1i , t 2i )]δ1i (1−δ 2 i ) ⋅ [ψ 1 (t1i , t 2i )]δ 2 i (1−δ1i ) i =1

⋅ [ F1 (t1i , t 2i )](1−δ1i )(1−δ 2 i ) ). (4.4) After substituting from (2.1), (2.2), (4.2) and (4.3) in (4.4) and taking the logarithm we get: ln L1 (α) = ∑ [δ1i ln α 13 + δ 2i ln α 2 + α 13 (1 − δ1i ) ln u1i (β, γ ) + (α 13 − 1)δ1i ln u1i (β, γ ) i∈I1

+ α 2 (1 − δ1i )(1 − δ 2i ) ln u 2i (β, γ ) + (α 2 − 1)δ 2i ln u 2i (β, γ ) + δ1i (1 − δ 2i ) ln[(u 2i (β, γ )) α 2 − (u1i (β, γ )) α 2 ] + δ1i ln(β + γt1i ) γ + δ 2i ln(β + γt 2i ) − β(δ1i t1i + δ 2i t 2i ) − (δ1i t12i + δ 2i t 22i )]. 2

(4.5) − ( βt ki +

γ

2 where u ki ( β , γ ) = 1 − e The Likelihood on the set I 2 is:

t ki2 )

k = 1, 2 , and

α = (α 1 , α 2 , α 3 ) .

n

L2 (α ) = ∏ ([ f 2 (t1i , t 2i )]δ1iδ 2 i ⋅ [ϕ 2 (t1i , t 2i )]δ1i (1−δ 2 i ) ⋅ [ψ 2 (t1i , t 2i )]δ 2 i (1−δ1i ) i =1

(4.6)

.[ F2 (t1i , t 2i )](1−δ1i )(1−δ 2 i ) ). After substituting from (2.1), (2.2), (4.2) and (4.3) in (4.6) and taking the logarithm we get:

Bivariate generalized linear failure rate distribution parameters

831

ln L2 (α) = ∑ [δ1i ln α 1 + δ 2i ln α 23 + α 23 (1 − δ 2i ) ln u 2i (β, γ ) + (α 1 − 1)δ1i ln u1i (β, γ ) i∈I 2

+ α 1 (1 − δ1i )(1 − δ 2i ) ln u1i (β, γ ) + (α 23 − 1)δ 2i ln u 2i (β, γ ) + δ 2i (1 − δ1i ) ln[(u1i (β, γ )) α1 − (u 2i (β, γ )) α1 ] + δ1i ln(β + γt1i ) γ + δ 2i ln(β + γt 2i ) − β(δ1i t1i + δ 2i t 2i ) − (δ1i t12i + δ 2i t 22i )]. 2

(4.7) and the Likelihood on the set I 3 is :

(

n

)

L3 (α) = ∏ [ f 3 (t i )]δ1i δ 2 i ⋅ [ϕ 3 (t i )]δ1i (1−δ 2 i ) ⋅ [ψ 3 (t i )]δ 2 i (1−δ1i ) ⋅ [ F3 (t i )](1−δ1i )(1−δ 2 i ) .

(4.8)

i =1

after substituting from (2.1), (2.2), (4.2) and (4.3) in (4.8) and taking the logarithm we get : ln L3 (α) = ∑ (δ1i δ 2i ln α 3 + δ1i (1 − δ 2i ) ln α 1 + δ 2i (1 − δ1i ) ln α 2 i∈I 3

+ [δ1i + δ 2i (1 − δ1i )](α 123 − 1) ln u i (β, γ ) + α 123 (1 − δ1i )(1 − δ 2i ) ln u i (β, γ )

(4.9)

+ [δ1i + δ 2i (1 − δ1i )] ln(β + γt1i ) γ − βt i [δ1i + δ 2i (1 − δ1i )] − t i2 [δ1i + δ 2i (1 − δ1i )]). 2 Now taking the logarithm of (4.1) and substitute from (4.5), (4.7) and (4.9) we get ln L(α ) ∝ [∑ δ 1i + ∑ δ 1i (1 − δ 2i )] ln α 1 + [∑ δ 2i + ∑ δ 2i (1 − δ 1i )] ln α 2 i∈I 2

i∈I 3

i∈I1

i∈I 3

+ [∑ δ 1i δ 2i ] ln α 3 + [∑ δ 1i ] ln(α 1 + α 3 ) + [∑ δ 2i ] ln(α 2 + α 3 ) i∈I 3

i∈I1

i∈I 2

+ (α 1 + α 3 )[∑ (1 − δ 1i ) ln u1i ( β , γ )] + (α 2 + α 3 )[∑ (1 − δ 2i ) ln u 2i ( β , γ )] i∈I1

i∈I 2

+ α 1 [∑ (1 − δ 1i )(1 − δ 2i ) ln u1i ( β , γ )] + α 2 [∑ (1 − δ 1i )(1 − δ 2i ) ln u 2i ( β , γ )] i∈I 2

i∈I1

+ (α 1 + α 2 + α 3 )[∑ (1 − δ 1i )(1 − δ 2i ) ln u i ( β , γ )] i∈I 3

+ (α 1 + α 2 + α 3 )(∑ [δ 1i + δ 2i (1 − δ 1i )] ln u i ( β , γ )) i∈I 3

+ (α 2 − 1)[∑ δ 2i ln u 2i ( β , γ )] + (α 1 + α 3 − 1)[∑ δ 1i ln u1i ( β , γ )] i∈I1

i∈I1

+ (α 2 + α 3 − 1)[∑ δ 2i ln u 2i ( β , γ )] + (α 1 − 1)[∑ δ 1i ln u1i ( β , γ )] i∈I 2

+ ∑ δ 1i (1 − δ 2i ) ln[(u 2i ( β , γ )) i∈I1

i∈I 2

α2

− (u1i ( β , γ )) α 2 ]

+ ∑ δ 2i (1 − δ1i ) ln[(u1i (β, γ )) α1 − (u 2i (β, γ )) α1 ]. i∈I 2

(4.10)

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H. M. Aly and H. Z. Muhammed

On differentiating (4.10) with respect to α 1 , α 2 , α 3 in turn and equating to zero, we obtain the following likelihood equations: 1 1 [ ∑ δ1i + ∑ δ1i (1 − δ 2i )] + ∑ δ1i + k1 + w1 (αˆ 1 , u1 , u 2 , δ1 , δ 2 ) = 0 , αˆ 1 i∈I 2 αˆ 1 + αˆ 3 i∈I1 i∈I 3 1 1 [∑ δ 2i + ∑ δ 2i (1 − δ1i )] + ∑ δ 2i + k 2 + w2 (αˆ 2 , u1 , u 2 , δ1 , δ 2 ) = 0 , αˆ 2 i∈I1 αˆ 2 + αˆ 3 i∈I 2 i∈I 3

And

1 αˆ 3

∑δ i∈I 3

1i

δ 2i +

1 αˆ 1 + αˆ 3

∑δ i∈I1

1i

+

1 αˆ 2 + αˆ 3

∑δ

i∈I 2

2i

+ k 3 = 0.

(4.11)

where

w1 (αˆ 1 , u1 , u 2 , δ1 , δ 2 ) = ∑ δ 2i (1 − δ1i ) i∈I 2

[u1i (β, γ )]αˆ 1 ln u1i (β, γ ) − [u 2i (β, γ )]αˆ 1 ln u 2i (β, γ ) , [u1i (β, γ )]αˆ 1 − [u 2i (β, γ )]αˆ 1

w2 (αˆ 2 , u1 , u 2 , δ1 , δ 2 ) = ∑ δ1i (1 − δ 2i ) i∈I1

[u 2i (β, γ )]αˆ 2 ln u 2i (β, γ ) − [u1i (β, γ )]αˆ 2 ln u1i (β, γ ) , [u 2i (β, γ )]αˆ 2 − [u1i (β, γ )]αˆ 2

k1 = ∑ ln u1i ( β , γ ) + ∑ ln u i ( β , γ ) + ∑ δ 1i δ 2i (1 − δ 2i ) ln u1i ( β , γ ), i∈I1

i∈I 3

i∈I 2

k 2 = ∑ ln u 2i ( β , γ ) + ∑ ln u i ( β , γ ) + ∑ δ 1i δ 2i (1 − δ 1i ) ln u 2i ( β , γ ), i∈I 2

i∈I 3

i∈I1

i∈I1

i∈I 2

i∈I 3

k 3 = ∑ ln u1i ( β , γ ) + ∑ ln u 2i ( β , γ ) + ∑ ln u i ( β , γ ) , γ − ( βt ki + t ki2 )

2 and u k = u ki (β, γ ) = 1 − e k = 1, 2. These three equations have not explicit form; therefore, their solutions are numerically obtained using Newton-Raphson method as will be seen in Section 5. They are solved simultaneously to obtain αˆ 1 , αˆ 2 and αˆ 3 . For simplification we assumed that β and γ are fixed known. The asymptotic variance-covariance matrix of (αˆ 1 , αˆ 2 , αˆ 3 ) is obtained by inverting the Fisher information matrix with elements that are negatives of expected values of the second order derivatives of logarithms of the likelihood function. In the present situation, it seems appropriate to approximate the expected values by their maximum likelihood estimates [see Cohen (1965)]. Accordingly; we have the following approximate variance-covariance matrix

⎡ ⎢ ⎢ ⎢⎣ where

a11

a12

a13

a 21

a 22

a 23

a31

a32

a33

⎤ ⎥ ⎥ ⎥⎦

−1

Bivariate generalized linear failure rate distribution parameters

a11 = −

∂ 2 ln L ∂α 12 αˆ ,αˆ 1

1 1 [ δ + ∑ δ 1i (1 − δ 2i )] + 2 ∑ 1i αˆ 1 i∈I 2 (αˆ 1 + αˆ 3 ) 2 i∈I 3

= 2 ,αˆ 3

∑δ i∈I1

833

1i

+ v1 (αˆ 1 , u1 , u 2 , δ 1 , δ 2 ), a 22 = −

∂ ln L ∂α 22 αˆ ,αˆ 2

1

1 1 [ δ + ∑ δ 2i (1 − δ 1i )] + 2 ∑ 2i αˆ 2 i∈I1 (αˆ 2 + αˆ 3 ) 2 i∈I 3

= 2 ,αˆ 3

∑δ

i∈I 2

2i

+ v 2 (αˆ 2 , u1 , u 2 , δ 1 , δ 2 ),

a33 = −

∂ ln L ∂α 32 αˆ ,αˆ 2

1

a12 = − a13 = − a 23 = −

∂ ln L ∂α 1 ∂α 2

= 2 ,αˆ 3

2

∂ 2 ln L ∂α 1 ∂α 3 ∂ 2 ln L ∂α 2 ∂α 3

1 αˆ 32

∑δ i∈I 3

1i

δ 2i +

1 (αˆ1 + αˆ 3 ) 2

∑δ i∈I1

1i

+

1 (αˆ 2 + αˆ 3 ) 2

∑δ

i∈I 2

2i

,

= 0, αˆ1 ,αˆ 2 ,αˆ 3

= αˆ1 ,αˆ 2 ,αˆ 3

1 (αˆ 1 + αˆ 3 ) 2

= αˆ1 ,αˆ 2 ,αˆ 3

1 (αˆ 2 + αˆ 3 ) 2

v1 (αˆ 1 , u1 , u 2 , δ1 , δ 2 ) = ∑ δ 2i (1 − δ1i ) i∈I 2

∑δ i∈I1

1i

∑δ

i∈I 2

2i

,

,

[u1i (β, γ )]αˆ 1 [u 2i (β, γ )]αˆ 1 [ln u1i (β, γ ) − ln u 2i (β, γ )]2 , ([u1i (β, γ )]αˆ 1 − [u 2i (β, γ )]αˆ 1 ) 2

and [u1i (β, γ )]αˆ 2 [u 2i (β, γ )]αˆ 2 [ln u1i (β, γ ) − ln u 2i (β, γ )] 2 . v 2 (αˆ 2 , u1 , u 2 , δ1 , δ 2 ) = ∑ δ1i (1 − δ 2i ) ([u1i (β, γ )]αˆ 2 − [u 2i (β, γ )]αˆ 2 ) 2 i∈I1 Now we state the asymptotic normality results to obtain the asymptotic confidence intervals of α 1 , α 2 and α 3 . It can be stated as follows: [ n (αˆ − α)] → N 3 (0 , I −1 (α)) as n → ∞

(4.12)

Where I (α) is the variance-covariance matrix, αˆ = (αˆ 1 , αˆ 2 , αˆ 3 ) , and α = (α 1 , α 2 , α 3 ) . −1

Since α is unknown in (4.12), I −1 (α ) is estimated by I −1 (αˆ ) ; the asymptotic variance-covariance matrix that defined above and this can be used to obtain the asymptotic confidence intervals of α 1 , α 2 and α 3 .

5. Numerical Results To illustrate the results, a numerical example is given where the MLEs, their approximate variance- covariance matrix, relative bias, mean square error and asymptotic confidence intervals are determined using Mathcad (2001) software. Thirty seven observations from BGLFR distribution ( X 1 , X 2 ) have been obtained

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H. M. Aly and H. Z. Muhammed

from Sarhan et al. (2011) and are listed in table (1). Two samples of thirty seven observations are selected at random from, exponential distribution with different parameters θ 1 = 2.5 and θ 2 = 3.5 where Yi ~ exp ( θ i ), i = 1, 2 are also listed in table(1). After censoring the sample observations with the generated times Yi , i = 1,2 , we observe (t1i , δ 1i , t 2i , δ 2i ) that defined in Section 4 as in table (2). From The likelihood equations (4.11), the Maximum likelihood estimators (MLEs) are obtained by iterative technique of Newton-Raphson using Mathcad (2001) software, their approximate variance- covariance matrix, relative bias, Mean Square Error (MSE) and a 95% asymptotic confidence intervals are listed in table (3).

Table (1): Samples observations from BGLFR distribution and Exponential distribution with θ 1 = 2.5 and θ 2 = 3.5

No. 1 2 3 4 5 6 7 8 9 10 11 12 13

x1 0.26 0.63 0.19 0.66 0.4 0.49 0.08 0.69 0.39 0.82 0.72 0.66 0.25

x2 y1 0.20 0.6 0.18 0.66 0.19 0.21 0.85 0.42 0.4 0.08 0.49 0.7 0.08 0.14 0.71 0.48 0.39 0.96 0.48 0.77 0.72 0.00462 0.62 0.85 0.09 1.89

y2 0.08 0.22 0.08 0.15 0.09 0.16 0.54 0.24 0.19 0.08 0.51 0.2 0.1

No. 20 21 22 23 24 25 26 27 28 29 30 31 32

x1 0.34 0.53 0.54 0.51 0.76 0.64 0.26 0.16 0.44 0.25 0.55 0.49 0.24

x2 y1 0.34 0.26 0.39 0.05 0.07 0.02 0.28 0.25 0.64 0.31 0.15 0.06 0.48 0.1 0.16 0.00128 0.6 0.2 0.14 0.53 0.11 0.07 0.49 0.39 0.24 0.16

y2 0.06 0.47 0.06 0.53 0.09 0.36 0.11 0.09 0.6 0.05 0.19 0.24 0.01

Table (1): Samples observations from BGLFR distribution and Exponential distribution with θ 1 = 2.5 and θ 2 = 3.5 (continued)

14 15 16 17 18 19

0.41 0.16 0.18 0.22 0.42 0.36

0.03 0.75 0.18 0.14 0.42 0.52

0.25 0.2 0.72 0.32 1.15 0.1

0.55 0.56 0.1 0.24 0.0097 0.54

33 34 35 36 37

0.44 0.42 0.27 0.28 0.02

0.30 0.03 0.47 0.28 0.02

1.89 0.52 0.21 0.07 0.29

0.17 0.21 0.05 0.22 0.0049

Bivariate generalized linear failure rate distribution parameters

835

Table (2): values of (t1i , δ 1i , t 2i , δ 2i ) after left random censoring

No. T1

T2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.20 0.22 0.19 0.85 0.4 0.49 0.54 0.71 0.39 0.48 0.72 0.62 0.1 0.55 0.75 0.18 0.24 0.42 0.54

0.60 0.66 0.21 0.66 0.4 0.7 0.14 0.69 0.96 0.82 0.72 0.85 1.89 0.41 0.2 0.72 0.32 1.15 0.36

δ1

δ2 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1

1 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0

No. T1

T2

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

0.34 0.47 0.07 0.53 0.64 0.36 0.48 0.16 0.6 0.14 0.19 0.49 0.24 0.30 0.21 0.47 0.28 0.02

0.34 0.53 0.54 0.51 0.76 0.64 0.26 0.16 0.44 0.53 0.55 0.49 0.24 1.89 0.52 0.27 0.28 0.29

δ1

δ2 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0

1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1

Table (3): The estimates, Relative Bias, MSE, Confidence interval and variancecovariance matrix

Parameter

α1 α2 α3

MLE 0.846 0.534 0.307

Relative MSE Confidence Variance-Covariance Bias Interval matrix 0.046 0.002152 (0.783-0.91) 0.039 0.0007 - 0.005 0.034 0.001169 (0.491-0.577) 0.0007 0.018 - 0.003 0.00741 0.000055 (0.274-0.341) - 0.005 - 0.003 0.011

6. Conclusion In this paper we have provide a review of the new bivariate generalized linear failure rate distribution whose marginals are generalized linear failure rate

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H. M. Aly and H. Z. Muhammed

distributions. This new distribution is a singular distribution, Sarhan et al. (2011) concluded that it can be used quit effectively instead of the Marshall-Olkin bivariate exponential model or the bivariate generalized exponential model when there are ties in the data.

We have obtained the moment generating function for the BGLFR distribution and then obtained different joint and marginal moments for this distribution. And we have considered the maximum likelihood estimators of the unknown parameters of the BGLFR distribution when both components of the bivariate variable are subject to left random censoring. The associated asymptotic variance-covariance matrix for the unknown parameters was derived. To illustrate the results, a numerical example was given where the MLEs, their approximate variance- covariance matrix, relative bias, mean square error and asymptotic confidence intervals were determined using computer facilities and Mathcad (2001) software.

References 1) C. Cohen, Maximum Likelihood Estimation in The Weibull Distribution Based On Complete and Censored Samples, Technometrics, 7(1965), 579588.

2) I. Dewan and S. Nandi, an EM Algorithm for Estimation of Parameters of Bivariate Generalized Exponential Distribution under Random Left Censoring. Technical report, Indian Statistical Institute, Delhi Center (2010).

3) W. Marshall and I. Olkin, a Multivariate Exponential Distribution. Journal of the American Statistical Association, 62 (1967), 30 -44.

4) M. Sarhan and D. Kundu Generalized Linear Failure Rate Distribution. Communications in Statistics – Theory and Methods, 38(5) (2009), 642660.

Bivariate generalized linear failure rate distribution parameters

837

5) M. Sarhan, D. C. Hamilton, B. Smith and D. Kundu, the Bivariate Generalized Linear Failure Rate Distribution and Its Multivariate Extension Computational Statistics and Data Analysis, 55 (2011), 644 – 654.

Received: October, 2011