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1982-03
Simple dependent pairs of exponential and
uniform random variables
Lawrance, A. J.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/30167
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LIDilHKT
RESEARCH REPORTS Dl
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SIMPLE DEPENDENT PAIRS OF EXPONENTIAL AND
UNIFORM RANDOM VARIABLES
by
A.
J.
Lawrance
P.
A.
W.
Lewis
March 1982
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SIMPLE DEPENDENT PAIRS OF EXPONENTIAL AND
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neqative dependency
correlation
Spearman
simulation
bivariate exponential
bivariate uniform
random coefficient
linear function
a ntitheti c s
20.
It
(Continue on reveree aide
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neceaaary and Identity by block number)
A random-coeficient linear function of two independent exponential variables
yielding a third exponential variable is used in the construction of simple,
By employing antithetic exponential
dependent pairs of exponential variables.
to encompass neaative dependency.
developed
are
variables, the constructions
yield simple multiconstructions
the
exponentiation,
By employing neqative
ranges of dependency
The
nairs.
uniform
dependent
plicative-based models for
both of the
calculations,
correlation
by
allowable in the models are assessed
DD
FORM
1
JAN
73
1473
EDITION OF
1
NOV
65
IS
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product moment and Spearman types; broad ranges within the theoretically
allowable ranges are found.
Because of their simpilicity, all models are
particularly suitable for simulation and are free of point and line concentrations of values.
S/N 0102-
LF-
014-6601
UNCLASSIFIED
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Simple Dependent Pairs of Exponential and Uniform
Random Variables
by
A.
J.
Lawrance
University of
Birmingham
and
P.
A.
W.
Lewis
Naval Postgraduate School
Monterey
*
Department of Statistics, University of Birmingham,
Birmingham, B15 2TT, U.K.
**
Department of Operations Research, Naval Postgraduate
School, Monterey, California 93940, U.S.A.
Summary
A random-coefficient linear function of two independent exponential
variables yielding
a
third exponential variable is used in the construction
of simple, dependent pairs of exponential variables.
By employing antithetic
exponential variables, the constructions are developed to encompass negative
dependency.
By employing negative exponentiation, the constructions yield
simple multiplicative-based models for dependent uniform pairs.
The ranges
of dependency allowable in the models are assessed by correlation calcula-
tions, both of the product moment and Spearman types; broad ranges within
the theoretically allowable ranges are found.
all
Because of their simplicity,
models are particularly suitable for simulation and are free of point
and line concentrations of values.
Key Words:
BIVARIATE EXPONENTIAL, BIVARIATE UNIFORM, RANDOM COEFFICIENT,
LINEAR FUNCTION, ANTITHETICS, NEGATIVE DEPENDENCY, CORRELATION,
SPEARMAN, SIMULATION
6/1/82
Contents
1.
Background.
2.
One parameter positively dependent exponential model:
3.
Negative exponential transformation to uniformity:
4.
Negative dependency using an antithetic approach:
5.
Two parameter versions of the models:
6.
More general four parameter models:
7.
Multivariate generalizations.
8.
Line Discontinuities and Reliability Problems.
References
UP1
EP1
model.
model.
EP2, UP2 models
EP3, UP3, EP4, UP4 models.
EP+, EP-, EP5, EP6 models.
1
.
BACKGROUND
This paper introduces some models for pairs of dependent exponential
random variables constructed as random-coefficient linear functions of pairs
of independent exponential
random variables.
The transformations used
occurred first in connection with time series models in exponential variables
(see Lawrance and Lewis [1981] for details of this work).
indicate that
a
These models
specific random-coefficient linear function of independent
exponential variables leads to further univariate and multivariate exponential
variables, somewhat analogously to linear functions of independent
Gaussian variables leading to further univariate and multivariate Gaussian
variables.
Negative exponentiation of the transformation then leads directly
to a random-power multiplicative transformation, and it is found that this
can similarly be employed in constructing pairs of dependent uniform random
variables from pairs of independent uniform variables.
Both transformations
are particularly suited to simulation work, the pairwise uniform giving
a
base from which to transform to many other bivariate distributions.
This paper then presents results on both dependent exponential pairs
and dependent uniform pairs.
The transformations used are necessarily
limited to situations of positive dependency, but broadening to negative
dependency is achieved by employing simple antithetic operations in the
constructions.
The distributions given are free from point and linear con-
centrations of probability.
in
If these are required,
as
is
sometimes the case
reliability studies, the concentrations of probability can be put in by
using mixtures.
There are, of course, many earlier bivariate exponential and uniform
distributions; for instance, the work of Marshall and
01 kin
[1967] which
involves choosing the minimum of two exponentials, and which can easily be
1
simulated, but which always has
a
line-concentration of probability.
Downton
[1970] and Hawkes [1972] developed bivariate exponentials employing the idea
of bivariate geometric compounding of independent exponentials; in simulation
this would require simulation of bivariate geometries.
a
Gaver [1972] proposed
bivariate exponential specifically for negative dependency, also allowed
in Hawkes'
extension, and which in simulation would need both the operations
of choosing minima and ordinary geometric compounding; further, it is
intrinsically non-reversible, unlike most bivariate exponential models.
Other types of bivariate exponentials could clearly be obtained via the
inverse probability-integral transform of other standard bivariate distributions; bivariate uniforms in this category are specifically considered by
Barnett [1980].
All
these methods involve a fair amount of complication,
particularly the inverse probability integral transform, and limited flexibility.
The methods proposed in this paper seem simpler than all previous
methods at least in the major respect that each bivariate dependent pair is
obtained by simple random-coefficient linear transformation of
a
correspond-
ing independent pair; furthermore they are flexible in their dependency
properties, and easy to simulate.
In summary, unless yery
required, e.g.
specific modelling or structural detail is
the linear regression of Downton's model, the present models
offer the following advantages:
(i)
(ii)
The constructions are sjery simple.
They require only two independent exponential random variables
and one or two binary random variables for their construction.
(iii)
There is
a
broad range of the attainable dependency, as measured
linearly by the product-moment correlation or monotonically by
the Spearman correlation.
(iv)
The models are analytically tractable so that, in some cases,
closed form joint probability density functions and regressions
can be obtained.
The scheme of the following sections of the paper is to examine the
properties of the very simplest, one parameter, models (EP1 and UP1
detail.
)
in
The properties include product-moment and Spearman-type correla-
tions, joint probability functions and regressions.
These models have
a
broad but, positive correlation range which is then extended to negative
correlation by using antithetic pairs.
In later sections
models with more
parameters which can attain correlations right up to the theoretical bounds
are introduced, but developed in less detail.
The most general of these,
the EP+ and EP- models, allow the complete range of possible correlations
for bivariate exponential pairs and the possibility of accommodating other
structural details; they include all the other exponential models.
development is given for bivariate uniform pairs.
Parallel
ONE PARAMETER POSITIVELY DEPENDENT EXPONENTIAL MODEL:
2.
EP1
MODEL
The random coefficient linear function referred to in Section
central
to this work and is presented first.
Let
a
1
is
pair of independent,
identically and exponentially distributed random variables be denoted by
(E-,,
a
E
?
);
for simplicity, unit means are assumed throughout.
binary random variable, independent of
P(I = 0) = 3
by
,
P(I = 1)
=
-
1
3
(E-,,
E
2 ),
Let
with distribution given
Then the random variable
•
be
I
X
,
given
by
+
X = 3E
IE
2
1
(2.1)
,
has an exponential distribution with unit mean; this can be seen by taking
moment generating transforms,
4>
x
(t)
=
E{exp(-tX)}
=
^L
The result can be extended by taking
function of
E-,
.
a
(3 + (1-3)
E
^}
=
^
to be a random-coefficient linear
2
further pair of exponential random variables independent of
However, (2.1) is all that is required here to construct bivariate
pairs of exponential random variables; the result (2.1) is implicit in the
construction of exponential time series models evolving out of the EAR(l)
model of Gaver and Lewis [1980].
The EP1 Model
The first and simplest bivariate exponential distribution of the paper
is
obtained by interchanging
member of
a
E-,
and
pair, and hence produce
E
?
in
(2.1) to give the second
(2.2)
X
=
E2
+
3E
1
]
,
(EP1)
(2.3)
=
X
2
3E
E
+
2
.
1
These will be called the EP1(3) pair; the pair is clearly dependent because
of the common
E-, ,
using different
E
2
3's
and
I
There are clear generalizations obtained by
.
and I's
for
X-j
which will be considered later.
and X
2
The dependence of
X«) can be assessed by their (product moment)
(X-,,
correlation coefficient as
Corr(X
r
X
2
)
=
E{(3E
+
IE
]
=
33(1-3)
2
)(3E
+
2
IE
IE
2
)E(3E
+
2
IE
)
]
(2.4)
1.
The correlation thus has the non-negative
range (0, 0.75) with maximum value 0.75 at
0.5.
+
]
,
which is sketched in Figure
3 =
E(3E
-
)}
]
3
=
0.5 and symmetry about
The product moment correlation emphasizes linear dependency,
taking maximal values of +
1
if and only if the variables are linearly
related which is not the case here;
a
more general and more appropriate
measure of non-linear dependency, Spearman's correlation, will be considered
in Section 3.
RHO
1.00
Figure 1.
Correlation functions for the EP1 (3) model
and the UP1(3) model as functions of 3; both models
exhibit only positive dependence.
The joint probability density of
considering the probability element for
3En + IE,
and
1
-
3
and
On letting
.
and
X-,
X-,
X
and
take its values
I
can be obtained by
2
X
1
in terms of
2
3E-,
+
IE
2
with probabilities
and
this becomes
3
+ E
(1-3)P{(3E
2
1
)
€(x
r
(3E
+ dx-,),
x
1
2
+
E^
2
€(x 2> x
6(x
2
,
x
£
+ dx )}
2
(2.5)
+ 3PI3E-,
€(x 15
+ dx >P{3E
x-j
1
2
+ dx )}
.
2
The first line of (2.5) can be expressed as a probability element statement
in terms of
E,
and
the transformation
X-,
and
X
2
as
separately, and using the Jacobian, (1-3
E
2
from
E-.
and
E
2
leads to the joint pdf of
2
),
of
f
X ,X (x ,x
2
1
1
2
(1-B)l(x
=
)
>
3x
1
2
,
x
3x-,)f
>
2
2
(1 _ 3
(x^x-,)
)E
(2.6)
x f
where
1(
otherwise, and
and
x
x
(x
r
-* l) + 3f
x
)
=
l(3x
2
<
2
f
1
if
)f
(x
3E 1
l
x
>
3E
(x
2
?
)
•
and
3x
2
]
denotes the density of the exponential
Simplifying (2.6) then gives the desired result
variable in its suffix.
f
W
is an indicator function taking the value
)
x ? > 3x,
z
{
0-fi )Z 9 *r**Z
<
x
]
3" 1 x
)(l+3)"
1
exp{-(x-,+x
2
2
)/(H3)}
(2.7)
-1
exp{-(x-!+x )/3}
+ 3
The joint density thus has
(
3x~ <
x-,
< 3
x
2
a
±
,
3
1
1
;
x-j
,x
2
wedge shaped sector of relatively high density
superimposed over an independent bivariate exponential
2)
density.
This function is isometrically plotted in Figure
For low
values of
3
x
10
2
for
3
=
2
AXIS
0.5.
the raised section spreads out over the quadrant and
-1
X
>
X-,
AXIS x 10
Figure 2.
Isometric plot of the joint probability density
function of the positively dependent EP(3) model for 3 = 0.5.
This value of 3 gives the maximum attainable correlation of 0.75;
the figure has been truncated for both variables at 0.5.
3=0;
leads to independence at
for high
values of
the raised section
3
3=1.
diminishes towards nothing and again gives independence at
This joint density function could be used to obtain a maximum likelihood
estimate of
from
3
bivariate sample; note that
a
a
unique moment estimator
can not be obtained from the correlation (2.2).
The joint moment generating function of the
(X-,
,X 9
pair,
)
<|>
Y
v
(t n ,t
),
is
easily obtained via the EP1 defining equation (2.3); thus
^X
^
X
t
i'
=
t 2^
E{ ex P(-t-| X
i
E{exp[-(3t
=
t X )}
2 2
+ It )E
2
]
=
3(1
1
+ 3t )"
Differentiating with respect to
allows the regression of
E(X |X
2
= x)
=
It )E ]}
2
1
2
)
2
)
_1
_1
(1-0)0
+ Bt
+ t
1
t-j
+
2
_1
+ 3t
(l
1
+
(3t
1
X
t
2
3x + 3 +
on X,
-
(1
+ t
+ 3t
1
at t« =
2
1
(l
2
(2.8)
.
)
and inverting with respect to
to be obtained as
+ x/3)exp{-(3
_1
-
1
)x}
(2.9)
.
]
This conditional expectation will be fairly linear except for small
the asymptotic conditional variance will
be linear for large
x
x
,
and
Note from
.
the joint moment generating function (2.8) that the pair (X-,,X 2 ) is reversible,
<|>
x
jX
(t,,t
2
)
=
<|>
x
jX
(t
2
^)
,
in particular,
An aspect of this reversibility is that
directly from the definition (2.3).
E(X |X
1
P{X
>
2
X
}
]
=
= x)
2
0.5
= E(X
|X
2
=
x)
.
]
which can be shown
THE NEGATIVE EXPONENTIAL TRANSFORMATION TO UNIFORMITY:
3.
known that for an exponential random variable
It is well
mean, both
(0,1).
e
-X
and
'
1-e
-X
UP!
X
MODEL
of unit
are uniform random variables, distributed over
'
These results are of double interest here.
the transform to both components of
Firstly, application of
dependent exponential pair will lead
a
to a pair of dependent uniforms, and secondly, which may be more important
to stress, the product-moment correlation in this pair of uniforms is a
population analogue of Spearman's rank correlation for the original dependent
exponential pair.
This was demonstrated generally by Kruskal [1958]; he
showed that the product-moment correlation in the transformed pair of
dependent uniforms, could be linearly related to the concordance (quadrant)
probability
P{X
-
X )(Y
2
]
where (X.,Y.:
i
=
-
Y
3
1
)
>
0}
,
1,2,3) are three independent pairs with the untransformed
bivariate distribution.
Estimation of this probability was based on con-
structing the empirical joint distribution of an observed sample, and led
Kruskal to estimating the required linear function of the concordance prob-
ability by Spearman's correlation.
product-moment correlation in
a
Thus, we have the useful
idea that
bivariate distribution after its transforma-
tion to uniform marginals represents a population analogue of Spearman's
correlation for the untransformed bivariate distribution.
suitable measure of monotonic dependency for
particular, the Spearman correlation is
a
a
As such,
it is a
bivariate distribution.
In
suitable measure for dependency in
bivariate non-Gaussian distributions because of their intrinsic non-linearity,
and the consequent limitations of the product moment correlation.
X
General Results for the Transformati on
exponential pair
In the case of an
transformation to
(X-,
,X
2
of unit means, the preferred
)
uniform pair is given by
a
=
Y
1
-
expC-X^,
=
Y
exp(-X
-
1
2
]
)
(3.1)
.
2
The correlation and regression for this pair are easily obtained from the
joint moment generating function,
of
(X-j.Xp)
$„
«,
(t, ,t
2
=
)
t^X^)}
-
E{exp(-t,X-,
First for the correlation we have
•
E{Y
Y
2
1
=
)
E{(l-exp(-X )(l-exp(-X
1
2
=
)}
^
fy
(1,1)
(3.2)
,
and thus the correlation result
Corr(Y
rY2
For the regression of
E
Now setting
by
f„
iw
Y
( Y-,
t
=
-
x
(x-,|x)
,
=
2
=
)
on
Y,
12<J>
Y
,
2
X
-
(1,1)
x
3
(3.3)
.
first write it in terms of
y) = E{l-exp(-X 1 )|X
=
log(l-y)}
-
2
X-j
(3.4)
.
and denoting the conditional p.d.f. of
log(l-y)
and X
X,
|
2
the regression is
oo
E[Y |Y
1
l-exp(-x)]
=
2
=
/
[l-exp(- Xl )]f
x
^
|x)dx
(x
]
(3.5)
.
1
The required result is obtained after multiplying both sides by exp(-tx)
and integrating
x
over (0,°°), so leading to
exp(-tx)E[Y 1 |Y_
/
o
=
1
-
exp(-x)]dx
L
'
10
=
1
t"
-
<J)
Y
X
Y
T X2
(l,t-l)
.
(3.6)
as
2
Thus, the regression of the uniforms follows from inverting the joint moment
generating
^
function of
x
x
,
(l,t-l)
of the exponential
and then replacing
by
x
pair with respect to
-
log(l-y)
as a
t
.
The UP1 Model
The negative exponential transformation to uniformity is now exemplified
using the EP1 (3) model.
variables from
a
To obtain
a
pair of independent uniform random
pair of independent exponentials it is simplest here to use
the monotonic decreasing version of (3.1), that is
= exp(-E.j)
U
U
,
1
It follows then
exp(-E
=
2
2
(3.7)
.
)
that the dependent uniform pair derived from the EP1
(
3)
exponential model via (3.1) is given by
l
(UP1)
=
Y
1
where
I
1-uf U z
=
Y
,
2
1-UJ b|
retains its definition from Section
cative aspect of the model is evident.
(3.8)
,
The random power multipli-
2.
The pair (Y,
,Y
2
)
of (3.8) will
be
termed the UP! (3) model; the correlation of this uniform pair follows imme-
diately from the joint moment generating function of
(X-,,X
?
)
as given at
(2.8), or by direct calculation, and is
Corr(Y 15 Y 9 )
1
This function of
for
3 s
0.32.
3
is
l
=
33(1-6)
8+73+32
.
0+3)
illustrated in Figure
The joint density of
11
Y-.
2
(2+3)
1;
(3.9)
2
its maximum value is 0.72
and Yo may be obtained by considering
P(U^ u\ < y
r
uf u{ <
y2 )
(3.10)
3P(U^
<
y^PCU^
< y )
2
+ (l-e)P(uf U
2
lyr
U-,
U
2
<y 2
)
.
Density for the last probability is restricted to the leaf shaped region
(yi
f
<
< y5.
y-i
Y 1 ,Y (y i
1' 2
y
=
2
)
UyJ
<y2 <
76
1)»
!^
and the final result takes the form
iy^)(i + e)"
1
(y y
2
1
)'
1
3/(1+3) +
b"
"^ 70
^)
(3.11)
(0 < y
r y2
The density is isometrically plotted in Figure
3
< 3 < 1)
< 1;
for
3 =
.
0.32, the value
giving maximum product moment correlation.
r4.00
1.00-1
-1
AXIS x 10
Y
Y
1
2
AXIS x 10
Figure 3.
Isometric plot of the joint probability density
function of the positively dependent UP1 ( 3) model for 3 = 0.32
This value of 3 gives the maximum attainable correlation of 0.75,
.
12
Regression for the UP1 model is most easily obtained via (3.6) using
the joint moment generating function of
X-,
and X
given at (2.8) and takes
?
the form
E(Y |Y
2
1
= y)
_1
=
1
-
(l+3)
(l-y)
(1 " 3)/3
_1
(l-y)
+ (2+3)
The regression has initial value
(2 " 3)/3
near monotonic concave way.
a
3=1
which is correct since
independent and
reversible,
P{U,
>
U
?
}
in
=
E( Yp)
1/2
=
1/2.
particular
E(Y
0.5; this can be
Note that the UP1
= y)
|Y
2
1
(l-y)
3
.
which is always less than 0.5 and
3/(1-3)
curves upwards towards 1.0 in
it has the value
_1
(2+3)
-
=
E(Y |Y
(
For
Yo and
Y-.
3=0
and
are then
pair is, like the EP1 pair,
2
= y)
.
Also
directly from the definition (3.8).
seen
13
NEGATIVE DEPENDENCY USING AN ANTITHETIC APPROACH:
4.
EP2, UP2 MODELS
The antithetic transformation of any random variable
first expressing it as
uniform (0,1) variable
function (the inverse probability integral) of
a
U
,
X
X
;
For
the antithetically transformed variable
,
has the same marginal distribution as
correlated with
a
and then by taking the same function of 1-U.
any nonnegative random variable
a(X)
is found by
X
and is maximally negatively
X
note that the Spearman correlation of
X
and
a(X)
is
minus one, but their product moment correlation will not necessarily be as
negative as this.
In the exponential
a(X)
=
-
case,
log(l-exp(-X))
(4.1)
and the maximum negative product moment correlation is
known after Moran [1967].
-
0.6449, as is well
As a pair of exponential random variables,
X
and
lie on the curve
a(X)
e
"x
+ e
" a(x)
=
1
and so are highly degenerate in the sense that
X
completely determines
a(X)
The EP2 Model
The aim here is to construct negatively dependent exponential pairs
which are not, like the antithetic pair, partially or completely degenerate;
the random-coefficient linear transformation (2.1) allows us to do this while
still making use of antithetic pairs.
The general approach is illustrated
with respect to the EP1 model; in (2.3) the
their antithetic versions a(E-,), a(E
I 1
as
2
),
and
E-,
I
and E~ of
in X,
X,,
are replaced by
and X~ is replaced by
and Ip respectively, these variables having the same marginal distribution
I
.
Thus the suggested negatively dependent model has the form
14
X
+1^2,
3E
=
1
]
(EP2)
(4.2)
3a(E
=
X
2
2
)
+
a^)
I
2
,
and the expression for its correlation is given by
Corr(X
r X2
23(1-3) (-0.6449) +
=
)
CovO^I^
This suggests that maximum negative correlation of
maximum negative covariance of
achieved for minimum E(L,I
2
),
I-,
and I«
I
2
3
or equivalently minimum p
Consideration of the general joint distribution of
h
and X
X-,
For fixed
.
=
I,
(4.3)
I
,
9
this will
,
=
requires
2
P(I,
,
be
I« = 1)
= 1,
that is
1
-l
1-3
1-3-P
p
(4.4)
23-1+p
1-3-P
1-3
3
3
1
shows that, since the probabilities must be non-negative, the (0,0) term
constrains
p
to the minimum value of
1-23 if
3
1
and
H.
if 3
:•
%
.
The
implied joint distribution is, not surprisingly, the antithetic one, and it
has covariance
2
-
3
for
< 3 <
back to (4.2), the model with
I,
h and -(1-3)
=
I,
2
I„ = a(I)
for ^
£
3
£
1
Thus, going
.
is of most interest in
connection with negative dependence and will be called the EP2(3) model; its
correlation
is
given by
f
Corr(X 1} X
2
)
=
-23(l-3)(0.6449)
-
2
13 1%,
3
(4.5)
-
-23(l-3)(0.6449)
15
-
(1-3)
h
< 3 <
1
.
This function is sketched in Figure
3 =
0.5 which, considering the model
factory.
(The Gaver model
RHO
4.
The minimum value is -0.5724 at
is never degenerate,
is fairly satis-
[Gaver, 1972] is also not degenerate and has
a
.00
Correlation functions for the EP2(3) model
Figure 4.
and the UP2(3) model as functions of 3; both models
exhibit negative dependence.
minimum correlation of -0.5).
value is given in Figure
5;
A simulation of EP2(3) pairs at the 3 = 0.5
the scatter is best described as
emulation of the antithetic concentration along the curve
Weakening of the dependency as the parameter
3
e
approaches
a
—x
'
non-degenerate
+ e
or
-J
v
=
1
1
.
fades
this characteristic scatter into the blander independent exponential picture.
The scatter in Figure 5 should be compared to the isometric plot of the joint
density in Figure
3.
The joint moment generating function of
moment generating function of
E.
and a(E.)
evaluated as
16
X,
and X~
requires the joint
which is straightforwardly
X2 5
XI
Figure 5.
Scatter plot of 50 simulated pairs of observations
from the negatively dependent EP2( 3) model for 3 = 0.5:
the solid
line represents the curve relating the antithetic bivariate
exponential pairs.
1
<Kt
r
t
2
)
=
which is the Beta function B(l +
(T-x)
?
^dx
(4.6)
,
o
t-,
,
1
p.. = P{I = i, a(I) = j},
distribution
t
t,
/ x
+ t
i,
2
)
j
.
=
Also required is the joint
1,0, available as remarked
previously following (4.4); hence the result, via (4.2),
*X ,X
1
=
(t
l»
=
t
2
)
2
p 11 cf>(et 1 ,t 2 )4>(t 1 ,et 2
p Q1
)
+ p
<j)(3t ,t )(i+3t )
1
10
2
2
1
,3t )(l+3t )"
(})(t
1
2
1
17
+ p
00
-i
1
1
(l+3t )" (l+3t )"
1
2
(4.7)
Actually the model (4.2) works for any independent bivariate exponential
pairs
and (E ,E
(E-|,E-|)
2
2
However, the antithetic pair is simplest to
.
)
generate and gives the greatest negative correlation in the (X,
,X
?)
pair.
The UP2 Model
Thus far we have the EP2 model giving
broad range of negative depen-
a
dency for exponential pairs; this is now transformed to the corresponding
uniform pair giving the UP2 model.
By negative exponentiation of (4.2) and
use of (3.1) and (3.6) we have
(UP2)
Y
-
-
1
Ufu|,
]
The correlation of Y
and Y
]
2
Y
2
«
1
-
aCl)
Cl-0
1
)
O-U 2 ) B
(4.8)
•
may be evaluated directly from (4.8) or by use
of (4.7) and (3.3) and gives the result
2
2
3
-36(4+56+6B +3 )/[(l+3) (2+3)
Corr(Y
r
Y
2
)
2
<
]
3 <
h
,
=
(4.9)
2
[
2
-3(l-3)(2+33-S )/[(l+e) (2+3)]
h <
3 <
1
.
A graph of this correlation function is given in Figure 2; its maximum
negative value is -13/15
=
-0.8667 which occurs at
3 =
0.5; this indicates for
the EP2 model a good negative range to the Spearman correlation, but no
Bivariate scatters from the UP2 model show
ultimate degeneracy.
a
strong
degree of linearity compared to the near antithetically curved scatter of
the corresponding exponential
EP(2) pair in Figure 5.
the dependency about 3 = 0.5 in the EP2 model
Spearman correlation of the UP2 model.
stronger for
the EP1 model
3 =
0.5
-
6
is more
Again, the asymmetry of
evident from the
Negative dependency is considerably
than the corresponding 3 = 0.5 +
6
.
This parallels
in which positive dependency is stronger for 3 = 0.5 - 6 than
18
for 3
of
3
=
0.5 +
•
6
these remarks are borne out in scatters for various values
;
Finally, it should be noted that the antithetic choice
(
I
,a(
I ) )
is
best in regard to obtaining negative dependency in the UP2 model, as was found
for the EP2 model.
Regressions are again not easily tractable.
Positive and Negative Dependence
Extension of the bivariate models given above to models which exhibit
both positive and negative product-moment correlation is achieved quite
simply by mixture arguments.
=
X
+
BE
1
h
1^2
]
f3E
For example, combining (2.3) and (4.2) let
+
I,
E]
2
w.p.
p
w.p.
1-p
(4.10)
-\
3a(E
2
)
+
I
2
a(E
)
]
This uses the fact that a probability mixture of two identically distributed
random variables (possibly dependent) has that same identical distribution.
These two parameter model contains both the EP1 model
model
(p = 0)
(p =
1)
and the EP2
and has product moment correlation ranging between -0.5724
and 0.75.
19
TWO PARAMETER VERSIONS OF THE MODELS:
5.
EP3, UP3, EP4, UP4 MODELS
Dependency in the models has thus far been determined by one parameter
3
but it may be desirable to have an additional parameter in the model so
,
that two aspects of the joint distribution can be modelled; for instance,
one dependency measure, such as the product-moment correlation, and a prob-
ability statement, such as
0.5 for all
allows
by
3-i
3
for
3
> X ),
2
P(X-,
which in the EP1 model was equal to
Two approaches will be discussed rather briefly.
.
in the randomly linear combination of
and
X-,
3
for Xp
2
linear operation is used:
while in Section
,
E-,
6
a
and
E
One
to be replaced
2
two parameter randomly-
this stems from the NEAR(l) model of Lawrance
[1980], Lawrance and Lewis [1981].
The EP3 Model
The first two parameter model, EPS, is defined by the equations
=
X
+
E
3-|
E
I-,
]
1
< B
,
2
<
1
,
<
1
.
1
(EP3)
(5.1)
X
2
=
The marginal distributions of
3
parameters so that
P{I
2
= 0}
=
1
-
P{I
P{I,
E
2
+
2
and
I-,
= 0}
1} = 3
=
2
3
=
1
I
-
I
E-j
2
2
<
»
3
2
must correspond to the different
P{I,
1} = $.
=
and
The product moment correlation of the
.
2
model is given by
Corr(X
r X2
)
=
3-,(l-3
2
)
The general joint distribution for
+ 3 (l-6-,
2
(I-,,I
2
)
)
+
Cov^,^)
is the obvious
.
(5.2)
generalization of
(4.4); the value of p-^ for maximum positive dependence is now min(l-3-, ,l-3 )
2
Hence the result
20
'2B-,
Corr(X
r X2
)
=
(l-3
2
)
+ 3 (l-3-,)
<
2
3
1
1
3
1
3-j
2
1
1
1
"I
»
«
3-|(l-3
2
)
+ 23
2
(l-3-,)
Contours of this function (Figure 6) show
< 3
a
2
region with
central
a
local
maximum of 0.75; the function decreasing from each side of the centre along
the line
3-,
=
3?
=
3,
=
the line
1
3-,
density of
3
2
t0 zer0 values at the corners where
3,
=
3
=
and
2
and the function increasing from each side of the centre along
,
=
+ 3
2
X-,
,X
?
1
is
to unit values at the other two corners.
simply obtainable but is now
instead of two, components.
a
The joint
mixture of three,
Thus maximum likelihood estimates of
8-j
can be obtained.
BETA2 1.0
Correlation function of the two-parameter positively
Figure 6.
unlabel led contours may be identified
dependent EP3(3, ,$«) model:
from their joins with the 3-| and 3 2 axes.
21
and
3
2
The edges ($, = 0) or (3 = 0) give a one-parameter bivariate exponen?
pair which corresponds to adjacent (exponential) variables in an
tial
Note that using the basic relation-
EAR(l) process [Gaver and Lewis, 1980].
ship (2.1) outside of the serial constraint posed by
time series gives
a
a
broader bivariate exponential pair.
The UP3 Model
The corresponding uniform model, UP3, is given by
i
-
? h
^
iyu
J
6
=
y
(up3)
1
l
l
uyiy,
=
y
i
-
2
1
(5.4)
2
and has correlation (the Spearman correlation of the EP3 model), given by
l-3
Corr(Y ,Y
1
2
)
=
12
3
2
(2+3j(l+3o)
+
-3
2
1
(2+3o)d+3 1
1
)
(5.5)
-3
+
Cl+3-,)(l+3
For
3,
>_
there is
3o
a
2
(0 <
)
symmetrical result.
3-,
£
3
2
< 1)
The contours of this function
represent
a
deformation of those just described for the EP3 model, symmetry
about
=
3o
3i
is
towards the origin.
preserved, while the area of local maximum is moved
Thus, asymmetry in the Spearman dependence is indicated,
The EP4 Model
The negatively dependent version of the EP3 model
is the two
parameter
version of (4.2), and will be called the EP4 model; this has correlation
function
22
Corr(X
r X2
=
)
0.6449L3-, (l-3
-
2
for
3^2
-
<
1
,
>
1
.
(5.6)
=
+ 3
2
)(l-3
2
1
for
)
a
+ 3
3
1
with
>
2
valley of lowest negative values along
a
central value of -0.5720 and corner values
a
For specified negative correlation down to
of maximum negativity, -0.6449.
about -0.5 there is
2
i
Contours of this function indicate
3-,
+ B
3
1
(1-3-,
the direction
^(l-^)]
+
)
wide choice of ($, ,3 ) pairs, in the two regions either
2
side of the valley along
+ 3
3-.
=
1
.
2
The UP4 Model
The final development of the two parameter models is to the negatively
dependent two parameter uniform model, the UP4 model.
(4.8) with
Y
?
and
3
replaced by
3
in
I
and
2
replaced by
Y,
a(I
2
from the 'antithetic' joint distribution for
?
u
3,
=
1
+ 3
2
-
>_
(3-,+3
1
2
when
)
3-,
+ 3
1
Corr(U
r
<
and
1
2
We then have, for the case
.
U
2
)
=
12
-
and
I,
and
3
I
in
Maximum negativity of dependence follows
.
)
and
3i
This corresponds to
(3-,+3
2
p
and
I,
Qo
=
+ 3
3-.
3-,
2
2
+ 3
£
which has
I
1
-
when
1
2
,
3
)
+
(l+3-|)(2+3-|)(l+3 )(2+3
2
2
)
1
d+3
)(l+3 )(2+3
2
1
)
1
(5.7)
+
(H3 )d+3 2 )(2+3 2
-
3
.
)
1
The corresponding result for
+ 3
3
]
>
1
is omitted.
The contours of this
2
function are quite similar to those of the EP4 model, but with
23
a
central
value of -0.867 and associated corner values of minus one; the other two
corner values are zero, as for the EP4 model; it is again useful to view
these as the Spearman correlations of the EP4 model.
From the point of view of simulation, the two parameter models are
hardly more complicated than the one parameter version and offer an extra
degree of flexibility.
The joint probability density functions and moment
generating functions are more complicated than for the one parameter models,
but can be derived if needed, for instance, for estimation.
possible to derive measures such as
P{X-,
24
>
X^}
,
It is also
which is not necessarily
)
MORE GENERAL FOUR PARAMETER MODELS:
6.
EP+, EP-MODELS
The NEAR(l) exponential time series models of Lawrance and Lewis [1981]
suggests a class of four-parameter models; these are likely to be over-
complicated for ordinary use, but place the earlier models in
setting and suggest
a
a
further class of two-parameter models.
a
general
They also give
bivariate exponential pair with full positive correlation, and without the
degeneracy which occurs for the case
=
(3-.
^
or
= 0)
in the EP3 model.
Being as brief as possible, this development gives the EP+ model as
V^
- 8
X,
I,E
2
,
(EP+)
(6.1
= 6 V E
+
2 2 2
X
2
where, for
1
i
w.p.
=
2
,
1,2
a.
1
V
l
w.p.
E-,
I
w.p.
(l^/D-O-ou)^.]
w.p.
a.3 /[l-(l-a )3 ]
i
i
i
i'\
(l-a.)3
1-a-
i
and the random variables (V,,V
2
are independent between pairs but
(I-i.Io)
),
This model has correlation structure of
usually dependent within pairs.
the form
Corr(X
r X2
)
=
3-,3
Cov(V-, ,V
2
2
)
+ Cov(I ls
I
2
)
(6.2;
+ a-|3-|(l-a 3
2
For
a
2
)
+ a 3 (l-a-|3-|
2
2
model of maximum positive dependency the appropriate joint distributions
of (V-,,V ? ) and (I-,,I 2 ) could be obtained using constructions as at (4.4).
A further two parameter model, EP5, is suggested by taking
3,
=
32 =
3 and V
=
]
V 2>
I-j
=
I
2
;
then (6.2) reduces to
25
a-j
=
a
=
2
a
Corr(X ls X
This has maximum value 0.75 at
2
3ag(l-a3)
=
)
=
a3
0.5
(6.3)
.
but does not attain the higher
,
correlations of the EP3 model, and is more complicated in its construction.
Both the EP1 and EP3 models can be obtained as special cases of the
is given by taking
EP+ model; the EP1 model
with
with
I,
- I«
(I, ,I«)
,
while the EP3 model has
=
ou
s
ou
=
a«
ou =
1
and
1
(3-i
>
»3
2
)
3,
=
Bo = 3
unchanged, and
of maximum possible dependency.
Corresponding to (6.1) there is
more general negatively dependent
a
four-parameter model, EP-, of the form
Wi
x
i
+
¥2
•
(6.4)
X
=
a(E
3 V
2
1
]
2
+
)
I
2
a(E
)
,
1
which has, by analogy, the correlation structure
Corr(X
r X2
)
=
3-,3
Cov(V
2
,V
1
2
CovO^I^
+
)
(6.5)
-
0.6449[a
3
1
1
(l-a 3 ) +
2 2
06
2
3
2
(
1 -cc-,
3-,
)D
It contains the EP2 and EP4 models, and would suggest the further two param-
eter model, EP6, by taking
cu
=
a
=
2
a, 3,
=
3
=
2
3
and (V-,,V ),
2
(I-,,I
2
)
as two antithetic pairs.
Any further details are omitted, as are the corresponding uniform models
of the four-parameter class, UP+, UP- given in Table 1, which summarizes all
the models.
26
MULTIVARIATE GENERALIZATIONS
7.
The possibility of multivariate generalizations is apparent by repeated
use of the random-coefficient linear functions of (2.1).
the way the EAR(l) process is constructed with
exponentials
E-,
,
E
2
Any
,
a
In fact this
serial chaining of i.i.d.
variables in an EAR(l) process (or an
k
NEAR(l) process) are a k-variate exponential random variable.
this serial context many possibilities suggest themselves.
3
to
$-,
,
to
I
and replacing
I-.
linear function of independent
X =
1
Here
I-,
and
I
?
2
]
E
E
+
I-,E
2
2
Thus changing
using
3
I^^
3o
and
I
2
,
gives
(7.1)
.
A triple of dependent exponential
must be independent.
variables could be constructed using
Outside of
by a similar random-coefficient
2
and
E
+ 3
E
e
is
E,
,
E
2
E
,
3
in three of the six
possible orders.
A simpler possibility would be to use the basic randomly-linear
operation on the pairs
(E 1S E
are not considered here.
2)
,
(E
2
»E
3
)
and
(£3^)
.
Such developments
The possibilities are legion but no 'natural'
simple method suggests itself above any other.
27
I
LINE DISCONTINUITIES AND RELIABILITY PROBLEMS
8.
The problem of discriminating amongst the many bivariate exponential
models is not simple, although there may be modelling or structural details
Most of these details
which recommend certain models in certain contexts.
are, however, tenuous and difficult to verify from data.
An important
mathematical scheme is given in Griffiths [1969], who found
expansion for bivariate exponential random variables.
forward here are simple to generate on
canonical
a
Again the models put
computer and are analytically
a
tractable.
Another property is that the bivariate distributions do not have line
However, as an illustration of modelling considerations,
discontinuities.
we note that in a reliability context thus is not necessarily
from
in a system can fail
is known that components
a
virtue.
It
common cause, which is
a
precisely what gives the line discontinuity in the bivariate distribution.
However, it is simple to put this in to the present models and it can be
done in at least three ways.
Thus, let (X, ,X
let
( I-.
,
2
)
2
denote any unit mean, bivariate exponential pair,
)
denote an indicator pair, possibly completely or partially
dependent, with marginal distributions
P{Io
=
1} =
P{Io = 0}
-
1
=
P
=
Z
(ii)
=
Z
1
+ (1-I
X
I
1
1
1
3E +
)E
Z
,
1
X
I
1
1
-
P{I-|
=
0} = P^
and
be an independent, unit mean
E
Three new bivariate exponential pairs are given
exponential random variable.
(i)
and let
,
2
1} =
=
P{I,
=
2
2
and
28
2
=
Z
,
]
I
X
2
3E +
+ (1-I
I
?
X
2
;
2
)E
;
(8.1)
(8.2)
(iii)
Z
=
min(X
1
In all
three cases there is
proportional to
E
a
r E)
=
Z
,
2
min(X
,E)
2
non-zero probability that
Z-,
and 1
?
are
.
The first pair uses the idea that mixtures of identically distributed
random variables have that same distribution, the second pair uses the basi
relationship (2.1) and the third uses the fact that the minimum of independent
exponential random variables is an exponential random variable.
if
X-,
and X~
are independent, then (8.3)
In fact,
is the Marshall-01 kin model.
ACKNOWLEDGMENTS
The work of Professor
Naval
P.
A.
W.
Lewis was supported by the Office of
Research under Grant NR-42-469.
29
J
TABLE
I
Summary of Models Considered (See text for relevant details)
EP1
X
=
0E
1
x
2
EP2
+ IE
1
= eE
X
=
X
2
Y
+
3a(E
2
)
UP2
E
T
1
2
+
I
Y
l
2
y
a{E^)
^
U
l
-
l-UJl|
"
l- U U
f 2
=
i-(i-u
2
1
1
=
Y
l
+ ie
2
*E
l
UP1
2
2
a CD
)
cl
_u
1
e
2)
B
EP3
X
l
X
2
=
¥l
+
E
T
1
= 6 E
+
2 2
UP3
2
1
Y
E
I
2
1U
-
I
2
1
2
1
l
=1 - U
Y
1-uJ^
3
EP4
=
X
1
1
+
3 E
I^g
UP4
l
1
I
1
= 1-U
U
YY
I)
U
2
l
'
B
I
X
EP+
=
2
h
=
X
=
+
3 a(E )
2
2
Wl
l
2
a{E^)
=
Y
2
2
1-O-u^) ^(1-U 2
e v
+
UP+
¥2
=
YY
l
1
'
i
i
- UJ
U
1
2
L
)
h
1
2
l
Io B 9 V 9
EP-
2
X
l
+
B V E
2 2
1
"
Wl
=
B-,V
T
2
=
Y
E
1-U-,
C
U
2
2
1
31 V1
+
UP-
¥2
=
Y
1-U/
1
I,
]
(1-U
2
1
)
e,v
i
X
2
2
a(E
2
)
+
Y
a(E.,)
I
2
=
2
30
l-Cl-^J
2
]
(1-U
2
)
2
REFERENCES
Barnett, V. D.
1980.
Some bivariate uniform distributions.
Theor. Meth. A 1(4)
453-461.
Commun. Statist.
,
Downton, F.
theory.
1970.
J.
R.
Bivariate exponential distributions in reliability
Statist. Soc. B, 32, 63-73.
Gaver, D. P.
1972.
Point process problems in reliability.
In Stochastic
Point Processes ed. P. A. W. Lewis, Wiley, New York, 775-800.
,
Gaver, D. P. and Lewis, P. A. W.
1980.
First order autoregressive sequences
and point processes.
Adv. Appl
Prob. 12, 727-745.
.
Griffiths, R. C.
1969.
The canonical correlation coefficients of bivariate
gamma distributions.
Ann. Math. Statist.
40, 1401-1408.
,
Hawkes, A.
1972.
A bivariate exponential distribution with applications in
reliability, J. R. Statist. Soc. B, 24, 129-131.
Kruskal
,
W.
Assoc
,
1958.
Ordinal measures of association.
53, 814-859.
J.
Amer. Stats.
Lawrance, A. J.
1980.
Some autoregressive models for point processes.
Point Processes and Queuing Problems (Colloquia Mathematica Societatis
Janos Bolyai 24 ), ed. P. Bartfai and J. Tomko, North Holland,
Amsterdam, 257-275.
Lawrance, A. J. and Lewis, P. A. W.
1981.
A new autoregressive time series
model in exponential variables (NEAR(l)).
Adv. Appl. Prob. 13
826-845.
,
Marshall, A. W. and Olkin, I.
1967.
A generalized bivariate exponential
distribution.
J. Appl. Prob. 4, 291-302.
Moran, P.
1967.
Testing for correlation between non-negative variables.
Biometrika 54, 385-394.
31
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1
Library, Code 55
Postgraduate School
Monterey, CA 93940
1
Naval
250
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Code 55Lw
Naval Postgraduate School
Monterey, CA 93940
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