In this section first we introduce Marshall-Olkin GE (MOGE) distribution similar to the MOE and MOWE distribution. The idea is very similar and it can be defined as follows.
Suppose {Xn;n = 1,2,· · ·} is a sequence of i.i.d. GE(α, λ) random variables and N is a geometric random variable with the PMF as given in (31). Moreover, N and {Xn;n = 1,2,· · ·} are independently distributed. Let us define a new random variable Y as
Y = max{X1, . . . , XN}.
The random variable Y is said to have GE-Geometric (GEG) distribution with parameter p, α, λ, and it will be denoted by GEG(p, α, λ).
The CDF of Y for y >0 can be obtained as
FY(y;, α, λ) = P(Y ≤y) = P(X1 ≤y,· · ·, XN ≤y) =p X∞ n=1
Pn(X1 ≤y)(1−p)n−1
= p(1−e−λx)α 1−(1−p)(1−e−λx)α. The corresponding PDF becomes
fY(y;p, α, λ) = pαλe−λx(1−e−λx)α−1
(1−(1−p)(1−e−λx)α)2 for y >0, and zero, otherwise.
Now we can define bivariate GEG (BGEG) along the same line as before. Suppose {(X1n, X2n);n = 1,2, . . .} is a sequence of i.i.d. random variables with parameters β1, β2, β3, λ having the joint PDF as given in (20) and
Y1 = max{X11, . . . , X1N} and Y2 = max{X21, . . . , X2N}, then (Y1, Y2) is said to have BGEG distribution with parameters β1, β2, β3, λ, p.
The joint CDF of (Y1, Y2) becomes:
P(Y1 ≤y1, Y2 ≤y2) = X∞ n=1
Pn(X11 ≤y1, X21≤y2)p(1−p)n−1
= pP(X11 ≤y1, X21≤y2) 1−(1−p)P(X11≤y1, X21 ≤y2)
=
( p(1−e−λy1)β1(1−e−λy2)β2+β3
1−(1−p)(1−e−λy1)β1(1−e−λy2)β2+β3 if y1 > y2 p(1−e−λy1)β1+β3(1−e−λy2)β2
1−(1−p)(1−e−λy1)β1+β3(1−e−λy2)β2 if y1 ≤y2.
It will be interesting to develop different properties of this distribution similar to the BWG distribution as it has been obtained by Kundu and Gupta [33]. Moreover, classical and Bayesian inference need to be developed for analyzing bivariate data sets with ties. This model also can be used for analyzing dependent complementary risks data. More work is needed along this direction.
8 Conclusions
In this paper we have considered the class of bivariate distributions with a singular com- ponent. Marshall and Olkin [44] first introduced the bivariate distribution with singular component based on exponential distributions. Since then an extensive amount of work has taken place in this direction. In this paper we have provided a comprehensive review of all the different models. It is observed that there are mainly two main approaches available in the literature to define bivariate distributions with singular components, and they produce two broad classes of bivariate distributions with singular components. We have shown that these two classes of distributions are related through their copulas under certain restriction.
We have provided very general EM algorithms which can be used to compute the MLEs of the unknown parameters in these two cases. We have provided the analysis of one data set to show how the EM algorithms perform in real life.
There are several open problems associated with these models. Although, in this paper we have mainly discussed different methods for bivariate distributions, all the methods can be generalized for the multivariate distribution also. Franco and Vivo [20] first obtained the multivariate Sarhan-Balakrishnan distribution and developed several properties, although no inference procedures were developed. Sarhan et al. [59] provided the multivariate gen- eralized failure rate distribution and Franco, Kundu and Vivo [19] developed multivariate modified Sarhan-Balakrishnan distribution. In both these papers, the authors developed
different properties and very efficient EM algorithms for computing the maximum likelihood estimators of the unknown parameters. More general multivariate distributions with pro- portional reversed hazard marginals can be found in Kundu, Franco and Vivo [31]. Note that all these models can be very useful to analyze dependent competing risks model with multiple causes. No work has been done along that like, more work is needed along these directions.
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0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
1 2 3 4 5 6
(a)
0 0.2 0.4 0.6 0.8 1
1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(b)
0 0.1 0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0 1
1.5 2 2.5 3 3.5 4
(c)
0 0.1 0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0 1
1.5 2 2.5 3 3.5 4
(d)
Figure 1: PDF plots of MOBE(α1, α2, α3) distribution for different (α1, α2, α3) values: (a) (2.0,2.0,1.0) (b) (1.0,1.0,1.0) (c) (1.0,2.0,1.0) (d) (2.0,1.0,1.0).
0 0.05
0.1 0.15
0.2 0.25 0
0.05 0.1 0.15 0.2 0.25 10 0
20 30 40 50 60 70 80 90
(a)
0 0.05 0.1 0.15 0.2 0.25
0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 1
2 3 4 5 6 7 8 9 10 11
(b)
0 0.2 0.4 0.6 0.8 1
1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(c)
0 0.2 0.4 0.6 0.8 1
1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0
0.5 1 1.5 2 2.5
(d)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.5 0 1
1.5 2 2.5 3 3.5 4
(e)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.5 0 1
1.5 2 2.5 3 3.5
(f)
Figure 2: PDF plots of MOBW(α1, α2, α3, θ) distribution for different (α1, α2, α3, θ) val- ues: (a) (1.0,1.0,1.0,0.5) (b) (1.0,1.0,1.0,0.75) (c) (2.0,2.0,1.0,1.5) (d) (2.0,2.0,1.0,3.0) (e) (3.0,3.0,3.0,1.5) (f) (3.0,3.0,3.0,3.0).
0.2 0 0.4 0.6
0.8 1 1.2 1.4
1.6 1.8
2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
(a)
0 0.2 0.4 0.6 0.8 1
1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0
0.4 0.6 0.8 1 1.2 1.4
(b)
0.2 0 0.4 0.6
0.8 1 1.2 1.4
1.6 1.8
2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(c)
0 0.5 1 1.5
2 2.5 3 0
0.5 1 1.5 2 2.5 3 0
0.05 0.1 0.15 0.2 0.25
(d)
Figure 3: PDF plots of BWEE(λ1, λ2, λ3, α) distribution for different (λ1, λ2, λ3, α) values:
(a) (1.0,1.0,1.0,0.5) (b) (1.0,1.0,2.0,1.0) (c) (1.5,1.5,1.5,1.5) (d) (0.5,0.5,0.5,0.5).
0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2
4 6 10 8 12 14 16 18
(a)
0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.5 1 1.5 2 2.5 3
(b)
0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1
2 3 4 5 6 7 8
(c)
0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0 1
1.5 2 2.5 3 3.5 4
(d)
0 0.1 0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5
10 15 20 25 30 35 40 45
(e)
0.1 0 0.3 0.2 0.5 0.4 0.7 0.6 0.9 0.8 1 0 0.1
0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9
1 0.2 0
0.4 0.6 0.8 1 1.2
(f)
Figure 4: PDF plots of BVK(α1, α2, α3, β) distribution for different (α1, α2, α3, β) val- ues: (a) (1.0,1.0,1.0,0.5) (b) (2.0,2.0,2.0,2.0) (c) (2.0,2.0,2.0,1.0) (d) (2.0,2.0,2.0,3.0) (e) (2.0,2.0,2.0,0.5) (f) (0.75,0.75,0.75,1.1).
0 0.2 0.4 0.6 0.8 1
1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0
0.05 0.1 0.15 0.2 0.25 0.3
(a)
0 0.5
1 1.5
2 2.5 3 0
0.5 1 1.5 2 2.5 3 0.02 0
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
(b)
0 0.5 1 1.5 2 2.5
3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.02 0
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
(c)
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0.02 0
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
(d)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
(e)
0.1 0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0 0.1
0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.15 0.1 0.2
0.25 0.3 0.35 0.4 0.45 0.5 0.55
(f)
Figure 5: PDF plots of MOBE(α1, α2, α3, θ) distribution for different (α1, α2, α3, θ) val- ues: (a) (1.0,1.0,1.0,0.5) (b) (1.0,1.0,1.0,0.75) (c) (2.0,2.0,1.0,1.5) (d) (2.0,2.0,1.0,3.0) (e) (3.0,3.0,3.0,1.5) (f) (3.0,3.0,3.0,3.0).