Applied Mathematical Sciences, Vol. 7, 2013, no. 80, 3947 - 3961
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2013.35268
Marshall-Olkin Extended Log-Logistic Distribution
and Its Application in Minification Processes
Wenhao Gui
Department of Mathematics and Statistics
University of Minnesota Duluth
Duluth MN 55812, USA
[email protected]
c 2013 Wenhao Gui. This is an open access article distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, a new class of Log-Logistic distribution using MarshallOlkin transformation is introduced. Its characterization and statistical
properties are obtained. The proposed model extends the Log-Logistic
distribution and is more flexible. Using the proposed model, we construct an autoregressive model and investigate its minification structure.
Mathematics Subject Classification: 60E05, 62F10
Keywords: Log-Logistic distribution, Marshall-Olkin distribution, Order
statistics, Minification process
1
Introduction
The Log-Logistic distribution (known as the Fisk distribution in economics) is
the probability distribution of a random variable whose logarithm has a logistic
distribution.
It has attracted a wide applicability in survival and reliability over the last
few decades, particularly for events whose rate increases initially and decreases
later, for example mortality from cancer following diagnosis or treatment, see
[10]. It has also been used in hydrology to model stream flow and precipitation, see [14] and [6], for modeling flood frequency , see [1]. [4] used this model
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to describe the thermal inactivation of Clostridium botulinum 213B at temperatures below 121.1 ◦ C. Furthermore, it is applied in economics as a simple
model of the distribution of wealth or income, see [8].
The cumulative distribution function F (x) and density function f (x) of
Log-Logistic distribution are given by
F (x) =
xβ
,
α β + xβ
x>0
(1)
and
αβ βxβ−1
, x>0
(2)
(xβ + αβ )2
Where α > 0 is the scale parameter and is also the median of the distribution,
β > 0 is the shape parameter. We denoted this by writing X ∼ LLD(α, β).
When β > 1, the Log-Logistic distribution is unimodal. It is similar in shape
to the log-normal distribution but has heavier tails. Its cumulative distribution
function can be written in closed form, unlike that of the log-normal. [7] used
the ratio of maximized likelihood to consider the discrimination procedure
between the two distribution functions.
On the other hand, [12] introduced a new family of survival functions which
is obtained by adding a new parameter γ > 0 to an existing distribution. The
new parameter will result in flexibility in the distribution. Let F̄ (x) = 1−F (x)
be the survival function of a random variable X. Then
f (x) =
Ḡ(x) =
γ F̄ (x)
1 − (1 − γ)F̄ (x)
(3)
is a proper survival function. Ḡ(x) is called Marshall-Olkin family of distributions. If γ = 1, we have that G = F . The density function corresponding to
(3) is given by
γf (x)
g(x) =
[1 − (1 − γ)F̄ (x)]2
and the hazard rate function is given by
h(x) =
hF (x)
1 − (1 − γ)F̄ (x)
where hF (x) is the hazard rate function of the original model with distribution
F.
By using the Marshall-Olkin transformation (3), several researchers have
considered various distribution extensions in the last few years. [12] generalized
the exponential and Weibull distributions using this technique. [2] introduced
Marshall-Olkin extended semi Pareto model for Pareto type III and estabilished its geometric extreme stability. Semi-Weibull distribution and generalized Weibull distributions are studied by [3]. Ghitany et al. (2005) conducted
Marshall-Olkin extended log-logistic distribution
3949
a detailed study of Marshall-Olkin Weibull distribution, that can be obtained
as a compound distribution mixing with exponential distribution, and apply
it to model censored data. Marshall-Olkin Extended Lomax Distribution was
introduced by [9]. [11] investigated Marshall-Olkin q-Weibull distribution and
its max-min processes.
In this paper, we use the Marshall-Olkin transformation to define a new
model, so-called the Marshall-Olkin Log-Logistic distribution, which generalizes the Log-Logistic model. We aim to reveal some statistical properties of
the proposed model and apply it to time series models.
The rest of this paper is organized as follows: in Section 2, we introduce
the new extended Log-Logistic distribution and investigate its basic properties, including the shape properties of its density function and the hazard
rate function, stochastic orderings and representation, moments and measurements based on the moments. Section 3 discusses the distributions of some
extreme order statistics. A Marshall-Olkin Log-Logistic minification process
is presented in Section 4. Our work is concluded in Section 5.
2
2.1
Marshall-Olkin Log-Logistic Distribution and
its Properties
Density and hazard function
Let X follows Log-Logistic distribution LLD(α, β), then its survival function
β
β
is given by F̄ (x) = 1 − αβx+xβ = αβα+xβ . Substituting it in (3) we obtain a
Marshall-Olkin Log-Logistic distribution denoted by M OLLD(α, β, γ) with
the following survival function
Ḡ(x) =
αβ γ
,
xβ + α β γ
α, β, γ, x > 0
(4)
The corresponding density function is given by
g(x) =
αβ βγxβ−1
,
(xβ + αβ γ)2
α, β, γ, x > 0
(5)
If γ = 1, we obtain the Log-Logistic distribution with parameter α, β > 0.
This distribution contains the Log-Logistic distribution as a particular case.
The following theorem gives some conditions under which the density function
(5) is decreasing or unimodal.
Theorem 2.1. The density function g(x) of the M OLLD(α, β, γ) distribution is decreasing (unimodal) if β ≤ 1(> 1), independent of α and γ.
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0.4
0.6
γ=1
γ=2
γ=3
γ=4
0.0
0.0
0.2
0.4
g(x)
0.6
γ=1
γ=2
γ=3
γ=4
0.2
g(x)
0.8
α = 1,β = 1
0.8
α = 1,β = 0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
x
2.0
2.5
3.0
x
α = 1,β = 1.5
α = 1,β = 6
1.0
1.5
γ=1
γ=2
γ=3
γ=4
0.5
g(x)
0.4
0.6
γ=1
γ=2
γ=3
γ=4
0.0
0.0
0.2
g(x)
1.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
x
1.0
1.5
2.0
2.5
3.0
x
Figure 1: Plots of Marshall-Olkin Log-Logistic density function for some parameter values
Proof. The first derivative of g(x) is given by
xβ−2 αβ βγ[αβ (β − 1)γ − xβ (β + 1)]
dg(x)
=
dx
(xβ + αβ γ)3
If β ≤ 1, dg(x)
< 0, the function g(x) is decreasing. If β > 1, the function
dx
g(x) has a unique mode at x = xm with g(x) increasing for all x < xm and
1
1
1
decreasing for all x > xm , where xm = α(β − 1) β (β + 1)− β γ β . Furthermore,
g(0) = g(∞) = 0.
Figure 1 shows some density functions of the M OLLD(α, β, γ) distribution
with various parameters.
The hazard rate function of the M OLLD(α, β, γ) distribution is given by
xβ−1 β
g(x)
= β
h(x) =
,
x + αβ γ
Ḡ(x)
α, β, γ, x > 0
(6)
The following theorem gives some conditions under which the hazard rate
function (6) is decreasing or unimodal.
Theorem 2.2. The hazard rate function h(x) of the M OLLD(α, β, γ) distribution is decreasing (unimodal) if β ≤ 1(> 1), independent of α and γ.
Proof. We can obtain the first derivative of h(x) as
dh(x)
xβ−2 β[αβ (β − 1)γ − xβ ]
=
dx
(xβ + αβ γ)2
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Marshall-Olkin extended log-logistic distribution
α = 1,β = 1
1.5
α = 1,β = 0.5
h(x)
0.8
γ=1
γ=2
γ=3
γ=4
0.0
0.0
0.5
0.4
h(x)
1.0
γ=1
γ=2
γ=3
γ=4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
x
1.5
2.0
2.5
α = 1,β = 6
4
α = 1,β = 1.5
3
h(x)
2
γ=1
γ=2
γ=3
γ=4
0
0.0
1
0.4
h(x)
0.8
γ=1
γ=2
γ=3
γ=4
0.0
0.5
1.0
1.5
3.0
x
2.0
2.5
3.0
x
0.0
0.5
1.0
1.5
2.0
2.5
3.0
x
Figure 2: Plots of Marshall-Olkin Log-Logistic hazard function for some parameter values
For β ≤ 1, it is clear that dh(x)
< 0, h(x) is decreasing in x. For β > 1,
dx
the function h(x) has a unique mode at x = x0 with h(x) increasing for all
1
1
x < x0 and decreasing for all x > x0 , where x0 = α(β − 1) β γ β . Furthermore,
h(0) = h(∞) = 0.
Figure 2 shows some shapes of the M OLLD(α, β, γ) hazard function with
various parameters.
2.2
Stochastic Orderings
In probability theory and statistics, a stochastic order quantifies the concept
of one random variable being “bigger” than another. It is an important tool
to judge the comparative behavior. Here are some basic definitions.
A random variable X is less than Y in the ususal stochastic order (denoted
by X ≺st Y ) if FX (x) ≥ FY (x) for all real x. X is less than Y in the hazard
rate order (denoted by X ≺hr Y ) if hX (x) ≥ hY (x), for all x ≥ 0. X is less
than Y in the likelihood ratio order (denoted by X ≺lr Y ) if fX (x)/fY (x)
increases in x over the union of the supports of X and Y . It is known that
X ≺lr Y ⇒ X ≺hr ⇒ X ≺st Y , see [13].
Theorem 2.3. If X ∼ M OLLD(α, β, γ1 ) and Y ∼ M OLLD(α, β, γ2 ), and
γ1 < γ2 , then X ≺lr Y , X ≺hr Y and X ≺st Y .
Proof. The density ratio is given by
γ1 xβ + αβ γ2 2
fX (x)
U (x) =
=
fY (x)
γ2 (xβ + αβ γ1 ) 2
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Wenhao Gui
Taking the derivative with respect to x,
2xβ−1 αβ βγ1 (γ1 − γ2 ) xβ + αβ γ2
U (x) =
γ2 (xβ + αβ γ1 ) 3
0
If γ1 < γ2 , U 0 (x) < 0, U (x) is a decreasing function of x. The results follow.
2.3
Stochastic representation
Let Ḡ(x|θ), −∞ < x < ∞, −∞ < θ < ∞, be the conditional survival function
of a continuous random variable X given a continuous random variable Θ.
Let Θ be a random variable with probability density function m(θ). Then a
distribution with survival function
Z ∞
Ḡ(x|θ)m(θ)dθ, −∞ < x < ∞
Ḡ(x) =
−∞
is called a compounding distribution with mixing density m(θ). Compounding
distribution provides a useful technique to get new class of distributions in
terms of existing ones. The following result shows that the M OLLD(α, β, γ)
distribution can be expressed as a compound distribution.
Theorem 2.4. Suppose that the conditional survival function of a continuous random variable X given Θ = θ is given by
h
x i
Ḡ(x|θ) = exp −θ( )β , x > 0, α > 0, β > 0, θ > 0.
(7)
α
Let Θ have an exponential distribution with density function
m(θ) = γe−γθ ,
γ > 0, θ > 0.
Then the random variable X has the M OLLD(α, β, γ) distribution.
Proof. For x > 0, the survival function of X is given by
Z ∞
Z ∞
x β
Ḡ(x) =
Ḡ(x|θ)m(θ)dθ = γ
e−θ( α ) e−γθ dθ =
0
0
αβ γ
xβ + α β γ
Which is the survival function of the M OLLD(α, β, γ) distribution.
In fact, Ḡ(x|θ) is a class of Weibull distributions with shape parameter β and scale paprameter αθ−1/β . Compounding a distribution belonging to this class with an exponential distribution for θ leads to a certain
M OLLD(α, β, γ) distribution. Next we will present another representation
of the M OLLD(α, β, γ) distribution.
Marshall-Olkin extended log-logistic distribution
3953
Theorem 2.5. Let {Xi , i ≥ 1} be a sequence of i.i.d. random variables with
a Log-Logistic distribution LLD(α, β). Let N be a geometric random variable
with parameter 0 < γ < 1 such that P (N = n) = γ(1 − γ)n−1 , n = 1, 2, ...,
which is independent of {Xi , i ≥ 1}. Then,
(1) min(X1 , ..., XN ) has a Marshall-Olkin Log-Logistic distribution with parameters α, β and γ.
(2) max(X1 , ..., XN ) has a Marshall-Olkin Log-Logistic distribution with parameters α, β and 1/γ.
Proof. The survival function of min(X1 , ..., XN ) is
∞
X
P (min(X1 , ..., XN ) > x) =
n=1
∞
X
=
P (X1 > x, ..., Xn > x)P (N = n)
[F̄ (x)]n γ(1 − γ)n−1
n=1
γ F̄ (x)
1 − (1 − γ)F̄ (x)
=
which is the survival function of the Marshall–Olkin Log-Logistic distribution
M OLLD(α, β, γ).
The survival function of max(X1 , ..., XN ) is
P (max(X1 , ..., XN ) > x) = 1 − P (max(X1 , ..., XN ) ≤ x)
∞
X
P (X1 ≤ x, ..., Xn ≤ x)P (N = n)
= 1−
n=1
= 1−
=
1
∞
X
[F (x)]n γ(1 − γ)n−1
n=1
1
F̄ (x)
γ
− (1 − γ1 )F̄ (x)
which is the survival function of the Marshall–Olkin Log-Logistic distribution
M OLLD(α, β, γ1 ).
2.4
Moments and Quantiles
Let X ∼ M OLLD(α, β, γ), for k = 1, 2, ..., the kth moment is given by
k
Z
µk = E(X ) = k
∞
x
k−1
Z
Ḡ(x)dx =
0
The kth moment exists if k < β.
0
∞
kπαk γ k/β csc( kπ
)
kxk−1 αβ γ
β
dx =
β
β
x +α γ
β
(8)
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When γ = 1, i.e., the Log-Logistic distribution, for k = 1, 2, ..., we have
k
E(X ) =
kπαk csc( kπ
)
β
β
For the standardized skewness coefficient
coefficient β2 =
µ4 −4µ1 µ3 +6µ21 µ2 −3µ41
(µ2 −µ21 )2
,
β>k
√
β1 =
µ3 −3µ1 µ2 +2µ31
(µ2 −µ21 )3/2
and kurtosis
, we have the following results.
Theorem 2.6. Let X ∼ M OLLD(α, β, γ), the skewness and kurtosis coefficients of X are given by
p
) csc( βπ ) + 3β 2 csc( 3π
)
2π 2 csc3 ( βπ ) − 6πβ csc( 2π
β
β
√
β1 =
2π
π 3/2
2
π[(2β csc( β ) − π csc ( β )]
(9)
for β > 3;
β2 =
) csc( βπ ) + 4β 3 csc( 4π
)
6π 2 β sec( βπ ) csc3 ( βπ ) − 3π 3 csc4 ( βπ ) − 12πβ 2 csc( 3π
β
β
π[π csc2 ( βπ ) − 2β csc( 2π
)]2
β
(10)
for β > 4.
Kurtosis β2
6
10
4
0
5
2
Skewness β1
15
8
20
10
Figure 3 shows the skewness and kurtosis coefficients for the Marshall-Olkin
Log-Logistic M OLLD(α, β, γ) model. The parameters α, γ do not affect the
two coefficents. The skewness and kurtosis coefficients decrease as β increases.
Also, as β goes to infinity, the skewness and kurtosis coefficients converge to
0 and 21
respectively.
5
5
10
15
20
25
30
5
β
10
15
20
25
30
β
(a) Skewness coefficient (b) Kurtosis coefficient
Figure 3: The plots for the skewness
√
β1 and kurtosis coefficient β2
The qth quantile xq = G−1 (q) of the M OLLD(α, β, γ) distribution is given
by
xq = [
qγαβ 1/β
] ,
(1 − q)
0≤q≤1
(11)
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Marshall-Olkin extended log-logistic distribution
Where G−1 (.) is the inverse distribution. In particular, the median of the
M OLLD(α, β, γ) distribution is given by median(X) = αγ 1/β .
Figure 4 displays the measures of central tendency (mean, median, mode)
of the the M OLLD(α, β, γ) distribution. From the figure, it is found that, as
expected, the mean is larger than the median and the median is larger tahn
the mode. The distribution has a right tail. All the measures converge to the
parameter α as β goes to infinity.
5
α = 1,γ = 5
5
α = 1,γ = 0.5
4
3
Value
1
0
4
6
8
10
2
4
6
β
α = 3,γ = 0.5
α = 3,γ = 5
6
Value
8
mean
median
mode
0
0
2
2
10
4
3
4
mean
median
mode
1
8
10
β
5
2
Value
mean
median
mode
2
3
2
0
1
Value
4
mean
median
mode
2
4
6
8
10
2
4
6
β
8
10
β
Figure 4: Plot of mean, median and mode of the M OLLD(α, β, γ) distribution
for various values
3
Distributions of Order Statistics
Let X1 , X2 , ..., Xn be a random sample of size n from M OLLD(α, β, γ) distribution. X1:n = min(X1 , ..., Xn ) and Xn:n = max(X1 , ..., Xn ) are the sample
minima and maxima respectively. These extreme order statistics represent the
life of series and parallel system and have important applications in probability
and statistics. In this section, we consider their limiting distributions.
Theorem 3.1. Let X1:n and Xn:n be the smallest and largest order statistics
from the M OLLD(α, β, γ) distribution. Then
1
1
β
(1) lim P (X1:n ≤ b∗n t) = 1 − e−t , t > 0, where b∗n = αγ β (n − 1)− β .
n→∞
1
−β
1
(2) lim P (Xn:n ≤ bn t) = e−t , t > 0, where bn = αγ β (n − 1) β .
n→∞
Proof. We use the following asymptotical results for X1:n and Xn:n (See [5]).
(1) For the smallest order statistic X1:n , we have
c
lim P (X1:n ≤ a∗n + b∗n t) = 1 − e−t ,
n→∞
t > 0, c > 0
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(of the Weibull type) where a∗n = F −1 (0) and b∗n = F −1 (1/n) − F −1 (0) if and
only if F −1 (0) is finite and for all t > 0 and c > 0,
lim+
→0
F (F −1 (0) + t)
= tc
F (F −1 (0) + )
For the M OLLD(α, β, γ) distribution, its cumulative distribution function is
xβ
given by G(x) = xβ +α
α, β, γ, x > 0.
βγ ,
−1
We have G (0) = 0 is finite. Furthermore,
lim+
→0
G(0 + t)
(t)β (β + αβ γ)
= lim+
= tβ
→0 [(t)β + αβ γ]β
G(0 + )
1
1
Therefore, we obtain that c = β, a∗n = 0 and b∗n = αγ β (n − 1)− β .
(2) For the largest order statistic Xn:n , we have
−d
lim P (Xn:n ≤ an + bn t) = e−t ,
n→∞
t > 0, d > 0
(of the Fréchet type) where an = 0 and b∗n = F −1 (1 − 1/n) if and only if
F −1 (1) = ∞ and there exists a constant d > 0 such that
1 − F (xt)
= t−d
x→∞ 1 − F (x)
lim
For the M OLLD(α, β, γ) distribution, G−1 (1) = ∞. Furthermore,
1 − G(xt)
xβ + α β γ
= lim
= t−β
x→∞ 1 − G(x)
x→∞ (xt)β + αβ γ
lim
1
1
Thus, we obtain d = β, an = 0 and bn = G−1 (1 − 1/n) = αγ β (n − 1) β .
Remark 3.2. If γ = 1, the results for the Log-Logistic distribution can be
1
1
obtained. The norming constants are b∗n = α(n − 1)− β and bn = α(n − 1) β .
Let Q∗ (t) and Q(t) denote the limiting distributions of the random variables
(X1:n − a∗n )/b∗n and (Xn:n − an )/bn respectively, then for i > 1, the limiting
distributions of (Xi:n − a∗n )/b∗n and (Xn−i+1:n − an )/bn are given by, See [5],
lim P (Xi:n ≤ a∗n + b∗n t) = 1 −
n→∞
i−1
X
(1 − Q∗ (t))
j=0
lim P (Xn−i+1:n ≤ an + bn t) =
n→∞
i−1
X
j=0
Q(t)
[− log(1 − Q∗ (t))]j
j!
[− log Q(t)]j
j!
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Marshall-Olkin extended log-logistic distribution
β
−β
For the M OLLD(α, β, γ) distribution, Q∗ (t) = 1 − e−t and Q(t) = e−t . For
i > 1, the limiting distributions of the ith and (n − i + 1)th order statistics
from the M OLLD(α, β, γ) distribution are obtained as follows
lim P (Xi:n ≤ α(n − 1)
n→∞
− β1
t) = 1 −
i−1
X
β
e−t
j=0
1
lim P (Xn−i+1:n ≤ α(n − 1) β t) =
n→∞
i−1
X
j=0
tβj
= 1 − P (W < i)
j!
−β
e−t
t−βj
= P (Y < i)
j!
Where W and Y follow the Poisson distributions with means tβ and t−β .
4
A Marshall–Olkin Log-Logistic Minification
Process
Here, we construct a minification process with the proposed Marshall–Olkin
Log-Logistic distribution. Consider a first order autoregressive minification
progress {Xn , n ≥ 0} as follows
(
n
with probability γ
Xn =
(12)
min(Xn−1 , n ) with probability 1 − γ
where 0 < γ < 1, {n , n ≥ 1} is a sequence of i.i.d. random variables and
independent of {Xn }.
Theorem 4.1. For the minification process given by (12) with X0 distributed as M OLLD(α, β, γ) distribution, {Xn , n ≥ 0} is a stationary Markovian autoregressive model with the marginals as Marshall–Olkin Log-Logistic
distribution M OLLD(α, β, γ) if and only if {n } has a Log-Logistic distribution LLD(α, β).
Proof. From (12), we have
F̄Xn (x) = γ F̄n (x) + (1 − γ)F̄Xn−1 (x)F̄n (x)
(13)
Using the fact that X0 has the M OLLD(α, β, γ) distribution and 1 has the
Log-Logistic distribution LLD(α, β), we have
F̄X1 (x) = γ F̄1 (x) + (1 − γ)F̄X0 (x)F̄1 (x)
αβ
αβ γ
αβ
= γ β
+
(1
−
γ)
×
x + αβ
xβ + α β γ xβ + α β
αβ γ
= β
x + αβ γ
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Wenhao Gui
Thus, X1 has the M OLLD(α, β, γ) distribution. By induction method, we
can prove that Xn has the M OLLD(α, β, γ) distribution. {Xn , n ≥ 0} is a
stationary Markovian autoregressive model and its marginals are Marshall–
Olkin Log-Logistic distribution.
On the other hand, let {Xn , n ≥ 0} be a stationary Markovian autoregressive model with Marshall-Olkin Log-Logistic distribution marginals, then the
equation of stationary equilibrium is given as follows
αβ
F̄X (x)
= β
x + αβ
γ + (1 − γ)F̄X (x)
F̄ (x) =
(14)
That is, n has the Log-Logistic distribution LLD(α, β).
Remark 4.2. Even if X0 has an arbitrary distribution FX0 , the minification
process is asymptotically stationary with M OLLD(α, β, γ). In fact,
F̄Xn (x) = γ F̄ (x)
n−1
X
(1 − γ)i F̄i (x) + (1 − γ)n F̄X0 (x)F̄n (x)
i=0
→
αβ γ
γ F̄ (x)
,
= β
x + αβ γ
1 − (1 − γ)F̄ (x)
n→∞
Figure 5 displays some sample paths of Marshall–Olkin Log-Logistic minification process for various values.
α = 1,β = 4,γ = 0.2
0
1
50
2
100
3
150
4
200
5
α = 1,β = 1,γ = 0.2
0
200
400
600
800
1000
0
400
600
800
1000
α = 2,β = 4,γ = 0.8
0
0
5
200
10
400
600
15
α = 2,β = 1,γ = 0.8
200
0
200
400
600
800
1000
0
200
400
600
800
1000
Figure 5: Sample paths of Marshall–Olkin Log-Logistic minification process
for various values
Marshall-Olkin extended log-logistic distribution
3959
Now we consider some properties of the Marshall–Olkin Log-Logistic minification process. For the joint survival function of (Xn , Xn−1 ), we have
F̄ (x, y) = P (Xn ≥ x, Xn−1 ≥ y)
= F̄ (x)[γ F̄X (y) + (1 − γ)F̄X (max(x, y))]
(
F̄ (x)[γ F̄X (y) + (1 − γ)F̄X (x))] 0 < y < x
=
F̄ (x)F̄X (y)
0<x<y
( α2β γ[(xβ +αβ )γ−yβ (γ−1)]
0<y<x
(xβ +αβ )(xβ +αβ γ)(y β +αβ γ)
=
α2β γ
0<x<y
(xβ +αβ )(y β +αβ γ)
The joint survival function is not an absolutely continuous, since
P (Xn = Xn−1 ) = (1 − γ)P (n ≥ Xn−1 )
Z ∞
= (1 − γ)
P (n ≥ x)fXn−1 (x)dx
0
1 − γ + γ log γ
∈ (0, 1)
=
1−γ
For the event {Xn > Xn−1 }, we have
P (Xn > Xn−1 ) = γP (n > Xn−1 )
Z ∞
= γ
P (n > x)fXn−1 (x)dx
0
=
1
γ(1 − γ + γ log γ)
∈ (0, )
2
(1 − γ)
2
Now let us consider the autocovariance between Xn and Xn−1 of the minificaπα csc( π
)
β
tion process. Let EXnk = µk and En =
= a, then
β
Cov(Xn , Xn−1 ) = E(Xn Xn−1 ) − µ21
= (1 − γ)E[min(Xn−1 , n )Xn−1 ] + γaµ1
and
E[min(Xn−1 , n )Xn−1 ]
2
= E[I(Xn−1 < n )Xn−1
] + E[I(Xn−1 > n )Xn−1 n ]
2
= µ2 − E[I(Xn−1 > n )Xn−1
] + E[I(Xn−1 > n )Xn−1 n ]
3960
Wenhao Gui
where I(.) is the indicator function.
2
AA ≡ E[I(Xn−1 > n )Xn−1
]
2
= E E[I(Xn−1 > n )Xn−1
|Xn−1 ]
2
)
= E(F (Xn−1 )Xn−1
Z ∞
=
x2 F (x)g(x)dx
0
=
πα2 γ[(β − 2γ + 2)γ 2/β − β] csc( 2π
)
β
β(γ − 1)2
,
β>2
and
BB ≡ E[I(Xn−1 > n )Xn−1 n ]
= E {E[I(Xn−1 > n )Xn−1 n |Xn−1 ]}
= E(F (Xn−1 )Xn−1 )
Z ∞
=
xF (x)g(x)dx
0
1
=
παγ[(β − γ + 1)γ β − β] csc( βπ )
β(γ − 1)2
Thus, Cov(Xn , Xn−1 ) = (1 − γ)(µ2 − AA + BB) + γaµ1 . Furthermore, the
first order process can be easily extended to high order process and the corresponding results can be derived.
5
Concluding Remarks
In this article, we have introduced the Marshall–Olkin Log-Logistic distribution. The properties, including the shape properties of its density function and
the hazard rate function, stochastic orderings and representation, moments
and measurements based on the moments are investigated. The distributions
of some extreme order statistics are derived. Using the proposed model, we
construced a minification process and discuss its covariance structure.
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Received: May 17, 2013