LIDilHKT
RESEARCH REPORTS Dl
NAVAL POSTGRADUATE
MONTEREY, CALIFORNI/
NP.S55-fi?-mi
NAVAL POSTGRADUATE SCHOOL
Monterey, California
SIMPLE DEPENDENT PAIRS OF EXPONENTIAL AND
UNIFORM RANDOM VARIABLES
by
A. J. Lawrance
P. A. W. Lewis
March 1982
Approved for public release; distribution unlimited
Prepared for:
Naval Postgraduate School
Monterey, CA 93940
FEDDOCS
D 208.14/2:
NPS-55-82-011
DUDLEY KNOX LIBRARY
NAVAL POSTGRADUATE SCHOOL
MONTEREY, CA 93943-5101
mw NAVAL POSTGRADUATE SCHOOL
MONTEREY, CALIFORNIA
Rear Admiral J. J. Ekelund D. A. Schrady
Superintendent Acting Provost
This report was prepared by:
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered;
REPORT DOCUMENTATION PAGE
READ INSTRUCTIONS
BEFORE COMPLETING FORM
1. REPORT NUMBER
NPS55-82-011
2. GOVT ACCESSION NO
3. RECIPIENT'S CATALOG NUMBER
4. TITLE (and Subtitle)
SIMPLE DEPENDENT PAIRS OF EXPONENTIAL AND
UNIFORM RANDOM VARIABLES
5. TYPE OF REPORT 6 PERIOD COVERED
TECHNICAL
6. PERFORMINO ORG. REPORT NUMBER
7. AUTHORfs;
A. J. Lawrance
P. A. W. Lewis
8 CONTRACT OR GRANT NUMBERftj
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Naval Postgraduate School
Monterey, CA 93940
10. PROGRAM ELEMENT. PROJECT. TASK
AREA * WORK UNIT NUMBERS
11. CONTROLLING OFFICE NAME AND ADDRESS
Naval Postgraduate School
Monterey, CA 93940
12. REPORT DATB
March 1982
13. NUMBER OF PAGES
30
14. MONITORING AGENCY NAME ft ADDRESSf// different from Controlling Otllce)
IS. SECURITY CLASS, (of thla report)
Unclassified
15a. DECLASSIFI CATION/ DOWNGRADING
SCHEDULE
16. DISTRIBUTION ST ATEM EN T (of thla Report)
Approved for public release; distribution unlimited
17. DISTRIBUTION STATEMENT (of the abetract entered In Block 20, It different from Report)
18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse aide It neceaaary and Identity by block number)
bivariate exponential
bivariate uniform
random coefficient
linear function
a ntitheti c s
neqative dependency
correlation
Spearman
simulation
20. ABSTRACT (Continue on reveree aide If 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,
dependent pairs of exponential variables. By employing antithetic exponential
variables, the constructions are developed to encompass neaative dependency.
By employing neqative exponentiation, the constructions yield simple multi-
plicative-based models for dependent uniform nairs. The ranges of dependency
allowable in the models are assessed by correlation calculations, both of the
DD
FORM
1 JAN 73
1473 EDITION OF 1 NOV 65 IS OBSOLETE
S/N 0102- LF-014-6601
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAOE (Whan Data Kntered)
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Dmta Enltr.d)
BLOCK #20.
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 concen-
trations of values.
S/N 0102- LF- 014-6601
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE(TWien Dale Entered)
Simple Dependent Pairs of Exponential and Uniform
Random Variables
by
A. J. Lawrance and P. A. W. Lewis
University of Naval Postgraduate School
Birmingham 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. Because of their simplicity,
all 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: EP1 model.
3. Negative exponential transformation to uniformity: UP1 model.
4. Negative dependency using an antithetic approach: EP2, UP2 models
5. Two parameter versions of the models: EP3, UP3, EP4, UP4 models.
6. More general four parameter models: EP+, EP-, EP5, EP6 models.
7. Multivariate generalizations.
8. Line Discontinuities and Reliability Problems.
References
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). These models
indicate that a specific random-coefficient linear function of independent
exponential variables leads to further univariate and multivariate exponen-
tial 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. If these are required, as is sometimes the case
in 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. Gaver [1972] proposed
a 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 distribu-
tions; 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 flexi-
bility. 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 specific modelling or structural detail is
required, e.g. the linear regression of Downton's model, the present models
offer the following advantages:
(i) The constructions are sjery simple.
(ii) 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 ) in
detail. 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. Parallel
development is given for bivariate uniform pairs.
2. ONE PARAMETER POSITIVELY DEPENDENT EXPONENTIAL MODEL: EP1 MODEL
The random coefficient linear function referred to in Section 1 is
central to this work and is presented first. Let a pair of independent,
identically and exponentially distributed random variables be denoted by
(E-,, E ? ); for simplicity, unit means are assumed throughout. Let I be
a binary random variable, independent of (E-,, E 2 ), with distribution given
by P(I = 0) = 3 , P(I = 1) = 1 - 3 • Then the random variable X , given
by
X = 3E 1 + IE 2 , (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 (3 + (1-3) ^} = ^ (2.2)
The result can be extended by taking E 2 to be a random-coefficient linear
function of a further pair of exponential random variables independent of
E-, . 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 E-, and E ? in (2.1) to give the second
member of a pair, and hence produce
X 1 = 3E ] + E 2 ,
(EP1) (2.3)
X 2 = 3E 2 + E 1 .
These will be called the EP1(3) pair; the pair is clearly dependent because
of the common E-, , E 2 and I . There are clear generalizations obtained by
using different 3's and I's for X-j and X 2 which will be considered later.
The dependence of (X-,, X«) can be assessed by their (product moment)
correlation coefficient as
Corr(X r X 2 ) = E{(3E ] + IE 2 )(3E 2 + IE ] )} - E(3E ] + IE 2 )E(3E 2 + IE ] )
= 33(1-3) , (2.4)
which is sketched in Figure 1. The correlation thus has the non-negative
range (0, 0.75) with maximum value 0.75 at 3 = 0.5 and symmetry about
3 = 0.5. 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 X-, and X 2 can be obtained by
considering the probability element for X-, and X 2 in terms of 3E-, + IE 2
and 3En + IE, . On letting I take its values 1 and with probabilities
1 - 3 and 3 this becomes
(1-3)P{(3E 1 + E 2 ) €(x r x 1 + dx-,), (3E 2 + E^ 6(x 2 , x £ + dx 2 )}
+ 3PI3E-, €(x 15 x-j + dx 1 >P{3E 2 €(x 2> x 2 + dx 2 )} .
(2.5)
The first line of (2.5) can be expressed as a probability element statement
in terms of E, and E 2 separately, and using the Jacobian, (1-3 2 ), of
the transformation from E-. and E 2 leads to the joint pdf of
X-, and X 2 as
f X 1 ,X 2 (x 1 ,x 2 ) = (1-B)l(x 1 > 3x 2 , x 2 > 3x-,)f (1 _ 3 2 )E (x^x-,)
x f 0-fi z )Z 9 { *r**Z W -* l) + 3f 3E 1 (x l )f 3E ? (x 2 ) •
where 1( ) is an indicator function taking the value 1 if x ] > 3x 2 and
x ? > 3x, and otherwise, and f denotes the density of the exponential
variable in its suffix. Simplifying (2.6) then gives the desired result
(2.6)
f x x (x r x 2 ) = l(3x 2 < x ] < 3" 1 x 2 )(l+3)" 1 exp{-(x-,+x 2 )/(H3)}
-1
+ 3 exp{-(x-!+x 2 )/3} ,
± 3 1 1 ; x-j ,x 2 >
(2.7)
The joint density thus has a wedge shaped sector of relatively high density
( 3x~ < x-, < 3 x 2 ) superimposed over an independent bivariate exponential
density. This function is isometrically plotted in Figure 2 for 3 = 0.5.
For low values of 3 the raised section spreads out over the quadrant and
X 2 AXIS x 10
X-, AXIS x 10
-1
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.
leads to independence at 3=0; for high values of 3 the raised section
diminishes towards nothing and again gives independence at 3=1.
This joint density function could be used to obtain a maximum likelihood
estimate of 3 from a bivariate sample; note that 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 - t 2 X 2 )}
= E{exp[-(3t ] + It 2 )E 1 - (3t 2 + It 1 )E 2 ]}
= 3(1 + 3t 1 )" 1 (l + 3t 2 ) _1
+ (1-0)0 + Bt 1 + t 2 ) _1 (l + t 1 + 3t 2 ) _1 . (2.8)
Differentiating with respect to t 2 at t« = and inverting with respect to
t-j allows the regression of X 2 on X, to be obtained as
E(X 2 |X ] = x) = 3x + 3 + 1 - (1 + x/3)exp{-(3 _1 - 1 )x} . (2.9)
This conditional expectation will be fairly linear except for small x , and
the asymptotic conditional variance will be linear for large x . 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, E(X 1 |X 2 = x) = E(X 2 |X ] = x) .
An aspect of this reversibility is that P{X 2 > X ] } = 0.5 which can be shown
directly from the definition (2.3).
3. THE NEGATIVE EXPONENTIAL TRANSFORMATION TO UNIFORMITY: UP! MODEL
It is well known that for an exponential random variable X of unit
-X -X
mean, both e ' and 1-e ' are uniform random variables, distributed over
(0,1). These results are of double interest here. Firstly, application of
the transform to both components of a dependent exponential pair will lead
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 2 )(Y 1 - Y 3 ) > 0} ,
where (X.,Y.: i = 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. Thus, we have the useful idea that
product-moment correlation in a bivariate distribution after its transforma-
tion to uniform marginals represents a population analogue of Spearman's
correlation for the untransformed bivariate distribution. As such, it is a
suitable measure of monotonic dependency for a bivariate distribution. In
particular, the Spearman correlation is a suitable measure for dependency in
bivariate non-Gaussian distributions because of their intrinsic non-linearity,
and the consequent limitations of the product moment correlation.
General Results for the Transformati
on
In the case of an exponential pair (X-, ,X 2 ) of unit means, the preferred
transformation to a uniform pair is given by
Y ] = 1 - expC-X^, Y 2 = 1 - exp(-X 2 ) . (3.1)
The correlation and regression for this pair are easily obtained from the
joint moment generating function, $„ «, (t, ,t 2 ) = E{exp(-t,X-, - t^X^)}
of (X-j.Xp) • First for the correlation we have
E{Y 1 Y 2 ) = E{(l-exp(-X 1 )(l-exp(-X 2 )} = fy ^ (1,1) , (3.2)
and thus the correlation result
Corr(Y r Y 2 ) = 12<J> X x (1,1) - 3 . (3.3)
For the regression of Y, on Y 2 , first write it in terms of X-j and X 2 as
E ( Y-, t Y 2 = y) = E{l-exp(-X 1 )|X 2 = - log(l-y)} . (3.4)
Now setting x = - log(l-y) and denoting the conditional p.d.f. of X, | X 2
by f„ iw (x-,|x) , the regression is
oo
E[Y 1 |Y 2 = l-exp(-x)] = / [l-exp(- Xl )]f x ^ (x ] |x)dx 1 . (3.5)
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_ = 1 - exp(-x)]dx = t" 1 - <J) Y Y (l,t-l) . (3.6)
o ' L X T X 2
10
Thus, the regression of the uniforms follows from inverting the joint moment
generating ^ x (l,t-l) of the exponential pair with respect to t as a
function of x , and then replacing x by - log(l-y) .
The UP1 Model
The negative exponential transformation to uniformity is now exemplified
using the EP1 (3) model. To obtain a pair of independent uniform random
variables from a pair of independent exponentials it is simplest here to use
the monotonic decreasing version of (3.1), that is
U
1 = exp(-E.j) , U 2 = exp(-E 2 ) . (3.7)
It follows then that the dependent uniform pair derived from the EP1 ( 3)
exponential model via (3.1) is given by
(UP1) Y 1 = 1-uf U l z , Y 2 = 1-UJ b| , (3.8)
where I retains its definition from Section 2. The random power multipli-
cative aspect of the model is evident. 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 ) = 33(1-6) 8+73+32 . (3.9)
1 l 0+3) 2 (2+3) 2
This function of 3 is illustrated in Figure 1; its maximum value is 0.72
for 3 s 0.32. The joint density of Y-. and Yo may be obtained by considering
11
P(U^ u\ < y r uf u{ < y 2 )
■ 3P(U^ < y^PCU^ < y 2 ) + (l-e)P(uf U 2 ly r U-, U 2 <y 2 ) .
(3.10)
Density for the last probability is restricted to the leaf shaped region
(yi < y-i < y5. <y 2 < 1)» and the final result takes the form
1' 2
f Y 1 ,Y (y i y 2 ) = UyJ 76 !^ iy^)(i + e)" 1 (y 1 y 2 )' 3/(1+3) + b" 1 ^) "^ 70
(3.11)
(0 < y r y 2 < 1; < 3 < 1) .
The density is isometrically plotted in Figure 3 for 3 = 0.32, the value
giving maximum product moment correlation.
1.00-1
Y 1 AXIS x 10
r4.00
Y 2 AXIS x 10
-1
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 2 |Y 1 = y)
= 1 - (l+3) _1 (l-y) (1 " 3)/3 + (2+3) _1 (l-y) (2 " 3)/3 - (2+3) _1 (l-y) 3 .
The regression has initial value 3/(1-3) which is always less than 0.5 and
curves upwards towards 1.0 in a near monotonic concave way. For 3=0 and
3=1 it has the value 1/2 which is correct since Yo and Y-. are then
independent and E( Yp) = 1/2. Note that the UP1 pair is, like the EP1 pair,
reversible, in particular E(Y 2 |Y 1 = y) = E(Y ( |Y 2 = y) . Also
P{U, > U ? } = 0.5; this can be seen directly from the definition (3.8).
13
4. NEGATIVE DEPENDENCY USING AN ANTITHETIC APPROACH: EP2, UP2 MODELS
The antithetic transformation of any random variable X is found by
first expressing it as a function (the inverse probability integral) of a
uniform (0,1) variable U , and then by taking the same function of 1-U. For
any nonnegative random variable X , the antithetically transformed variable
a(X) has the same marginal distribution as X and is maximally negatively
correlated with 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 case,
a(X) = - log(l-exp(-X)) (4.1)
and the maximum negative product moment correlation is - 0.6449, as is well
known after Moran [1967]. As a pair of exponential random variables, X and
a(X) lie on the curve
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 E-, and E~ of X,, are replaced by
their antithetic versions a(E-,), a(E 2 ), and I in X, and X~ is replaced by
I 1 and Ip respectively, these variables having the same marginal distribution
as I . Thus the suggested negatively dependent model has the form
14
(EP2)
X 1 = 3E ] +1^2,
X 2 = 3a(E 2 ) + I 2 a^) ,
(4.2)
and the expression for its correlation is given by
Corr(X r X 2 ) = 23(1-3) (-0.6449) + CovO^I^
(4.3)
This suggests that maximum negative correlation of X-, and X 2 requires
maximum negative covariance of I-, and I« . For fixed 3 , this will be
achieved for minimum E(L,I 2 ), or equivalently minimum p = P(I, = 1, I« = 1)
Consideration of the general joint distribution of I, , I 9 , that is
h =
1
I 2 -l
p
1-3-P
1-3-P
23-1+p
1-3
3
1-3
3
1
(4.4)
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 H. and if 3 :• % . The
implied joint distribution is, not surprisingly, the antithetic one, and it
has covariance - 3 2 for < 3 < h and -(1-3) 2 for ^ £ 3 £ 1 . Thus, going
back to (4.2), the model with I, = I, I„ = a(I) is of most interest in
connection with negative dependence and will be called the EP2(3) model; its
correlation is given by
f -23(l-3)(0.6449) - 3 2
Corr(X 1} X 2 ) = -
-23(l-3)(0.6449) - (1-3)
13 1%,
h < 3 < 1 .
(4.5)
15
This function is sketched in Figure 4. The minimum value is -0.5724 at
3 = 0.5 which, considering the model is never degenerate, is fairly satis-
factory. (The Gaver model [Gaver, 1972] is also not degenerate and has a
RHO .00
Figure 4. Correlation functions for the EP2(3) model
and the UP2(3) model as functions of 3; both models
exhibit negative dependence.
minimum correlation of -0.5). A simulation of EP2(3) pairs at the 3 = 0.5
value is given in Figure 5; the scatter is best described as a non-degenerate
—x - v
emulation of the antithetic concentration along the curve e ' + e J = 1 .
Weakening of the dependency as the parameter 3 approaches or 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 X, and X~ requires the joint
moment generating function of E. and a(E.) which is straightforwardly
evaluated as
16
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 t, t ?
<Kt r t 2 ) = / x (T-x) ^dx ,
o
(4.6)
which is the Beta function B(l + t-, , 1 + t 2 ) . Also required is the joint
distribution p.. = P{I = i, a(I) = j}, i, j = 1,0, available as remarked
previously following (4.4); hence the result, via (4.2),
*X 1 ,X 2 (t l» t 2 ) =
= p 11 cf>(et 1 ,t 2 )4>(t 1 ,et 2 ) + p 10 <j)(3t 1 ,t 2 )(i+3t 2 )
-i
p Q1 (})(t 1 ,3t 2 )(l+3t 1 )" 1 + p 00 (l+3t 1 )" 1 (l+3t 2 )" 1
(4.7)
17
Actually the model (4.2) works for any independent bivariate exponential
pairs (E-|,E-|) and (E 2 ,E 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 a broad range of negative depen-
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|, Y 2 « 1 - Cl-0 1 ) aCl) O-U 2 ) B • (4.8)
The correlation of Y ] and Y 2 may be evaluated directly from (4.8) or by use
of (4.7) and (3.3) and gives the result
Corr(Y r Y 2 ) =
-36(4+56+6B 2 +3 3 )/[(l+3) 2 (2+3) 2 ] < 3 < h ,
(4.9)
[ -3(l-3)(2+33-S 2 )/[(l+e) 2 (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
ultimate degeneracy. Bivariate scatters from the UP2 model show a strong
degree of linearity compared to the near antithetically curved scatter of
the corresponding exponential EP(2) pair in Figure 5. Again, the asymmetry of
the dependency about 3 = 0.5 in the EP2 model is more evident from the
Spearman correlation of the UP2 model. Negative dependency is considerably
stronger for 3 = 0.5 - 6 than the corresponding 3 = 0.5 + 6 . This parallels
the EP1 model in which positive dependency is stronger for 3 = 0.5 - 6 than
18
for 3 = 0.5 + 6 ; these remarks are borne out in scatters for various values
of 3 • 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. For example, combining (2.3) and (4.2) let
X 1 = BE ] + 1^2
f3E 2 + I, E]
h -\
w.p. p
(4.10)
3a(E 2 ) + I 2 a(E ] ) w.p. 1-p
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 (p = 1) and the EP2
model (p = 0) and has product moment correlation ranging between -0.5724
and 0.75.
19
5. TWO PARAMETER VERSIONS OF THE MODELS: 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 P(X-, > X 2 ), which in the EP1 model was equal to
0.5 for all 3 . Two approaches will be discussed rather briefly. One
allows 3 in the randomly linear combination of E-, and E 2 to be replaced
by 3-i for X-, and 3 2 for Xp , while in Section 6 a two parameter randomly-
linear operation is used: 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 1 = 3-| E ] + I-, E 2 , < B 1 < 1 ,
(EP3) (5.1)
X 2 = 3 2 E 2 + I 2 E-j » < 3 2 < 1 .
The marginal distributions of I-, and I 2 must correspond to the different
3 parameters so that P{I, = 0} = 1 - P{I, = 1} = $. and
P{I 2 = 0} = 1 - P{I 2 = 1} = 3 2 . The product moment correlation of the
model is given by
Corr(X r X 2 ) = 3-,(l-3 2 ) + 3 2 (l-6-, ) + Cov^,^) . (5.2)
The general joint distribution for (I-,,I 2 ) is the obvious generalization of
(4.4); the value of p-^ for maximum positive dependence is now min(l-3-, ,l-3 2 )
Hence the result
20
Corr(X r X 2 ) = «
'2B-, (l-3 2 ) + 3 2 (l-3-,)
3-|(l-3 2 ) + 23 2 (l-3-,)
< 3 1 1 3 2 1 1 »
< 3 2 1 3-j 1 "I
Contours of this function (Figure 6) show a central region with a local
maximum of 0.75; the function decreasing from each side of the centre along
the line 3, = 3 2 t0 zer0 values at the corners where 3, = 3 2 = and
3-, = 3 ? = 1 , and the function increasing from each side of the centre along
the line 3-, + 3 2 = 1 to unit values at the other two corners. The joint
density of X-, ,X ? is simply obtainable but is now a mixture of three,
instead of two, components. Thus maximum likelihood estimates of 8-j and 3 2
can be obtained.
BETA2 1.0
Figure 6. Correlation function of the two-parameter positively
dependent EP3(3, ,$«) model: unlabel led contours may be identified
from their joins with the 3-| and 3 2 axes.
21
The edges ($, = 0) or (3 ? = 0) give a one-parameter bivariate exponen-
tial pair which corresponds to adjacent (exponential) variables in an
EAR(l) process [Gaver and Lewis, 1980]. Note that using the basic relation-
ship (2.1) outside of the serial constraint posed by a time series gives a
broader bivariate exponential pair.
The UP3 Model
The corresponding uniform model, UP3, is given by
6 1 J l l ? h
(up3) y 1 = i - uyiy, y 2 = i - iyu 2 ^ (5.4)
and has correlation (the Spearman correlation of the EP3 model), given by
Corr(Y 1 ,Y 2 ) = 12
l-3 2 3 2 -3 1
(2+3j(l+3o) + (2+3o)d+3 1 )
1
(5.5)
+ Cl+3-,)(l+3 2 )
-3 (0 < 3-, £ 3 2 < 1)
For 3, >_ 3o there is a symmetrical result. The contours of this function
represent a deformation of those just described for the EP3 model, symmetry
about 3i = 3o is preserved, while the area of local maximum is moved
towards the origin. Thus, asymmetry in the Spearman dependence is indicated,
The EP4 Model
The negatively dependent version of the EP3 model is the two parameter
version of (4.2), and will be called the EP4 model; this has correlation
function
22
Corr(X r X 2 ) = - 0.6449L3-, (l-3 2 ) + ^(l-^)]
- i
3^2
for 3 1 + B 2 < 1 ,
(1-3-, )(l-3 2 ) for 3 1 + 3 2 > 1 .
(5.6)
Contours of this function indicate a valley of lowest negative values along
the direction 3-, + 3 2 = 1 > with a central value of -0.5720 and corner values
of maximum negativity, -0.6449. For specified negative correlation down to
about -0.5 there is a wide choice of ($, ,3 2 ) pairs, in the two regions either
side of the valley along 3-. + 3 2 = 1 .
The UP4 Model
The final development of the two parameter models is to the negatively
dependent two parameter uniform model, the UP4 model. This corresponds to
(4.8) with 3 and I in Y, replaced by 3i and I, and 3 and I in
Y ? replaced by 3 2 and a(I 2 ) . Maximum negativity of dependence follows
from the 'antithetic' joint distribution for I, and I 2 which has
? u = 1 - (3-,+3 2 ) when 3-, + 3 2 < 1 and p Qo = 3-, + 3 2 - 1 when
3 ■, + 3 2 >_ 1 . We then have, for the case 3-. + 3 2 £ 1 ,
Corr(U r U 2 ) = 12
1 - (3-,+3 2 ) 3 1
(l+3-|)(2+3-|)(l+3 2 )(2+3 2 ) + d+3 1 )(l+3 2 )(2+3 1 )
+ (H3 1 )d+3 2 )(2+3 2 )
- 3 .
(5.7)
The corresponding result for 3 ] + 3 2 > 1 is omitted. The contours of this
function are quite similar to those of the EP4 model, but with a central
23
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. It is also
possible to derive measures such as P{X-, > X^} , which is not necessarily
24
6. MORE GENERAL FOUR PARAMETER MODELS: 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 a general
setting and suggest a further class of two-parameter models. They also give
a 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
(EP+)
where, for i = 1,2
X, - 8 V^ ♦ I,E 2 ,
X 2 = 6 2 V 2 E 2 + I 2 E-, ,
(6.1
V
1 w.p. a.
w.p. 1-a-
1 w.p. (l^/D-O-ou)^.]
l i'\
(l-a.)3 i w.p. a.3 i /[l-(l-a i )3 i ]
and the random variables (V,,V 2 ), (I-i.Io) are independent between pairs but
usually dependent within pairs. This model has correlation structure of
the form
Corr(X r X 2 ) = 3-,3 2 Cov(V-, ,V 2 ) + Cov(I ls I 2 )
+ a-|3-|(l-a 2 3 2 ) + a 2 3 2 (l-a-|3-| )
(6.2;
For a 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 a-j = a 2
3, = 3 2 = 3 and V ] = V 2> I-j = I 2 ; then (6.2) reduces to
= a
25
Corr(X ls X 2 ) = 3ag(l-a3) . (6.3)
This has maximum value 0.75 at a3 = 0.5 , 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
EP+ model; the EP1 model is given by taking ou = a« = 1 and 3, = Bo = 3
with I, - I« , while the EP3 model has ou s ou = 1 > (3-i »3 2 ) unchanged, and
with (I, ,I«) of maximum possible dependency.
Corresponding to (6.1) there is a more general negatively dependent
four-parameter model, EP-, of the form
x i ■ Wi + ¥2 •
X 2 = 3 1 V ] a(E 2 ) + I 2 a(E 1 ) ,
which has, by analogy, the correlation structure
Corr(X r X 2 ) = 3-,3 2 Cov(V 1 ,V 2 ) + CovO^I^
- 0.6449[a 1 3 1 (l-a 2 3 2 ) + 06 2 3 2 ( 1 -cc-, 3-, )D
(6.4)
(6.5)
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
7. MULTIVARIATE GENERALIZATIONS
The possibility of multivariate generalizations is apparent by repeated
use of the random-coefficient linear functions of (2.1). In fact this is
the way the EAR(l) process is constructed with a serial chaining of i.i.d.
exponentials E-, , E 2 , Any k variables in an EAR(l) process (or an
NEAR(l) process) are a k-variate exponential random variable. Outside of
this serial context many possibilities suggest themselves. Thus changing
3 to $-, , I to I-. and replacing E 2 by a similar random-coefficient
linear function of independent E 2 and E 3 using 3o and I 2 , gives
X = e 1 E ] + 3 2 I-,E 2 + I^^ . (7.1)
Here I-, and I ? must be independent. A triple of dependent exponential
variables could be constructed using 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 2 ) , (E 2 »E 3 ) and (£3^) . Such developments
are not considered here. The possibilities are legion but no 'natural'
simple method suggests itself above any other.
27
8. LINE DISCONTINUITIES AND RELIABILITY PROBLEMS
The problem of discriminating amongst the many bivariate exponential
models is not simple, although there may be modelling or structural details
which recommend certain models in certain contexts. Most of these details
are, however, tenuous and difficult to verify from data. An important
mathematical scheme is given in Griffiths [1969], who found a canonical
expansion for bivariate exponential random variables. Again the models put
forward here are simple to generate on a computer and are analytically
tractable.
Another property is that the bivariate distributions do not have line
discontinuities. However, as an illustration of modelling considerations,
we note that in a reliability context thus is not necessarily a virtue. It
is known that components in a system can fail from a common cause, which is
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 2 ) denote any unit mean, bivariate exponential pair,
let ( I-. , I 2 ) denote an indicator pair, possibly completely or partially
dependent, with marginal distributions P{I, = 1} = 1 - P{I-| = 0} = P^ and
P{Io = 1} = 1 - P{Io = 0} = P 2 , and let E be an independent, unit mean
exponential random variable. Three new bivariate exponential pairs are given
(i) Z 1 = I 1 X 1 + (1-I 1 )E , Z 2 = I 2 X 2 + (1-I 2 )E ; (8.1)
(ii) Z 1 = 3E + I 1 X ] , Z 2 = 3E + I ? X 2 ; (8.2)
and
28
(iii) Z 1 = min(X r E) , Z 2 = min(X 2 ,E)
In all three cases there is a non-zero probability that Z-, and 1 ? are
proportional to E .
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. In fact,
if X-, and X~ are independent, then (8.3) is the Marshall-01 kin model.
ACKNOWLEDGMENTS
The work of Professor P. A. W. Lewis was supported by the Office of
Naval Research under Grant NR-42-469.
29
TABLE I
Summary of Models Considered (See text for relevant details)
EP1
X 1 = 0E 1 + IE 2
x 2 = eE 2 + ie 1
UP1
Y l = ^ U l
Y 2 - l-UJl|
EP2
X l ■ * E 1 + T 1 E 2
X 2 = 3a(E 2 ) + I 2 a{E^)
UP2
Y l " l- U f U 2
y 2 = i-(i-u 1 ) a CD cl _u 2) e
EP3
X l = ¥l + T 1 E 2
X 2 = 6 2 E 2 + I 2 E 1
UP3
B l I
Y 1 =1 - U 1 1U 2
Y 2 - 1-uJ^
EP4
X 1 = 3 1 E 1 + I^g
X 2 = 3 2 a(E 2 ) + l 2 a{E^)
UP4
3 1 I
Y = 1-U I)
Y l ' U l U 2
I 2 B 2
Y 2 = 1-O-u^) ^(1-U 2 ) L
EP+
h = Wl + ¥2
X 2 = B 1 V 2 E 2 + T 2 E 1
UP+
e i v i h
Y = 1 - 1 J 1 J
Y l ' U l U 2
Io B 9 V 9
Y 2 = 1-U-, U 2 C
EP-
X l " Wl + ¥2
X 2 = B-,V 2 a(E 2 ) + I 2 a(E.,)
UP-
3 1 V 1 I,
Y 1 = 1-U/ 1 (1-U 2 ) ]
i 2 e,v 2
Y 2 = l-Cl-^J 2 (1-U 2 ) ] 2
30
REFERENCES
Barnett, V. D. 1980. Some bivariate uniform distributions. Commun. Statist.
Theor. Meth. A 1(4) , 453-461.
Downton, F. 1970. Bivariate exponential distributions in reliability
theory. J. R. 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. 1958. Ordinal measures of association. J. Amer. Stats.
Assoc , 53, 814-859.
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
DISTRIBUTION LIST
NO. OF COPIES
Library, Code 0142 4
Naval Postgraduate School
Monterey, CA 93940
Dean of Research 1
Code 01 2A
Naval Postgraduate School
Monterey, CA 93940
Library, Code 55 1
Naval Postgraduate School
Monterey, CA 93940
Professor P. A. W. Lewis 250
Code 55Lw
Naval Postgraduate School
Monterey, CA 93940
32
DUDLEY KNOX LIBRARY - RESEARCH REPORTS
5 6853 01068021 8