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On a bivariate generalized inverted Kumaraswamy distribution Hiba Z. Muhammed Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt
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Article history: Received 6 April 2019 Received in revised form 20 August 2019 Available online xxxx Keywords: Kumaraswamy distribution Inverted Kumaraswamy distribution Generalized inverted Kumaraswamy distribution Maximum likelihood estimation
a b s t r a c t In this paper, a bivariate generalized inverted Kumaraswamy distribution is presented. Different properties of this distribution are discussed. The maximum likelihood estimates for the unknown parameters of this distribution and their approximate variance– covariance matrix are obtained. Bayesian estimators are also obtained for the unknown parameters of this model explicitly. Some simulations to see the performances of the MLEs are performed. One data analysis also has been performed for illustrative purpose. © 2020 Published by Elsevier B.V.
1. Introduction The Kumaraswamy distribution has been considered by Jones [1]. He pointed out some similarities and differences between the beta and Kumaraswamy (Kum) distributions. He introduced the cdf and pdf of the Kum distribution respectively, as following FK (t ; α, β) = 1 − (1 − t α )β fK (t ; α, β) = αβ t α−1 (1 − t α )β−1 ,
0 < t < 1.
Recently, Abd AL-Fattah et al. [2] derived the inverted Kumaraswamy (IKum) distribution from the Kum distribution using the transformation Z = T −1 − 1 Where T ∼ Kum (α, β ) and α and β are shape parameters, then Z has an IKum (α, β ) with cdf and pdf respectively FIK (z ; α, β) = [1 − (1 + z)−α ]β fIK (z ; α, β) = αγ (1 + z)−α−1 [1 − (1 + z)−α ]β−1 . where z > 0, α > 0 and β > 0. Iqbal et al. [3] generalized the IKum distribution by using power transformation X = Z γ Where Z ∼ IKum (α, β ), then X has a generalized inverted Kumaraswamy [denoted by X ∼ GIKum (α, β, γ )] with the following cdf and pdf respectively, FGIK (x; α, γ , β) = [1 − (1 + xγ )−α ]β fGIK (x; α, γ , β) = αβγ xγ −1 (1 + xγ )−α−1 [1 − (1 + xγ )−α ]β−1 For x > 0, α > 0, β > 0 and γ > 0. Iqbal et al. [3] concluded that this model is flexible enough to accommodate both monotonic as well as non-monotonic failure rates. The pdf of GIKum is positively skewed for all parameters values and for β = 0.5 the pdf of GIKum is monotonically decreasing. E-mail address:
[email protected]. https://doi.org/10.1016/j.physa.2020.124281 0378-4371/© 2020 Published by Elsevier B.V.
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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In this paper, the bivariate version of the GIKum is introduced for the first time using an idea similar to that of Theorem 3.2 proposed by Marshall and Olkin [4]. These authors introduced a multivariate exponential distribution whose marginals have exponential distributions and proposed a bivariate Weibull distribution. Contribution to this idea, Sarhan and Balakrishnan [5] introduced a bivariate distribution that is more flexible than the bivariate exponential distribution. Using the maximum instead of the minimum in the Marshall and Olkin scheme, Kundu and Gupta [6,7] and Sarhan et al. [8] introduced the bivariate generalized exponential, Bivariate proportional reversed hazard and bivariate generalized linear failure rate distributions, respectively. And Muhammed [9–11] introduced the bivariate inverse Weibull distribution, bivariate Dagum Distribution and bivariate generalized Burr distributions respectively. The proposed bivariate GIKum distribution (BGIKum) is constructed from three independent GIKum 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. The BGIKum model can be interpreted as Competing risk, Shock, Stress and Maintenance Model as following Competing risks model: Assume a system has two components, labeled 1 and 2, and the survival time of component i is denoted by Xi , 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 can fail at the same time due to cause 3. Let Ui be the lifetime of cause i, i = 1, 2, 3. If U1 , U2 and U3 follow a GIKum distribution, then (X1 , X2 ) follow the BGIKum 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 destroys component 2 immediately, while if the shock from source 3 hits the system it destroys both components immediately. Let Ui denote the inter-interval times between the shocks in source i, i = 1, 2, which follow the distribution GIKum. If X1 , X2 denote the survival times of the components, then (X1 , X2 ) follows the BGIKum model. Stress Model: Suppose a system has two components. Each component is subject to individual independent stress say U1 and U2 respectively. The system has an overall stress U3 which has been transmitted to both the components equally, independent of their individual stresses. Therefore, the observed stress at the two components are X1 = max(U1 , U3 ) and X2 = max(U2 , U3 ) respectively. Maintenance Model: Suppose a system has two components and it is assumed that each component has been maintained independently and also there is an overall maintenance. Due to component maintenance, suppose the lifetime of the individual components is increased by Ui amount and because of the overall maintenance, the lifetime of each component is increased by U3 amount. Therefore, the increased lifetimes of the two components are X1 = max(U1 , U3 ) and X2 = max(U2 , U3 ) respectively. The paper is organized as follows: In Section 2, the BGIKum distribution is introduced and representations for the cumulative distribution function (cdf) and probability density function (pdf) are obtained. Some basic properties of this model are presented in Section 3. Point and interval estimation for BGIKum distribution are provided in Section 4. A simulation study is discussed in Section 5. An absolutely continuous BGIKum distribution is introduced in Section 6. For illustrative purpose an empirical application is presented in Section 7. Finally conclude the paper in Section 8. 2. Model description The Marshal–Olkin bivariate inverted Kumaraswamy distribution will be defined as follows: Assume that U1 , U2 and U3 are three independent random variables such that Ui ∼ GIK (α, γ , βi ) i = 1, 2, 3. Define Xi = Max (Ui , U3 ) i = 1, 2. Then the bivariate vector (X1 , X2 ) has BGIKum distribution with parameters (β1 , β2 , β3 , α, γ ), denoted by BGIKum (β1 , β2 , β3 , α, γ ). Then, the joint cdf of (X1 , X2 ) is given as follows FX1 ,X2 (x1 , x2 ) = FGIK (x1 ; β1 ) FGIK (x2 ; β2 ) FGIK (x3 ; β3 ) γ
γ
γ
FX1 ,X2 (x1 , x2 ) = [1 − (1 + x1 )−α ]β1 [1 − (1 + x2 )−α ]β2 [1 − (1 + x3 )−α ]β3 where x3 = min(x1 , x2 ). Proposition 1. If (X1 , X2 ) ∼ BGIKum (β1 , β2 , β3 , α, γ ). Then, the joint cdf of (X1 , X2 ) can be written as
⎧ ⎨FGIK (x1 ; β13 ) FGIK (x2 ; β2 ) , FX1 ,X2 (x1 , x2 ) = FGIK (x1 ; β1 ) FGIK (x2 ; β23 ) , ⎩ FGIK (x; β123 ) ,
x1 < x2 x1 > x2 x1 = x2
where β13 = β1 + β3 , β23 = β2 + β3 and β123 = β1 + β2 + β3 . Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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Proposition 2. If (X1 , X2 ) ∼ BGIKum (β1 , β2 , β3 , α, γ ). Then, the joint pdf of (X1 , X2 ) is given as
⎧ f (x ; β ) f (x ; β ) , ⎪ ⎪ GIK 1 13 GIK 2 2 ⎨ fGIK (x1 ; β1 ) fGIK (x2 ; β23 ) , fX1 ,X2 (x1 , x2 ) = ⎪ β3 ⎪ ⎩ fGIK (x; β123 ) , β123
x1 < x2 x1 > x2 x1 = x2
Consequently, when γ = 1 the joint pdf of BIKum distribution is given as a special case from BGIKum as follows
⎧ fIK (x1 ; β13 ) fIK (x2 ; β2 ) , ⎪ ⎪ ⎨ fIK (x1 ; β1 ) fIK (x2 ; β23 ) , fX1 ,X2 (x1 , x2 ) = ⎪ β ⎪ ⎩ 3 fIK (x; β123 ) , β123
x1 < x2 x1 > x2 x1 = x2
3. Basic properties In this section, some properties of BGIKum distribution such as some reliability functions, some statistical measures, marginal and conditional densities and product moments will be discussed. 3.1. Reliability and reversed hazard functions The joint survival function of BGIKum distribution is
⎧ ⎪ ⎨FGIK (x1 ; β13 ) [FGIK (x2 ; β2 ) − 1] + 1 − FGIK (x2 ; β23 ) , x1 < x2 SX1 ,X2 (x1 , x2 ) = FGIK (x2 ; β23 ) [FGIK (x1 ; β1 ) − 1] + 1 − FGIK (x1 ; β13 ) , x1 > x2 ⎪ ⎩ 1 − FGIK (x2 ; β123 ) , x1 = x2 The reversed hazard function of BGIKum distribution is
⎧ ⎨r1(x1 ,x2 ) , r (x1 , x2 ) = r2(x1 ,x2 ) , ⎩ r3 (x) ,
x1 < x2 x1 > x2 x1 = x2
where γ −1 γ −1
r1 (x1 , x2 ) =
(αγ )2 β13 β2 x1
x2
γ
γ
(1 + x1 )−α−1 (1 + x2 )−α−1
γ x )−α ][1
γ x )−α ]
− (1 + 2 [1 − (1 + 1 γ )−α−1 γ )−α−1 ( γ −1 γ −1 ( 1 + x2 1 + x1 x β β x (αγ ) 1 23 1 2 ] ] [ [ r2 (x1 , x2 ) = ( ( γ )−α γ )−α 1 − 1 + x2 1 − 1 + x1 2
r3 (x) =
αγ β3 xγ −1 (1 + xγ )−α−1 [1 − (1 + xγ )−α ]
3.2. Factorization property It should be mentioned that the BGIKum distribution has both an absolute continuous part and a singular part. The joint cdf of the BGIKum can be factorized into absolutely continuous part and singular part as follows in the following form FX1 ,X2 (x1 , x2 ) =
β12 β3 Fa (x1 , x2 ) + Fs (x3 ) β123 β123
where x3 = min (x1 , x2 ), Fs (x3 ) = FGIK (x; β123 ) and Fa (x1 , x2 ) =
β123 β3 FGIK (x1 ; β1 ) FGIK (x2 ; β2 ) FGIK (x3 ; β3 ) − FGIK (x; β123 ) . β12 β12
Note that: Fs (., .) and Fa (., .) are the singular and the absolutely continuous part respectively. Accordingly, the pdf of the BGIKum can be factorized into absolutely continuous part and singular part as follows fX1 ,X2 (x1 , x2 ) =
β12 β3 fa (x1 , x2 ) + fs (x3 ) β123 β123
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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Fig. 1. Surface plot for BGIKum for different parameters values.
where
{ β123 fGIK (x1 ; β13 ) fGIK (x2 ; β2 ) , fa (x1 , x2 ) = β12 fGIK (x1 ; β1 ) fGIK (x2 ; β23 ) ,
x1 < x2 x1 < x2
and fs (x3 ) = fGIK (x; β123 ). Clearly, here fa (x1 , x2 ) and fs (x3 ) are the absolutely continuous and singular parts respectively. Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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Fig. 1 showed different surface plots of the absolutely continuous part of the joint pdf of the BGIKum distribution for different parameters values. 3.3. The mode and median The median for the absolutely continuous BGIKum distribution is given as follows
)1/α ( − 1}1/γ { 1 − 2−1/β123 The mode for the absolutely continuous BGIKum distribution for γ = 1 is given as follows
{( } ( )1/α ( )1/α )1/α ( )1/α 1 + αβ2 1 + αβ1 1 + αβ23 1 + αβ13 − 1, − 1} and − 1, −1 . { α+1 α+1 α+1 α+1 3.4. Marginal and conditional densities Proposition 3. If (X1 , X2 ) ∼ BGIKum (β1 , β2 , β3 , α, γ ). Then, (i) Xi ∼ GIK (βi3 ) , such that βi3 = βi + β3 and i = 1, 2. (ii) max (X1 , X2 ) ∼ GIK (β123 ). (iii) The conditional density of Xi given Xj = xj , i ̸ = j is as follows
) ⎧ (1) ( fi/j xi /xj , ⎪ ⎨ ) ( ) ( fi/j xi /xj = fi/(2j ) xi /xj , ⎪ ⎩ (3) fi (xi ) ,
xi < xj xi > xj xi = xj
where (1)
(
(2)
(
(3)
(xi ) =
fi/j fi/j fi
αγ β13 β2 γ −1 γ γ xi (1 + xi )−α−1 [1 − (1 + xi )−α ]β13 −1 [1 − (1 + xj )−α ]β3 −1 β23
xi / xj =
)
γ −1
xi /xj = αγ β1 xi
)
γ −1
β3 xi
γ
(1 + xi )−α−1 [1 − (1 + xi )−α ]β1 −1 γ
(1 + xi )−α−1 [1 − (1 + xi )−α ]β123 −1
γ −1
β23 xj
γ
(1 + xj )−α−1 [1 − (1 + xj )−α ]β123 −1
.
3.5. Product moments According to Proposition 3 the marginal distributions of the vector (X1 , X2 ) are GIKum distributions, then the moments of X1 and X2 can be obtained directly from the following marginals: [β13 −1]
∑
E X1 = αβ13
( r)
(−1)
i
(
β13 − 1 i
) ( B
γ
) (
r
i=0 [β23 −1]
∑
E X2r = αβ23
( )
(−1)i
(
β23 − 1 i
i=0
r
B
γ
+ 1, α (1 + i) − + 1, α (1 + i) −
r
) and
γ r
γ
)
.
where B(., .) is the beta function. Now the product moments of BGIKum will be presented Proposition 4. The rth and sth joint moments of the X1 and X2 , denoted by µ′r ,s is given by [β2 −1] [β13 −1]
E X1r X2s =
(
)
∑ ∑ j=0
(
Kij B s + r + γ −
i=0
(
· 3 F2 s + r + γ −
1
r
γ
, −α (1 + j) − s − r − 1 − γ +
j=0
)
γ )
r
i=0
[β123 −1]
+
i=0
− 1, −α (i + 1) ;
1
, −α (j + 1) − 1; 1 γ γ γ ( ) [β23 −1] [β1 −1] ∑ ∑ 1 1 ˇ Kij B s + r + γ − , −α (1 + j) − s − r − 1 − γ + + γ γ
∑
,
1
K´ i B
(
s+r
γ
+ 1, α (1 + i) −
s+r
γ
)
.
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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where
( )( ) α 2 γ β2 β13 β2 − 1 i+j β13 − 1 ( − 1) r i j −1 γ ( )( ) 2 α γ β1 β23 β23 − 1 i+j β1 − 1 = s (−1) i j −1 γ ( ) β −1 = β3 α (−1)i 123
Kij = Kˇ ij K´ i
i
And B(., .) is the beta function. 4. Estimation of BGIKum distribution The estimation of the unknown parameters for the BGIKum distribution is considered using maximum likelihood and Bayesian estimation in the following two sub sections 4.1. Maximum likelihood estimation In this section, the maximum likelihood estimation for the BGIKum distribution will be considered; maximum likelihood has not been considered for this distribution before. Suppose {(x11 , x21 ), . . . , (x1n , x2n )} is a random sample from BGIKum (β1 , β2 , β3 , α, γ ) distribution. Consider the following notation I1 = {i; x1i < x2i },
|I1 | = n1 ,
I2 = {i; x1i > x2i },
I = I1 ∪ I2 ∪ I3 ,
I3 = {x1i = x2i = xi }, |I3 | = n3 , and n1 + n2 + n3 = n.
| I 2 | = n2 ,
The log-likelihood function of the sample of size n is given by
∑
L(θ ) = ln l(θ ) =
∑
ln f1 (x1i , x2i ) +
i∈I1
ln f2 (x1i , x2i ) +
∑
ln f3 (xi )
i∈I3
i∈I2
L (θ) = 2n1 ln γ + 2n1 ln α + 2n2 ln γ + 2n2 ln α + 2n3 ln α + n3 ln α + n3 ln γ
+ n1 ln β13 + n1 ln β2 + n1 ln β2 + n2 lnβ23 + n2 ln β1 + n3 ln β3 ] ] ∑ [ ∑ [ ( ( γ )−α γ )−α + (β13 − 1) + (β2 − 1) ln 1 − 1 + x2i ln 1 − 1 + x1i I1
I1
+ (β1 − 1)
∑
[
γ )−α
(
ln 1 − 1 + x1i
]
+ (β23 − 1)
∑
[
γ )−α
(
ln 1 − 1 + x2i
]
I
I2
⎤ ⎡2 ] ∑ ∑ ∑ ∑ [ ( ) γ −α ln xi ⎦ ln x2i + ln x1i + ln 1 − 1 + xi + (γ − 1) ⎣ + (β123 − 1) I3
I1 ∪I2
I1 ∪I2
I3
⎤ ∑ ( ) ∑ ( ) ) ∑ ( γ γ γ ln 1 + x2i + ln 1 + xi ⎦ . − (α + 1) ⎣ ln 1 + x1i + ⎡
I3
I1 ∪I2
I1 ∪I2
The likelihood equations are as follows ∑ ( ) ∑ ( ) n1 n2 + + A x1i ; α, ˆ γˆ + A xi ; α, ˆ γˆ = 0
βˆ 13
βˆ 1
I1 ∪I2
n1
n2
∑ (
+
βˆ 2 n1
βˆ 13
βˆ 23
+
n2
βˆ 23
+
I3
A x2i ; α, ˆ γˆ +
)
∑ (
I1 ∪I2
+
n3
βˆ 3
2n1 + 2n2 + n3
A xi ; α, ˆ γˆ = 0
)
I3
+
∑ (
A x1i ; α, ˆ γˆ +
)
∑ (
I1
B x1i ; γˆ −
αˆ + (βˆ 2 − 1)
∑ (
)
I1 ∪I2
B x2i ; γˆ −
C x2i ; α, ˆ γˆ + (βˆ 1 − 1)
+ (βˆ 123 − 1)
∑ (
A xi ; α, ˆ γˆ = 0
)
I3
∑ (
)
I1 ∪I2
)
I1
∑ (
(
)∑ (
B xi ; γˆ + βˆ 13 − 1
)
I3
C x1i ; α, ˆ γˆ
)
I
( )∑ ( 1 ) ˆ C x1i ; α, ˆ γˆ + β23 − 1 C x2i ; α, ˆ γˆ
∑ ( I2
∑ (
)
I2
∑ (
−
A x2i ; α, ˆ γˆ +
)
I2
C xi ; α, ˆ γˆ = 0
)
I3
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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⎡
⎤ ∑ ( ) ∑ ( ) ∑ ( ) 2n1 + 2n2 + n3 − (α + 1) ⎣ D x1i ; γˆ − D x2i ; γˆ − D xi ; γˆ ⎦ γˆ I1 ∪I2 I1 ∪I2 I3 )∑ ( ∑ ( ∑ ( ) ) ( ) ˆ ˆ ˆ + (β2 − 1) E x2i ; α, ˆ γˆ + (β1 − 1) E x1i ; α, ˆ γˆ + β23 − 1 E x2i ; α, ˆ γˆ I1
+ (βˆ 123 − 1)
I
I2
( )2 ∑ ( ) ˆ E xi ; α, ˆ γˆ + β13 − 1 E x1i ; α, ˆ γˆ + M = 0
∑ (
)
I3
I1
where A (x; α, γ ) = ln[1 − (1 + xγ )−α ], B (x; γ ) = ln(1 + xγ )−α , C (x; α, γ ) = D (x; γ ) =
(1 + xγ )−α ln(1 + xγ )−α
xγ ln x 1 + xγ
1 − (1 + xγ )−α
,
, γ −α−1
α xγ (1+x ) ln x γ −α 1 − (1 + x ) ∑ ∑ ∑ and M = ln x1i + ln x2i + ln xi . E (x; α, γ ) =
I1∪ I2
I1∪ I2
I3
The numerical solutions for these equations will be considered to obtain βˆ 1 , βˆ 2 , βˆ 3 , αˆ and γˆ as will be shown in Section 5. The asymptotic variance–covariance matrix obtained as
⎡
a11 ⎢a21 ⎢ ⎢a31 ⎣a 41 a51
a12 a22 a32 a42 a52
a13 a23 a33 a43 a53
a14 a24 a34 a44 a54
⎤−1
a15 a25 ⎥ ⎥ a35 ⎥ a45 ⎦ a55
where
⏐ ⏐ ∂ 2 ln L ⏐⏐ n1 n2 ∂ 2 ln L ⏐⏐ = 0, = + , a = − 12 ∂β1 ∂β2 ⏐βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ ∂β12 ⏐βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ (βˆ 1 + βˆ 3 )2 βˆ 12 ⏐ ⏐ ∂ 2 ln L ⏐⏐ n2 n1 ∂ 2 ln L ⏐⏐ n1 − = = , a = − + 2, 22 ∂β1 ∂β3 ⏐βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ ∂β22 ⏐βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ (βˆ 1 + βˆ 3 )2 (βˆ 2 + βˆ 3 )2 βˆ 2 ⏐ ∑ ∑ ∂ 2 ln L ⏐⏐ − = −[ C (x1i ; α, ˆ γˆ ) + C (xi ; α, ˆ γˆ )] ⏐ ∂β1 ∂α βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ I1 ∪I2 I3 ⏐ ∑ ∑ ∂ 2 ln L ⏐⏐ − = −[ E(x1i ; α, ˆ γˆ ) + E(xi ; α, ˆ γˆ )] ⏐ ∂β1 ∂γ βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ I1 ∪I2 I3 ⏐ ∂ 2 ln L ⏐⏐ n2 − = ∂β2 ∂β3 ⏐βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ (βˆ 2 + βˆ 3 )2 ⏐ ∑ ∑ ∂ 2 ln L ⏐⏐ − = −[ C (x2i ; α, ˆ γˆ ) + C (xi ; α, ˆ γˆ )] ⏐ ∂β2 ∂α
a11 = − a13 = a14 =
a15 =
a23 = a24 =
βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ
I ∪I
I
β1 ,β2 ,β3 ,γˆ ,αˆ
I1
I2
1 2 3 ⏐ ∑ ∑ ∂ 2 ln L ⏐⏐ a25 = − = −[ E(x2i ; α, ˆ γˆ ) + E(xi ; α, ˆ γˆ )] ∂β2 ∂γ ⏐βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ I1 ∪I2 I3 ⏐ ∂ 2 ln L ⏐⏐ n2 n2 n1 a33 = − = + + 2, 2 ⏐ 2 2 ∂β3 βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ (βˆ 1 + βˆ 3 ) (βˆ 2 + βˆ 3 ) βˆ 2 ⏐ 2 ∑ ∑ ∑ ⏐ ∂ ln L ⏐ a34 = − = −[ C (x1i ; α, ˆ γˆ ) + C (x2i ; α, ˆ γˆ ) + C (xi ; α, ˆ γˆ )] ⏐ ∂β3 ∂α ˆ ˆ ˆ
I3
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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⏐ ∑ ∑ ∑ ∂ 2 ln L ⏐⏐ a35 = − = −[ E(x1i ; α, ˆ γˆ ) + E(x2i ; α, ˆ γˆ ) + E(xi ; α, ˆ γˆ )] ⏐ ∂β3 ∂γ βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ I1 I2 I3 ⏐ ∑ ∑ 2n1 + 2n2 + n3 ∂ 2 ln L ⏐⏐ − (β13 − 1) G(x1i ; α, γ ) − (β2 − 1) G(x2i ; α, γ ) = a44 = − ⏐ 2 2 ∂α βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ α I1 I1 ∑ ∑ ∑ − (β1 − 1) G(x1i ; α, γ ) − (β23 − 1) G(x2i ; α, γ ) − (β123 − 1) G(xi ; α, γ ) I2
I2
I3
⏐ ∑ ∑ ∑ ∂ 2 ln L ⏐⏐ a45 = − = −[ D(x1i ; α, ˆ γˆ ) + D(x2i ; α, ˆ γˆ ) + D(xi ; α, ˆ γˆ )] ⏐ ∂α∂γ βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ I1 ∪I2 I1 ∪I2 I3 ∑ ∑ ∑ − (β13 − 1) H(x1i ; α, γ ) − (β2 − 1) H(x2i ; α, γ ) − (β1 − 1) H(x1i ; α, γ ) I1
− (β23 − 1)
I1
∑
H(x2i ; α, γ ) − (β123 − 1)
I2
∑
I2
H(xi ; α, γ )
I3
⏐ ∑ ∑ 2n1 + 2n2 + n3 ∂ 2 ln L ⏐⏐ Q (x2i ; α, γ ) = Q (x1i ; α, γ ) − (β2 − 1) − (β13 − 1) ⏐ 2 2 ∂γ βˆ 1 ,βˆ 2 ,βˆ 3 ,γˆ ,αˆ γ I1 I1 ∑ ∑ ∑ − (β1 − 1) Q (x1i ; α, γ ) − (β23 − 1) Q (x2i ; α, γ ) − (β123 − 1) Q (xi ; α, γ )
a55 = −
I2
+ (α + 1)[
I2
∑
P(x1i ; γˆ ) +
I1 ∪I2
G (x; α, γ ) =
∑
P(x2i ; γˆ ) +
I1 ∪I2
−α−1
γ
x (ln x)
[
ln (1 + xγ ) 1 − (1 + xγ )−α
] −1
2
(1 + xγ )2
Q (x, α, γ ) =
P(xi ; γˆ )].
I3
−(1 + xγ ) [ln(1 + xγ )]2 [1 − (1 + xγ )−α ]2
H (x, α, γ ) = −αγ xγ −1 (1 + xγ ) P (x; γ ) =
I3
∑
and
α xγ (ln x)2 (1 + xγ )−α−1 [1 − xγ (α + 1) (1 + xγ )−1 − (2α + 1)xγ (1 + xγ )−α−1 ] [1 − (1 + xγ )−α ]2
The Asymptotic Confidence Interval Now, The asymptotic normality results will be considered to obtain the asymptotic confidence intervals of β1 , β2 , β3 , α and γ , It can be stated as follows
√
n[(αˆ − α ), (γˆ − γ ), (βˆ 1 − β1 ), (βˆ 2 − β2 ), (βˆ 3 − β3 )] → N5 (0, I(θ )−1 ) as n → ∞
where I −1 (θ ) is the variance–covariance matrix, θˆ = (βˆ 1 , βˆ 2 , β3 , α, ˆ γˆ ). and θ = (β1 , β2 , β3 , α, γ ). I −1 (θ ) is estimated by I −1 (θˆ ); The asymptotic variance–covariance matrix that defined above and this can be used to obtain the asymptotic confidence intervals of β1 , β2 , β3 , α and γ . 4.2. Bayesian estimation In this section the Bayesian analysis for the BGIKum distribution is considered. The explicit Bayes estimators under the squared error loss function are obtained. When the shape parameters α , and γ is known, we assume the same conjugate prior on β1 , β2 and β3 as considered by Kundu and Gupta [12] as follows: Assume β1 , β2 and β3 are independent, and distributed as gamma as following
πi (βi ) =
ba i
Γ (ai )
a −1 −bi βi
βi i
e
, i = 1, 2, 3, βi > 0
The joint prior density of β1 , β2 and β3 is given as follows
π0 (β1 , β2 , β3 ) =
3 ∏ bai a −1 β i e−bi λi Γ (ai ) i i=1
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
H.Z. Muhammed / Physica A xxx (xxxx) xxx
9
Posterior Analysis and Bayesian Estimation Suppose {(x11 , x21 ), . . . , (x1n , x2n )} is a random sample from BGIKum (β1 , β2 , β3 , α, γ ) distribution. Consider the following notation D = {(x11 , x21 ), . . . , (x1n , x2n )}, Θ = (β1 , β2 , β3 ) and n = n1 + n2 + n3 . Then the Likelihood function can be written as L(D\Θ ) = Exp(log L(D\Θ )) n
n
n
n
n
L (D\Θ ) = γ 2n1 +2n2 +n3 α 2n1 +2n2 +n3 β131 β232 β2 1 β1 2 β3 3
· Exp((β13 − 1)Z1 (α, γ ) + (β2 − 1)Z2 (α, γ ) + (β1 − 1)Z3 (α, γ ) + (β23 − 1)Z4 (α, γ ) + (β123 − 1)Z5 (α, γ ) + (γ + 1)Z6 − (α + 1)Z7 (γ )). n1 n2 ( ) ( ) ∑ ∑ n1 n2 i+n j+n L (D\Θ ) ∝ β1 2 β2 1 β3n−i−j Exp(β1 T1 + β2 T2 + β3 T3 ). i
j
(
γ )−α
i=1 j=1
where Z1 (α, γ ) =
∑
[
ln 1 − 1 + x1i
]
, Z2 (α, γ ) =
∑
I1
∑
Z5 (α, γ ) =
∑
[
(
γ )−α
]
[
(
γ )−α
]
ln 1 − 1 + x1i
Z4 (α, γ ) =
I2
]
,
∑
[
(
γ )−α
ln 1 − 1 + x2i
]
I2
ln 1 − 1 + x1i
,
∑
Z6 =
I3
∑
γ )−α
(
I1
Z3 (α, γ ) =
Z7 (γ ) =
[
ln 1 − 1 + x2i
ln x1i +
I1 ∪I2
γ
(
)
ln 1 + x1i +
∑
γ
(
)
ln 1 + x2i +
ln x2i +
∑
I1 ∪I2
γ)
(
ln 1 + xi
ln xi ,
I3
,
I3
I1 ∪I2
I1 ∪I2
∑
∑
T1 = Z1 (α, γ ) + Z3 (α, γ ) + Z5 (α, γ ) , T2 = Z2 (α, γ ) + Z4 (α, γ ) + Z5 (α, γ ) and T3 = Z1 (α, γ ) + Z4 (α, γ ) + Z5 (α, γ ) since f (D, Θ ) = π0 (Θ ) L(D\Θ ) andf (D) =
∫
f (D\Θ ) dΘ =
∫
π0 (Θ ) L(D\Θ )dΘ
Hence the joint posterior density function of Θ = (β1 , β2 , β3 ) given the data D, denoted by π1 ( Θ \D) can be written as f (D, Θ )
π1 ( Θ \D) = π1 ( Θ \D) ∝
f (D) n2 n1 ∑ ∑
Aij Gamma [β1 ; a1i , b1 + T1 ] · Gamma β2 ; a2j , b2 + T2
[
]
i=1 j=1
· Gamma[β3 ; a3ij , b3 + T3 ]
( )( )
Cij
where Aij = ∑n1 ∑n2 i=1
C j=1 ij
, and Cljks =
n1 j
n2 j
. [b Γ+(aT1i])a1i . [b 1
1
Γ (a2j ) 2 +T2 ]
a2j
. [b
Γ (a3ij ) 3 +T3 ]
a3ij
. a1i = a1 + i + n2 , a2j = a2 + j + n1 and
a3ij = a3 + n + i + j. Therefore, under the assumption of independence of β1 , β2 and β3 and α , γ are assumed to be known. It is possible to get the Bayes estimators of β1 , β2 and β3 explicitly under the square error loss function as follows:
βˇ 1 = βˇ 2 =
1 b1 + T 1 1 b2 + T 2
n1 n2 ∑ ∑
Aij a1i ,
i=1 j=1 n1 n2 ∑ ∑
Aij a2j ,
i=1 j=1
And
βˇ 3 =
1 b3 + T 3
n1 n2 ∑ ∑
Aij a3ij .
i=1 j=1
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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H.Z. Muhammed / Physica A xxx (xxxx) xxx Table 1 logL, AIC, BIC, CAIC and HQIC for different bivariate distributions. Model
log L
AIC
BIC
CAIC
HQIC
BGIKum BVD BGB BGE BE
−157.043 −30.459 −43.509 −20.59 −44.56
167.043 70.918 97.018 49.18 96.12
175.098 78.973 105.073 48.40 95.46
168.978 72.853 98.954
169.883 73.758 99.858
96.11
97.62
Table 2 The AE, MSE, RAB and CP of β1 , β2 , β3 , γ and α for BGIKum distribution with real parameter values (0.5, 0.2, 0.1, 0.1, 1). n
Parameters
AE
MSE
RAB
CP
30
β1 β2 β3 γ α
0.298 0.164 0.159 0.14 1.028
0.041 0.0013 0.0003 0.002 0.0008
0.404 0.182 0.159 0.398 0.028
0.87 0.92 0.88 0.90 0.84
40
β1 β2 β3 γ α
0.284 0.189 0.096 0.15 1.022
0.046 0.0001 0.00002 0.002 0.0005
0.431 0.053 0.039 0.499 0.022
0.89 0.93 0.90 0.89 0.93
60
β1 β2 β3 γ α
0.3 0.168 0.087 0.143 1.019
0.04 0.0001 0.0002 0.002 0.0004
0.399 0.053 0.132 0.428 0.019
0.91 0.92 0.89 0.92 0.95
80
β1 β2 β3 γ α
0.286 0.173 0.099 0.145 1.018
0.046 0.0001 0.000009 0.002 0.0003
0.427 0.053 0.009 0.452 0.018
0.90 0.95 0.91 0.94 0.96
100
β1 β2 β3 γ α
0.285 0.179 0.093 0.142 1.027
0.046 0.0001 0.00005 0.002 0.0007
0.431 0.053 0.069 0.424 0.027
0.90 0.94 0.89 0.92 0.98
5. Simulation study In this section, the results of a Monte Carlo simulation study testing the performance of MLE of the model Parameters will be introduced. The evaluation of the MLEs was performed based on the following quantities for each sample size: the Average Estimates (AE), the Mean Squared Error, (MSE) Relative Absolute Bias (RAB) and Coverage probability (CP) are estimated from R = 1000 replications for βˆ 1 , βˆ 2 , βˆ 3 , αˆ and γˆ the sample size has been considered at n = 30, 40, 60, 80 and 100, and some values for the parameters β1 , β2 , β3 , α and γ have been considered. Algorithm to generate from BGIKum distribution Step 1. Generate U1 , U2 (and U3 from ) U(0, 1). 1/β1
Step 2. Compute Z1 = { 1 − U1 1/β3
(
−1/α
( )−1/α 1/β − 1}1/γ , Z2 = { 1 − U2 2 − 1}1/γ ,
)−1/α
− 1}1/γ . Step 3. Obtain X1 = max(Z1 , Z3 ) and X2 = max(Z2 , Z3 ).
and Z3 = { 1 − U3
Step 4. Define the indicator functions as
δ1i =
{
1; 0;
x1i < x1i ,δ = other w ise 2i
{
1; 0;
x1i > x1i and δ3i = other w ise
{
1; 0;
x1i = x1i . other w ise
Step 5. The corresponding sample size n must satisfy n = n1 + n2 + n3 ∑n ∑n ∑n Such that n1 = and n3 = i=1 δ1i , n2 = i=1 δ2i i=1 δ3i . For different choices of sample sizes a 1000 data set is generated using the MATHCAD program that employed to solve the nonlinear likelihood equations. It can be noted from Tables 2 through 5 that the estimates are work well and MSE and RAB decreases as the sample size increases. Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
H.Z. Muhammed / Physica A xxx (xxxx) xxx
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Table 3 The AE, MSE, RAB and CP of β1 , β2 , β3 , γ and α for BGIKum distribution with real parameter values (1.1, 1.1, 1.1, 0.1, 1). n
Parameters
AE
MSE
RAB
CP
30
β1 β2 β3 γ α
0.296 0.28 0.431 0.291 0.972
0.647 0.673 0.447 0.037 0.0008
0.731 0.746 0.608 1.912 0.028
0.93 0.92 0.93 0.91 0.97
40
β1 β2 β3 γ α
0.297 0.314 0.423 0.309 0.959
0.645 0.617 0.458 0.044 0.002
0.73 0.714 0.615 2.094 0.041
0.94 0.93 0.93 0.90 0.95
60
β1 β2 β3 γ α
0.274 0.277 0.456 0.3 0.988
0.683 0.678 0.415 0.04 0.0014
0.715 0.748 0.586 2.005 0.012
0.95 0.94 0.98 0.94 0.95
80
β1 β2 β3 γ α
0.261 0.281 0.439 0.299 0.983
0.704 0.671 0.438 0.04 0.0003
0.763 0.745 0.601 1.99 0.017
0.98 0.97 0.98 0.99 0.96
100
β1 β2 β3 γ α
0.265 0.274 0.453 0.309 0.952
0.697 0.682 0.419 0.043 0.0022
0.759 0.751 0.589 2.086 0.048
0.97 0.99 0.97 0.93 0.96
Table 4 The AE, MSE, RAB and CP of β1 , β2 , β3 , γ and α for BGIKum distribution with real parameter values (0.2, 0.2, 0.2, 0.1, 1). n
Parameters
AE
MSE
RAB
CP
30
β1 β2 β3 γ α
0.134 0.131 0.207 0.14 1.002
0.0043 0.0048 0.000054 0.0016 0.000006
0.328 0.345 0.037 0.402 0.0024
0.92 0.89 0.94 0.94 0.96
40
β1 β2 β3 γ α
0.154 0.138 0.221 0.139 0.989
0.0021 0.0039 0.00042 0.0015 0.0001
0.231 0.312 0.103 0.39 0.011
0.88 0.87 0.92 0.93 0.94
60
β1 β2 β3 γ α
0.13 0.127 0.204 0.134 0.994
0.0049 0.0053 0.00002 0.0011 0.000039
0.35 0.365 0.019 0.336 0.0062
0.90 0.92 0.87 0.89 0.95
80
β1 β2 β3 γ α
0.131 0.132 0.211 0.136 1.046
0.0048 0.0046 0.00012 0.0013 0.0021
0.345 0.341 0.056 0.357 0.046
0.94 0.96 0.92 0.94 0.95
100
β1 β2 β3 γ α
0.128 0.133 0.219 0.142 0.998
0.0052 0.0045 0.00035 0.0018 0.000004
0.361 0.355 0.094 0.421 0.002
0.99 0.97 0.94 0.93 0.91
6. Absolutely continuous bivariate GIKum Based on the idea of Block and Basu [13], an absolutely continuous bivariate generalized inverted Kumaraswamy (BGIKac ) distribution will be introduced by removing the singular part from the Marshal–Olkin bivariate generalized inverted Kumaraswamy and remaining only the absolutely continuous part. Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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H.Z. Muhammed / Physica A xxx (xxxx) xxx Table 5 The AE, MSE, RAB and CP of β1 , β2 , β3 , γ and α for BGIKum distribution with real parameter values (0.4, 0.4, 0.4, 0.1, 1.5). n
Parameters
AE
MSE
RAB
CP
30
β1 β2 β3 γ α
0.163 0.166 0.262 0.172 0.95
0.056 0.055 0.019 0.0051 0.302
0.593 0.586 0.345 0.717 0.367
0.92 0.94 0.90 0.95 0.93
40
β1 β2 β3 γ α
0.128 0.141 0.254 0.172 0.93
0.074 0.067 0.021 0.0051 0.324
0.68 0.647 0.365 0.717 0.38
0.95 0.94 0.92 0.95 0.94
60
β1 β2 β3 γ α
0.147 0.145 0.261 0.172 0.934
0.064 0.065 0.019 0.0052 0.321
0.632 0.638 0.348 0.719 0.934
0.97 0.98 0.96 0.99 0.98
80
β1 β2 β3 γ α
0.155 0.16 0.25 0.171 0.941
0.06 0.058 0.022 0.0049 0.313
0.613 0.601 0.374 0.706 0.373
0.93 0.99 0.98 0.92 0.96
100
β1 β2 β3 γ α
0.182 0.167 0.251 0.166 0.933
0.048 0.054 0.022 0.0044 0.321
0.546 0.582 0.373 0.662 0.378
0.97 0.95 0.99 0.96 0.91
A random vector (Y1 , Y2 ) follows a BGIKac distribution if its pdf is given by fY1 ,Y2 (y1 , y2 ) = c ·
{
fGIK (y1 ; β13 ) · fGIK (y2 ; β2 )
if
y 1 < y2
fGIK (y1 ; β1 ) · fGIK (y2 ; β23 )
if
y 1 > y2
,
β
where c = β 12 is normalizing constant 123 We denote (Y1 , Y2 ) ∼ BGIKac (β1 , β2 , β3 , α, γ ) if (X1 , X2 ) has a BGIKum distribution, then (X1 , X2 ) given X1 ̸ = X2 has a BGIKac distribution. Proposition 5. Let (Y1 , Y2 ) ∼ BGIKac (β1 , β2 , β3 , α, γ ), then i. The associated failure function is FY1 ,Y2 (y1 , y2 ) =
β3 β123 FGIK (y1 ; β1 ) FGIK (y2 ; β2 ) FGIK (y; β3 ) − FGIK (y; β123 ) ; β12 β12
where y = min(y1 , y2 ). Furthermore, ii. The marginal failure functions are given by
β123 FGIK (y1 ; β13 ) − β12 β123 FY2 (y2 ) = FGIK (y2 ; β23 ) − β12
FY1 (y1 ) =
β3 FGIK (y1 ; β123 ) β12 β3 FGIK (y2 ; β123 ) β12
iii. The marginal pdfs associated with the cdf function given above are as follows fY1 (y1 ) = cfGIK (y1 ; β13 ) − c
β3 fGIK (y1 ; β123 ) , β123
fY2 (y2 ) = cfGIK (y2 ; β23 ) − c
β3 fD (y2 ; β123 ) , β123
y1 > 0
and y2 > 0.
Note that: Unlike those of the BGIKum distribution, the marginals of the BGIKac distribution are not GIKum distributions. If β3 → 0+ , then Y1 and Y2 follow GIKum distributions and in this case, Y1 and Y2 become independent. Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
H.Z. Muhammed / Physica A xxx (xxxx) xxx
13
Proposition 6. The product moments of (Y1 , Y2 ) ∼ BGIKac (β1 , β2 , β3 , α, γ ) are given by [β2 −1] [β13 −1]
E X1r X2s =
)
(
∑ ∑ j=0
(
Cij B s + r + γ −
i=0
(
· 3 F2 s + r + γ −
1
1
γ
, −α (1 + j) − s − r − 1 − γ +
r
1
)
γ )
r
, − 1, −α (i + 1) ; , −α (j + 1) − 1; 1 γ γ γ ) ( [β23 −1] [β1 −1] ∑ ∑ 1 1 + Cˇ ij B s + r + γ − , −α (1 + j) − s − r − 1 − γ + γ γ j=0
i=0
where
( )( ) α 2 γ β2 β13 β2 − 1 i+j β13 − 1 ( − 1) r i j −1 γ ( )( ) 2 β23 − 1 ˇCij = c α γ β1 β23 (−1)i+j β1 − 1 . s i j −1 γ Cij = c
Proposition 7. Let (Y1 , Y2 ) ∼ BGIKac (β1 , β2 , β3 , α, γ ). Then i. The Stress–Strength parameter has the following form; R = P (Y1 < Y2 ) =
β1 , β12
ii. Max (Y1 , Y2 ) ∼ GIK (β123 ). 7. Data analysis For illustrative purposes one data set has been analyzed to see how the BGIKum model works in practice. The data set has been obtained from [14]. The data represent the football (soccer) data where at least one goal scored by the home team and at least one goal scored directly from a penalty kick, foul kick or any other direct kick (all of them together will be called as kick goal) by any team have been considered. Here X1 represents the time in minutes of the first kick goal scored by any team and X2 represents the first goal of any type scored by the home team. In this case all possibilities are open, for example X1 < X2 or X1 > X2 or X1 = X2 = X . These data were analyzed by Meintanis [14], who considered the Marshall–Olkin bivariate exponential distribution, and by many authors such as [6,9–11] they considered the bivariate generalized exponential, bivariate inverse Weibull, bivariate Dagum and bivariate generalized Burr distributions, respectively. Here, these data will be analyzed using the BGIKum distribution. The Kolmogorov–Smirnov distances between the fitted distribution and the empirical distribution function for X1 , X2 and max (X1 , X2 ) with GIKum (2.594, 2.266, 0.742), GIKum (3.58, 2.266, 0.742) and GIKum (4.424, 2.266, 0.742) are (0.544), (0.662) and (0.726) respectively. That gives an indication that the BGIKum model may be used to analyze this data set. The asymptotic variance–covariance matrix is as follows 0.038
−0.04 −0.014 −0.0035 0.0026
−0.04 0.0088 −0.18 −0.058 0.02
−0.014 −0.18 0.126 −0.017 0.0069
−0.00355 −0.058 −0.017 0.017 0.013
0.0026 0.02 0.0069 0.013 0.00288
Also a 95% confidence intervals of β1 , β2 , β3 , α, γ are computed and they are as follows; (0.781, 0.907), (1.8, 1.861), (1.635, 1.864), (0.008, 0.092), (0.725, 0.759). With correspondence lengths (0.126, 0.06, 0.229, 0.085, 0.035). Using the Akaike information criterion (AIC), Bayesian information criterion (BIC), the consistent Akaike information criterion (CAIC) and Hannan–Quinn information criterion (HQIC) the BGIKum distribution can be compared with another bivariate distributions such as: bivariate Dagum (BVD) distribution [10], bivariate generalized Burr (BGB) distribution [11],bivariate generalized exponential (BGE) distribution [6] and bivariate exponential (BVE) distribution [14] as shown in Table 1. Now from the confidence intervals, from the log-likelihood values and also from the Kolmogorov–Smirnov distances, it is clear that GIKum can be representing this case. Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.
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H.Z. Muhammed / Physica A xxx (xxxx) xxx
8. Conclusion In this paper, a bivariate distribution called BGIKum distribution whose marginals are GIKum distributions have been proposed. The BGIKum distribution is a singular distribution and it has an absolute continuous part and a singular parts. Since the joint distribution function and the joint density function are in closed forms, therefore this distribution can be used in practice for non-negative and dependent random variables. This model has five unknown parameters the maximum likelihood estimates for the five unknown parameters and their approximate variance–covariance matrix have been obtained and performed some simulations to see the performances of the MLEs. Explicit Bayesian estimators are also obtained for the unknown parameters of this model. One data set has been re-analyzed for illustrative purpose. Along the same line as [13] bivariate exponential model, an absolute continuous version of the BGIKum also obtained and several of its properties are presented. References [1] M.C. Jones, Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages, Stat. Methodol. 6 (2009) 70–81. [2] A.M. Abd AL-Fattah, A.A. El-Helbawy, G.R. Al-Dayian, Inverted Kumaraswamy distribution: Properties and estimation, Pak. J. Stat. 33 (1) (2017) 37–61. [3] Z. Iqbal, M.M. Tahir, N. Riaz, A.A. Ali, M. Ahmed, Generalized inverted Kumaraswamy distribution: Properties and application, Open J. Stat. 7 (2017) 645–662. [4] A.W. Marshall, I. Olkin, A multivariate exponential distribution, J. Amer. Statist. Assoc. 62 (1967) 30–44. [5] A. Sarhan, N. Balakrishnan, A new class of bivariate distribution and its mixture, J. Multivariate Anal. 98 (2007) 1508–1527. [6] D. Kundu, R.D. Gupta, Bivariate generalized exponential distribution, J. Multivariate Anal. 100 (2009) 581–593. [7] D. Kundu, R.D. Gupta, A class of bivariate models with proportional reversed hazard marginals, Sankhya B 72 (2010) 236–253. [8] A. Sarhan, D.C. Hamilton, B. Smith, D. Kundu, The bivariate generalized linear failure rate distribution and its multivariate extension, Comput. Stat. Data Anal. 55 (2011) 644–654. [9] H.Z. Muhammed, Bivariate inverse Weibull distribution, J. Stat. Comput. Simul. 86 (12) (2016) 1–11. [10] H.Z. Muhammed, Bivariate dagum distribution, Int. J. Reliab. Appl. 18 (2) (2017a) 65–82. [11] H.Z. Muhammed, Bivariate generalized burr and related distributions: Properties and estimation, J. Data Sci. 17 (3) (2019) 532–548. [12] D. Kundu, A.K. Gupta, Bayes estimation for the Marshal – Olkin bivariate Weibull distribution, Comput. Stat. Data Anal. 57 (2013) 271–281. [13] H. Block, A.P. Basu, A continuous bivariate exponential extension, J. Amer. Statist. Assoc. 69 (1974) 1031–1037. [14] S.G. Meintanis, Test of fit for Marshall–Olkin distributions with applications, J. Stat. Plan. Inference 137 (2007) 3954–3963.
Please cite this article as: H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A (2020) 124281, https://doi.org/10.1016/j.physa.2020.124281.