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Full text of "Simple dependent pairs of exponential and uniform random variables"

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 



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SIMPLE DEPENDENT PAIRS OF EXPONENTIAL AND 
UNIFORM RANDOM VARIABLES 



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7. AUTHORfs; 

A. J. Lawrance 
P. A. W. Lewis 



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Naval Postgraduate School 
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March 1982 



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Approved for public release; distribution unlimited 



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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 



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FORM 
1 JAN 73 



1473 EDITION OF 1 NOV 65 IS OBSOLETE 

S/N 0102- LF-014-6601 



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UNCLASSIFIED 



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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 



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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. 



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