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