CHAPTER 2 Marshall-Olkin Beta Distribution and its Applications 2.1 Introduction The majority of continuous distributions are defined on infinite intervals. But beta distribu- tion is a continuous distribution defined on a bounded interval and its probability density function exhibits a wide range of shapes. This flexibility encourages its empirical use in a large variety of applications. For many years the use of beta distribution has been as ‘prior’ distribution for binomial proportions. In recent years beta distributions have been used in modelling distributions of hydrological variables, in operation research, in risk analysis etc. See Nadarajah (2007a) for some references. Recently many authors have obtained generalizations of beta distribution using various Some results included in this chapter have appeared in the paper Jose et al. (2009) 25
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CHAPTER 2
Marshall-Olkin Beta Distribution and its
Applications
2.1 Introduction
The majority of continuous distributions are defined on infinite intervals. But beta distribu-
tion is a continuous distribution defined on a bounded interval and its probability density
function exhibits a wide range of shapes. This flexibility encourages its empirical use in a
large variety of applications. For many years the use of beta distribution has been as ‘prior’
distribution for binomial proportions. In recent years beta distributions have been used in
modelling distributions of hydrological variables, in operation research, in risk analysis etc.
See Nadarajah (2007a) for some references.
Recently many authors have obtained generalizations of beta distribution using various
Some results included in this chapter have appeared in the paper Jose et al. (2009)
25
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
techniques. McDonald and Xu (1995) introduced a five-parameter beta distribution. Var-
ious properties of a family of generalized beta distributions including beta-normal, beta-
exponential, beta-Weibull etc are recently studied by different authors. Sepanski and Kong
(2007) applied these distributions to the modelling of the size distribution of income and
computed the maximum likelihood estimates of parameters. They also compared these
estimates with those of the widely used generalized beta distributions of the first and sec-
ond kind in terms of measures of goodness of fit.
New parameters can be introduced to expand families of distributions for added flex-
ibility. In this paper we generalize the beta distribution using the method proposed by
Marshall and Olkin (1997). They introduced a method of obtaining a family of distributions
given by
G(x) =F (x)
α+ (1− α)F (x), x ∈ R (2.1.1)
where F is a distribution function and α > 0. If α = 1 then we have G = F . The
distribution function G has the density given by
g(x) =αf(x)
(α+ (1− α)F (x))2
and the hazard rate
r(x) =rF (x)
α+ (1− α)F (x),
where rF (x) is the hazard rate of the distribution function F . In the past decade, many dis-
tributions which belong to the Marshall-Olkin family of distributions have been investigated.
Marshall and Olkin (1997) defined exponential and Weibull. Alice and Jose (2003), (2004)
(2005a) and (2005b) introduced and studied the various properties of semi-Weibull, bi-
variate semi-Pareto, Pareto and logistic distributions. Marshall-Olkin q-Weibull distribution
was introduced by Jose, Naik and Ristic (2008) and they discussed its application in time
series modelling. Semi-Burr distribution was studied by Jayakumar and Thomas (2007).
A physical interpretation of the Marshall-Olkin model using odds function was given by
Sankaran and Jayakumar (2007).
26
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
By a minification process of order 1 we mean a sequence {Xn, n ≥ 0} of nonnegative
random variables having the structure
Xn = kmin(Xn−1, εn), n ≥ 1
where {εn} is a sequence of independent and identically distributed (i.i.d) random variables
called innovations and k > 1. The study on autoregressive processes having minification
structure began with the work of Tavares (1980) and subsequently many authors devel-
oped minification processes with different marginal distributions. See Sim (1986), (1994),
Yeh et al. (1988), Lewis and Mckenzie (1991) and Pillai (1991). Balakrishna and Jacob
(2003) discussed the parameter estimation of a univariate minification process and stud-
ied properties such as ergodicity and uniform mixing. Ed Mckenzie (1985) presented two
stationary first order autoregressive processes with beta marginal distributions.
In section 2 the Marshall-Olkin beta distribution is introduced and some shape prop-
erties are discussed. In section 3 maximum likelihood estimation of the parameters are
considered. The expression for the moments are derived in section 4. In section 5 a minifi-
cation process with Marshall-Olkin beta distribution is introduced and some of its properties
are obtained. In section 6 two real data sets are analyzed and the new distribution is found
a better fit than the standard beta distribution.
2.2 A Marshall-Olkin Beta Distribution and its Properties
In this section we introduce a Marshall-Olkin beta distribution. Let
F (x) =B
(m,n, x−a
b−a
)B(m,n)
=1
B(m,n)
∫ x−ab−a
0
tm−1(1− t)n−1dt, a < x < b,
be the distribution function of a beta distribution with parameters m > 0, n > 0, a and b.
Replacing F in (2.1.1) we obtain the distribution function of a Marshall-Olkin beta distribu-
tion as
G(x) =B
(m,n, x−a
b−a
)αB(m,n) + (1− α)B
(m,n, x−a
b−a
) . (2.2.1)
27
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
The corresponding probability density function has the form
g(x) =αB(m,n)(x− a)m−1(b− x)n−1
(b− a)m+n−1(αB(m,n) + (1− α)B
(m,n, x−a
b−a
))2 .
The Marshall-Olkin beta distribution with parameters m, n, a, b and α is denoted by MO-
Beta (m,n, a, b, α). If α = 1 then the MOBeta (m,n, a, b, α) distribution reduces to the
four parameter beta distribution.
If m and n are positive integers, then the distribution function G is given by
G(x) =
m+n−1∑r=m
(m+n−1
r
)(x− a)r(b− x)m+n−r−1
α(b− a)m+n−1 + (1− α)m+n−1∑
r=m
(m+n−1
r
)(x− a)r(b− x)m+n−r−1
and the probability density function has the form
g(x) =α(b− a)m+n−1(x− a)m−1(b− x)n−1
B(m,n)
(α(b− a)m+n−1 + (1− α)
m+n−1∑r=m
(m+n−1
r
)(x− a)r(b− x)m+n−r−1
)2 .
Now we discuss some shape properties of the MOBeta (m,n, a, b, α) distribution. The
parameters m, n and α are the shape parameters. The first derivative of log g(x) is
1. If 0 < m,n < 1, then t(x) is an increasing function with unique root x0. Thus it
28
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
follows that g(x) decreases at (a, x0] and increases at (x0, b) with g(a) = g(b) = ∞.
2. Let 0 < n < 1 < m and α < 1.
(a) If m + n > 2 and t(x) has two real roots x1 and x2, then t(x) is positive at
(a, x1] ∪ (x2, b) and negative at (x1, x2]. Thus it follows that g(x) increases at
(a, x1] ∪ (x2, b) and decreases at (x1, x2] with g(a) = 0 and g(b) = ∞.
(b) If m+ n < 2, t′(x) and t(x) have two real roots, where x1 and x2 are the roots
of t(x), then g(x) has the same shape as in 2(a).
(c) Otherwise, g(x) is an increasing function with g(a) = 0 and g(b) = ∞.
3. If 0 < n < 1 < m and α > 1, then t(x) is a positive function. Thus it follows that
g(x) is an increasing function with g(a) = 0 and g(b) = ∞.
4. If 0 < m < 1 < n and α < 1, then t(x) is a negative function. Thus it follows that
g(x) is a decreasing function with g(a) = ∞ and g(b) = 0.
5. Let 0 < m < 1 < n and α > 1.
(a) If m + n > 2 and t(x) has two real roots x1 and x2, then t(x) is negative at
(a, x1] ∪ (x2, b) and positive at (x1, x2]. Thus it follows that g(x) decreases at
(a, x1] ∪ (x2, b) and increases at (x1, x2] with g(a) = ∞ and g(b) = 0.
(b) If m+ n < 2, t′(x) and t(x) have two real roots, where x1 and x2 are the roots
of t(x), then g(x) has the same shape as in 5(a).
(c) Otherwise, g(x) is an decreasing function with g(a) = ∞ and g(b) = 0.
6. If m,n > 1, then t(x) is a decreasing function with unique root x0. Thus it follows
that g(x) is a unimodal function with mode at x0 and g(a) = g(b) = 0.
Now, we define the hazard rate function. The hazard rate function of the Marshall-Olkin
beta distribution is given by
r(x) =B(m,n)(x− a)m−1(b− x)n−1(
αB(m,n) + (1− α)B(m,n, x−a
b−a
)) (B(m,n)−B
(m,n, x−a
b−a
))29
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
Figure 2.1: Graph of the probability density function of the Marshall-Olkin beta distribution fora = 0, b = 1 and various values of the parameters m, n and α.
Now we establish some properties of the Marshall-Olkin beta distribution.
Theorem 2.2.1. Let {Xn, n ≥ 1} be a sequence of i.i.d. random variables with a
MOBeta(m,n, a, b, α) distribution. Let N be a geometric r.v. with parameter p and
suppose that N and Xi are independent. Then:
i) min(X1, X2, . . . , XN) has a MOBeta(m,n, a, b, αp) distribution.
ii) max(X1, X2, . . . , XN) has a MOBeta(m,n, a, b, α/p) distribution.
Proof. : (i) Survival function of U = min(X1, X2, . . . , XN) is given by
P (U > u) = p
∞∑s=1
[G(u)
]s(1− p)s−1
=αpB
(n,m, b−u
b−a
)αpB(m,n) + (1− αp)B
(m,n, u−a
b−a
) .This is the survival function of a MOBeta (m,n, a, b, αp) distribution.
30
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
Figure 2.2: Graph of the hazard rate function of the Marshall-Olkin beta distribution for a = 0,b = 1 and different values of the parameters m, n and α.
(ii) To prove this consider the distribution function of Xn given by (2.2.1).
Let V = max(X1, X2, . . . , XN). Then distribution function of V is given by
P (V ≤ v) = p∞∑
s=1
[G(v)]s (1− p)s−1
=B
(m,n, v−a
b−a
)αpB(m,n) +
(1− α
p
)B
(m,n, v−a
b−a
)This is a MOBeta (m,n, a, b, α/p) distribution.
Corollary 2.2.1. Let {Xn} be a sequence of i.i.d. random variables with a beta dis-
tribution with parameters m, n, a and b. Let N be a Geometric(p) r.v. and suppose
that N and Xi are independent. Then:
i) min(X1, X2, . . . , XN) has a MOBeta(m,n, a, b, p) distribution.
ii) max(X1, X2, . . . , XN) has a MOBeta(m,n, a, b, 1/p) distribution.
31
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
2.3 Estimation
We consider estimation of the unknown parameters by the method of maximum likelihood.
The log-likelihood function for some realizations x1, x2, . . . , xN is
logL(m,n, a, b, α) = N logα+N logB(m,n) + (m− 1)N∑
i=1
log(xi − a)
+(n− 1)N∑
i=1
log(b− xi)−N(m+ n− 1) log(b− a)
−2N∑
i=1
log
(αB(m,n) + (1− α)B
(m,n,
xi − a
b− a
)).
The first derivatives of logL(m,n, a, b, α) with respect to m, n, a, b and α are
∂ logL
∂m= N(ψ(m)− ψ(m+ n)) +
N∑i=1
log(xi − a)−N log(b− a)
−2N∑
i=1
1
αB(m,n) + (1− α)B(m,n, xi−a
b−a
)[αB(m,n)(ψ(m)− ψ(m+ n))
+(1− α)
(B
(m,n,
xi − a
b− a
)log
(xi − a
b− a
)−
(xi − a
b− a
)m
Γ2(m)
× 3F2
(m,m, 1− n;m+ 1,m+ 1;
xi − a
b− a
))]∂ logL
∂n= N(ψ(n)− ψ(m+ n)) +
N∑i=1
log(b− xi)−N log(b− a)
−2N∑
i=1
1
αB(m,n) + (1− α)B(m,n, xi−a
b−a
)[B(m,n)(ψ(n)− ψ(m+ n))
+(1− α)
(Γ2(n)
(b− xi
b− a
)n
3F2
(n, n, 1−m;n+ 1, n+ 1;
b− xi
b− a
)− log
(b− xi
b− a
)B
(n,m,
b− xi
b− a
)) ]
32
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
∂ logL
∂a= (1−m)
N∑i=1
1
xi − a+N(m+ n− 1)
b− a+
2(1− α)
(b− a)m+n
×N∑
i=1
(xi − a)m−1(b− xi)n
αB(m,n) + (1− α)B(m,n, xi−a
b−a
)∂ logL
∂b= (n− 1)
N∑i=1
1
b− xi
− N(m+ n− 1)
b− a+
2(1− α)
(b− a)m+n
×N∑
i=1
(xi − a)m(b− xi)n−1
αB(m,n) + (1− α)B(m,n, xi−a
b−a
)∂ logL
∂α=
N
α− 2
N∑i=1
B(n,m, b−xi
b−a
)αB(m,n) + (1− α)B
(m,n, xi−a
b−a
)where
3F2(a1, a2, a3; b1, b2; z) =1
Γ(b1)Γ(b2)
∞∑k=0
(a1)k(a2)k(a3)k
(b1)k(b2)k
· zk
k!.
The maximum likelihood estimates can be obtained setting the above expressions to zero
and solving them numerically. Function nlm from the statistical software R can be used for
finding the maximum likelihood estimates.
2.4 Moments
We derive the moments of a random variable X with MOBeta(m,n, 0, 1, p) distribution,
since if X d= MOBeta(m,n, 0, 1, p) then Y = a + (b − a)X has a MOBeta(m,n, a, b, p)
distribution. Let us first consider the case p < 1. Let X1,N = min(X1, X2, . . . , XN).
From Corollary(2.2.1) it follows that X d= X1,N . Then the kth order moment of the random
variable X is
E(Xk) = E(Xk1,N) = p
∞∑r=1
(1− p)r−1E(Xk1,r), (2.4.1)
where X1,r = min(X1, X2, . . . , Xr). Nadarajah (2007b) derived the kth order moments of
beta order statistics as
33
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
18, 13, 24, 16, 13, 23, 36, 7, 14, 30, 14, 18, 20. Beta distribution and Marshall-Olkin beta
distribution are fit to the data. The results are presented in Table 2.2. The PP plots for the
two distributions are shown in Figure 2.4. We conclude that Marshall-Olkin distribution is
a much better model for the data set 2.
Table 2.2: Estimated values, log-likelihood, Kolmogorov-Smirnov statistics fordata set 2.
Distribution Estimates − log L K-SBeta m = 0.9677 548.3921 0.0808
n = 2.8426a = 1
b = 168.9589Marshall-Olkin beta m = 1.6243 529.7715 0.0567
n = 0.3073a = 0.9271
b = 168α = 0.0165
39
CHAPTER 2. MARSHALL-OLKIN BETA DISTRIBUTION AND ITS APPLICATIONS
Figure 2.4: PP plots for the fit of beta and Marshall-Olkin beta for data set 2 .
2.8 Conclusions
We have developed a new beta distribution called Marshall-Olkin beta distribution and its
various properties are discussed. We have dealt with the estimation problem also. Some
data analysis is also done. Also we have developed a time series model with minification
structure and derived its autocovariance structure.
Appendix 1
Here we present a program which can be used to compute the moments of a randomvariable with MOBeta(m,n, 0, 1, α) distribution. The program is written in R.