An Introduction to Copulas (Springer Series in Statistics)

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We now examine the extension of these families to find all copulas with cubic sections in both u and v. That is, we seek copulas C u,v which can be written both as 3. As a consequence of Theorem 3. Suppose that C has cubic sections in both u and v, i. Note that when C has cubic sections in both u and v, the functions a, b, g and c in 3. The two families of iterated FGM copulas in Example 3. For the Johnson and Kotz family of copulas given by 3. C is symmetric, i.

The next two examples show the ease with which the procedures outlined in the above theorems can be use to construct families of copulas. Several subfamilies are of interest. Applications of this family are discussed in Lee Additional subfamilies of 3. As a consequence of Theorem 2. Now suppose that d is any diagonal. Is there a copula C whose diagonal section is d? The answer is provided in the following theorem. The proof of the above theorem, which is a somewhat technical but straightforward verification of the fact that C is 2-increasing, can be found in Fredricks and Nelsen a.

Copulas of the form given by C u,v in 3. The diagonal copula constructed from this diagonal is M, as it must be as the only copula whose diagonal section is the identity is M [see Exercise 2. The diagonal copula constructed from this diagonal is not W but rather the shuffle of M given by M 2, I2 , 2,1 ,1 , i. The diagonal copula constructed from this diagonal is the singular copula from Example 3. Such is indeed the case. Suppose X and Y are continuous random variables with copula C and a common marginal distribution function. Suppose U and V are random variables whose joint distribution function is the copula C.

Assume C is a diagonal copula, i. Assuming that C is symmetric, we will show that C must be a diagonal copula [for the proof in the general case, see Fredricks and Nelsen b ]. For a thorough treatment of the properties of Bertino copulas, including a characterization similar to Theorem 3. Also show that the FGM family is positively ordered. Generate two independent uniform 0,1 variates u, t; 2.

Are there other such families? Regions for a and b in Exercise 3. Let Cq be the diagonal copula constructed from d q. Let X and Y be continuous random variables with a joint distribution function H, and margins F and G, respectively. Margins of the two roots in 3. Now let Cq denote the root in 3. This is tedious but elementary—but also shows that each of these copulas is absolutely continuous. Thus we have the Plackett family of copulas: So it is not surprising that Plackett family 92 3 Methods of Constructing Copulas copulas have been widely used both in modeling and as alternatives to the bivariate normal for studies of power and robustness of various statistical tests Conway ; Hutchinson and Lai One estimator is the maximum likelihood estimator, which must be found numerically.

See Mardia for details, and for an efficient estimator that is asymptotically equivalent to the maximum likelihood estimator. It is easy to check that the boundary conditions 2. As noted in Exercise 2. These copulas are similar to the FGM copulas, in that they can only model weak dependence. It is motivated by the following problem.

We answer this question by finding the distribution function H n and the copula C n of X n and Y n. If C is a copula and n a positive integer, then the function C n given by 3. Let C be a member of the Marshall-Olkin family 3. The above examples motivate the following definition: Let Cq be a member of the Gumbel-Hougaard family 2. If two or three of the numbers a, b, c, d coincide, the proof is similar. But continuous mid-convex functions must be convex Roberts and Varberg , which completes the proof.

See Durante and Sempi for details. The limit of the sequence C n leads to the notion of an extreme value copula. Note that if the pointwise limit of a sequence of copulas exists at each point in I2 , then the limit must be a copula as for each rectangle in I2 , the sequence of C-volumes will have a nonnegative limit.

A copula is max-stable if and only if it is an extreme value copula. Clearly every max-stable copula is an extreme value copula. Let C be a max-stable copula, and let X and Y be standard exponential random variables whose 98 3 Methods of Constructing Copulas survival copula is C. Define a function A: For the right side of 3. Thus the graph of A must lie in the shaded region of Fig. See Joe for details. Regions containing a the graph of A in 3. So when the point a,b lies in the shaded region in Fig. It is the only harmonic copula, because for any other harmonic copula C, C - P would also be harmonic and equal to 0 on the boundary of I2 and hence equal to 0 on all of I2.

Closely related notions are subharmonic and superharmonic copulas. For example, it is an elementary calculus exercise to show that if Cq is a FGM copula given by 3. Thus Cq is homogeneous of degree 2 - q. Suppose C is homogeneous of degree k. Hence we consider weaker versions of these properties. Suppose that only the vertical or the horizontal sections of a copula C are concave. As we shall see in Sect. We now weaken the notions in Definition 3. In the other direction, assume La is not convex for some a in [0,1 , so that the set L a is not convex. Suppose C is quasi-convex.

Thus the Schurconcavity of a copula can be interpreted geometrically as follows: It is easy to show that M, W, and P are Schur-concave and that any convex linear combination of Schur-concave copulas is a Schurconcave copula. The next example shows that Schur-concavity neither implies nor is implied by quasi-concavity: Some of the contours of C are illustrated in Fig.

These contours are the graphs of the functions Lt in Theorem 3. The contours of Ca ,b can be readily seen see Figs. But Ca ,b is not symmetric, hence it is not Schur-concave. For further properties of Schur-concave copulas and additional examples, see Durante and Sempi Some contours of the copulas in Example 3. Constructing n-copulas is difficult. Few of the procedures discussed earlier in this chapter have n-dimensional analogs. In this section, we will outline some of the problems associated with the construction of n-copulas and provide references to some techniques.

Most of our illustrations will be for 3-copulas; however, when appropriate, we will provide the n-dimensional version of the relevant theorems. Then, as shown in Example 2. Note that each of the 2-margins of W 3 is W, and it is impossible in set of three random variables X, Y, and Z for each random variable to be almost surely a decreasing function of each of the remaining two. Every 2-copula C is directly compatible with P; 2. The only 2-copula directly compatible with M is M; 3.

The only 2-copula directly compatible with W is M; 4. M is directly compatible with every 2-copula C; 5. W is directly compatible only with P; and 6. An important class of copulas for which this procedure—endowing a 2-copula with a multivariate margin—often succeeds is the class of Archimedean copulas. Archimedean n-copulas are discussed in Sect. Can we extend this procedure to higher dimensions by replacing F and G by multivariate distributions functions?

The following theorem Schweizer and Sklar presents related results for the cases when the 2-copula C in the preceding theorem is P or M, and the multidimensional distribution functions F and G are copulas or, if the dimension is 1, the identity function: Let C1 be an m-copula, and C2 an n-copula. The results in the preceding theorems illustrate some aspects of what has become known as the compatibility problem.

If they are, then these k-copulas ments of C equal to 1, then the result is one of the nk k-margins of C. The compatibility problem has a long history. To facilitate our discussion, let C 3 C12 denote the class of 3-copulas of continuous random variables X, Y, and Z such that the 2-copula of X and Y is C12 i. Note that parts 4, 5, and 6 of Theorem 3. For the more general problem of constructing a trivariate joint distribution function given the three univariate margins and one bivariate margin, see Joe Necessary and sufficient conditions for a 3-copula C to have specified 2-margins C12 and C13 i.

We conclude this section with an n-dimensional extension of one of the families discussed earlier in this chapter. The FGM family 3. Hence the density will be nonnegative on In if and only if it is nonnegative at each of the 2 n vertices of In , which leads to the following 2 n constraints for the parameters Cambanis These copulas find a wide range of applications for a number of reasons: As mentioned in the Introduction, Archimedean copulas originally appeared not in statistics, but rather in the study of probabilistic metric spaces, where they were studied as part of the development of a probabilistic version of the triangle inequality.

For an account of this history, see Schweizer and the references cited therein. But in the past chapter we saw cases in which a function of H does indeed factor into a product of a function of F and a function of G. The copula Cq from 2.

This can be done as follows: Let j, j [ -1] and C satisfy the hypotheses of Lemma 4. Hence assume that C satisfies 4. Then the function C from I2 to I given by 4. Proof Alsina et al. We have already shown that C satisfies the boundary conditions for a copula, and as a consequence of the preceding lemma, we need only prove that 4. In the other direction, assume j [ -1] is convex. Adding these inequalities yields 4.

The function j is called a generator of the copula. To be precise, the function j is an additive generator of C. In the sequel, we will deal primarily with additive generators. Thus P is a strict Archimedean copula. Hence W is also Archimedean. Strict a and non-strict b generators and inverses Example 4. We conclude this section with two theorems concerning some algebraic properties of Archimedean copulas. Let C be an Archimedean copula with generator j.

C is symmetric; i. C is associative, i. The proof of Theorem 4. Furthermore, it is also easy to show see Exercise 4. As noted earlier, one reason for the usefulness of Archimedean copulas in statistical modeling is the variety of dependence structures present in the various families. In each case, we used sample points and the algorithm in Exercise 4. Additional scatterplots for members of the families in Table 4. In the following example, we show how one family of Archimedean copulas arises in a statistical setting.

We now find the copula C1,n of X 1 and X n. Invoking part 2 of Theorem 2. Recall the Archimedean axiom for the positive real numbers: An Archimedean copula behaves like a binary operation on the interval I, in that the copula C assigns to each pair u,v in I a number C u,v in I.

It is one of only two families the other is 4.

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Also see Example 4. We encountered this family in Exercise 2. Also see Genest and Rivest Some of the statistical properties of this family were discussed in Nelsen ; Genest As noted above, this is one of two comprehensive families in the table. Although many authors refer to these copulas as another Gumbel family, Hutchinson and Lai call it the Gumbel-Barnett family, as Barnett first discussed it as a family of copulas, i. Scatterplots for copulas 4. The version of the Archimedean axiom for I,C is, For any two numbers u, v in 0,1 , there exists a positive integer n such that uCn 4.

Let u, v be any elements of 0,1. The nth C-power uCn of u is readily seen to be j [ -1] n j u. In the next theorem, we set the groundwork for determining which Archimedean copulas are absolutely continuous, and which ones have singular components. Indeed, the graph of this copula is one-quarter of a circular cone whose vertex is one unit above 1,1. This must be the case for all Archimedean copulas but not all copulas—see Example 3. Graphs of some level curves, the zero set, and the zero curve of the Archimedean copula in 4.

The level curves of an Archimedean copula are convex. For t in [0,1 , the level curves of C are given by 4. Let C be an Archimedean copula generated by j in W. Let q be in 0,1], and let jq be the piecewise linear function in W whose graph connects 0, 2 - q to q 2,1 - q 2 to 1,0 , as illustrated in part a of Fig. The slopes of the two line segments in the graph are - 2 - q q and —1. If Cq is the Archimedean copula generated by jq , then it follows from 4. The generator and support of the copula in Example 4. Let n be a fixed positive integer, and consider the same partitions of [t,1] and [0,w] as appear in the proof of Theorem 4.

Proceeding as in Theorems 4. For such copulas, Genest and MacKay a,b proved part 2 of Theorem 4. Thus the members of this family aside from W have both a singular and an absolutely continuous component. The following corollary presents a probabilistic interpretation of Theorem 4. Let U and V be uniform 0,1 random variables whose joint distribution function is the Archimedean copula C generated by j in W. Then the function KC given by 4. The next theorem Genest and Rivest is an extension of Corollary 4. An application of this theorem is the algorithm for generating random variates from distributions with Archimedean copulas given in Exercise 4.

Under the hypotheses of Corollary 4. Hence S and T are independent, and S is uniformly distributed on 0,1. We present a proof for the case when C is absolutely continuous. For a proof in the general case, see Genest and Rivest The Farlie-Gumbel-Morgenstern family of copulas was introduced in Example 3. Are any members of this family Archimedean?

If so, they must be associative. But it is easy to show that if Cq is given by 3. The Ali-Mikhail-Haq family of copulas was derived by algebraic methods in Sect. It is easy but tedious to show that when Cq is given by 3. As a consequence of Example 3. For further examples, see Joe Prove that if C is Archimedean, then for u in 0,1 , d C u 4. Using the copulas in Example 3. Generate two independent uniform 0,1 variates s and t; 2. Let C1 and C2 be the members of the Gumbel-Barnett family 4.

Hence the Gumbel-Barnett family of copulas is negatively ordered. Let C1 and C2 be the members of family 4. For Archimedean copulas, the situation is often simpler in that the concordance order is determined by properties of the generators. For the first of these results we need the notion of a subadditive function: Let C1 and C2 be Archimedean copulas generated, respectively, by j 1 and j 2 in W.

Then C1 p C2 if and only if j 1 o j [] is subadditive. Now suppose that C1 p C2. Applying j 1 to both sides of 4.

Conversely, if f satisfies 4. So we now present several corollaries that give sufficient conditions for the subadditivity of j 1 o j [] , and hence for the copula C1 to be smaller than C2. The first requires the following lemma from Schweizer and Sklar , which relates subadditivity to concavity. Under the hypotheses of Theorem 4.

Let C1 and C2 be members of the Gumbel-Hougaard family 4. Hence the Gumbel-Hougaard family is positively ordered. Hence f is subadditive, which completes the proof. Let C1 and C2 be members of family 4. Hence this family is also positively ordered. Yet another test—often the easiest to use—is the following, an extension of a result in Genest and MacKay a. The proof is from Alsina et al. Hence the family 4. The direction of the order can be easily changed by reparameterization.

In the preceding four examples, we have seen that four of the families of Archimedean copulas from Table 4. However, there are families of Archimedean copulas that are not ordered, as the next example demonstrates. The family of Archimedean copulas 4. Because M is not Archimedean, we treat it separately in the second theorem, which is also from Genest and MacKay a —the proof is from Alsina et al.

Then from Corollaries 4. Fix an arbitrary t in 0,1 and choose e in 0,t. The converse follows by reversing the argument. For the family of Archimedean copulas given by 4. In the second subsection, we consider a two-parameter family that contains every Archimedean copula that is a rational function on the complement of its zero set.

From such a j, we can create parametric families of generators, which can then, in turn, be used to create families of Archimedean copulas. If a is in 0,1], then ja ,1 is an element of W. The proof is elementary and consists of a straightforward verification that the two compositions of j with the power function are decreasing and convex for the specified values of the parameters a and b.

We let Ca ,1 and C1,b denote the copulas generated by ja ,1 and j 1,b , respectively. Other interior power families include 4. This family also appears in Genest and Rivest Are there other Archimedean extreme value copulas? The answer is no Genest and Rivest Assume j generates an Archimedean extreme value copula C. From part 3 of Theorem 4. As the examples in Sect. Let j be in W, and let ja ,1 and j 1,b be given by 4. Further assume that ja ,1 and j 1,b generate copulas Ca ,1 and C1,b , respectively.

Part 1 follows from Corollary 4. Part 2 follows from Theorem 4. Hence by Lemma 4. This is definitely not the case for interior power families—recall Example 4. An examination of Table 4. Appealing to Theorems 4. This family has been used as a family of survival copulas for a bivariate Weibull model, see Lu and Bhattacharyya for details. Four subfamilies of 4. As with the preceding example, this family is also positively ordered by both parameters. We call such copulas rational, and answer the question affirmatively by constructing a two-parameter family of all rational Archimedean copulas.

Because Archimedean copulas must be symmetric and associative recall Theorem 4. Let R be a rational 2-place real function reduced to lowest terms, i. Now let C be a function with domain I2 given by 4. In order for C to be a copula, we must impose further restrictions on the six coefficients in 4. In order to find the values of a and b so that Ca ,b in 4. To find a candidate for ja ,b , we appeal to Theorem 4.

Hence we have Theorem 4. The curve in the first quadrant will play a role when we discuss the generators of Ca ,b. The parameter space for Ca , b given by 4. Thus we have Corollary 4. Indeed, all the level curves of Ca ,b for b in [0,1 are portions of hyperbolas—see Exercise 4. In order to obtain an explicit expression for the generator of Ca ,b , we need only integrate both sides of equation 4.

But within the parameter space for a and b illustrated in Fig. It is now a simple matter to exhibit the generators ja ,b explicitly: This leads naturally to the following generalization of 4. The functions C n in 4. But note that this technique of composing copulas generally fails, as was illustrated in Sect. What additional properties of j and j [ -1] will insure that C n in 4.

One answer involves the derivatives of j [ -1] , and requires that those derivatives alternate in sign. A function g t is completely monotonic on an interval J if it is continuous there and has derivatives of all orders that alternate in sign, i. See also Schweizer and Sklar , Alsina et al.

If C n is the function from In to I given by 4. The following corollary guarantees that this must occur when j -1 is completely monotonic. If the inverse j -1 of a strict generator j of an Archimedean copula C is completely monotonic, then C f P. As a consequence of Exercise 4. But this inequality holds for completely monotonic functions Widder If g is completely monotonic and f is absolutely monotonic, i.

If f and g are completely monotonic, so is their product fg; 4. If f is completely monotonic and g is a positive function with a completely monotone derivative, then fog is completely monotonic. In particular, e - g is completely monotonic. Although all the generators of this family are strict, we must, as a consequence of Corollary 4. But because e - x is completely monotonic and t 1 q is a positive function with a completely monotonic derivative, jq-1 is completely monotonic.

The procedure in the preceding example can be generalized to any exterior power family of generators associated with a strict generator j whose inverse is completely monotonic. The two-parameter family of copulas presented in Example 4. The arguments in Alsina et al. Conclude that the converse of Corollary 4.

As Jogdeo notes, Dependence relations between random variables is one of the most widely studied subjects in probability and statistics. The nature of the dependence can take a variety of forms and unless some specific assumptions are made about the dependence, no meaningful statistical model can be contemplated.

An Introduction to Copulas (Springer Series in Statistics) - PDF Free Download

There are a variety of ways to discuss and to measure dependence. The focus of this chapter is an exploration of the role that copulas play in the study of dependence. Dependence properties and measures of association are interrelated, and so there are many places where we could begin this study. A note on terminology: To be more 5 Dependence precise, let x i , y i and x j , y j denote two observations from a vector X,Y of continuous random variables.

Let X1,Y1 and X2 ,Y2 be independent and identically distributed random vectors, each with joint distribution function H. We then show that this function depends on the distributions of X1 ,Y1 and X2 ,Y2 only through their copulas. Let Q denote the difference between the probabilities of concordance and discordance of X1,Y1 and X2 ,Y2 , i. Let C1 , C2 , and Q be as given in Theorem 5. Q is symmetric in its arguments: Q is nondecreasing in each argument: Copulas can be replaced by survival copulas in Q, i.

The function Q is easily evaluated for pairs of the basic copulas M, W and P. Now let C be an arbitrary copula. Let X and Y be continuous random variables whose copula is C. Note that the integral that appears in 5. Let Cq be a member of the Farlie-Gumbel-Morgenstern family 3. Indeed, one of the reasons that Archimedean copulas are easy to work with is that often expressions with a one-place function the generator can be employed rather than expressions with a two-place function the copula.

See Genest and Rivest, ; Nelsen et al. The equivalence of 5. Let C1 and C2 be copulas. When the copulas are absolutely continuous, 5. In this case the left-hand side of 5. The proof in the general case proceeds by approximating C1 and C2 by sequences of absolutely continuous copulas. See Li et al. Let Ca ,b be a member of the Marshall-Olkin family 3. To obtain the population version of this measure Kruskal ; Lehmann , we now let X1,Y1 , X2 ,Y2 , and X3 ,Y3 be three independent random vectors with a common joint distribution function H whose margins are again F and G and copula C.

A numeric measure k of association between two continuous random variables X and Y whose copula is C is a measure of concordance if it satisfies the following properties again we write k X ,Y or k C when convenient: As a consequence of Definition 5. Let k be a measure of concordance for continuous random variables X and Y: For both tau and rho, the first six properties in Definition 5. For the seventh property, we note that the Lipschitz condition 2. The fact that measures of concordance, such as r and t, satisfy the sixth criterion in Definition 5. Because U and V each have mean 1 2 and 5 Dependence variance 1 12, the expression for rC in 5.

We shall exploit this observation in Sect. In this section, we will determine just how different r and t can be. We begin with a comparison of r and t for members of some of the families of copulas that we have considered in the examples and exercises in the preceding sections. The next theo- 5. Our proof is adapted from Kruskal However, the subscripts on X and Y can be permuted cyclically to obtain the following symmetric forms for t and r: The next theorem gives a second set of universal inequalities relating r and t. It is due to Durbin and Stuart ; and again the proof is adapted from Kruskal Let X, Y, t, and r be as in Theorem 5.

If p denotes the probability that some pair of the three vectors is concordant with the third, then, e. Bounds for r and t for pairs of continuous random variables Can the bounds in Corollary 5. To give a partial answer to this question, we consider two examples. Thus every point on the linear portion of the lower boundary of the shaded region in Fig. Thus every point on the linear portion of the upper boundary of the shaded region in Fig. Hence selected points on the parabolic portion of the upper boundary of the shaded region in Fig. Hence selected points on the parabolic portion of the lower boundary of the shaded region in Fig.

Supports of some copulas for which r and t lie on the boundary of the t-r region We conclude this section with several observations. As a consequence of Example 5. However, the copulas in Example 5. All the copulas illustrated in Fig. Hutchinson and Lai describe the pattern in Fig. The relationship between r and t in a one-parameter family of copulas can be exploited to construct a large sample test of the hypothesis that the copula of a bivariate distribution belongs to a particular family. See Carriere for details. In the following theorem, we show that g, like r and t, is a measure of association based upon concordance.

We show that 5. Using the symmetry of Q from the first part of Corollary 5. Under the hypotheses of Theorem 5. We conclude this section with one additional measure of association based on concordance. Suppose that, in the expression 5. If we reparameterize the expressions in Exercise 5. The proof of the following theorem is analogous to that of Theorem 5. The same is often true when we know the value of a measure of association. These bounds can be evaluated explicitly, see Nelsen et al.

For further details, including properties of the six bounds in Theorem 5. Thus X and Y are independent precisely when H belongs to a particular subset of the set of all joint distribution functions, the subset characterized by the copula P see Theorem 2. Just as the property of independence corresponds to the subset all of whose members have the copula P and similarly for monotone functional dependence and the copulas M and W , many dependence properties can be described by identifying the copulas, or simple properties of the copulas, which correspond to the distribution functions in the subset.

See Barlow and Proschan ; Drouet Mari and Kotz ; Hutchinson and Lai ; Joe ; Tong and the references therein for further discussion of many of the dependence properties that we present in this section. Let X and Y be random variables. Negative quadrant dependence is defined analogously by reversing the sense of the inequalities in 5. Intuitively, X and Y are PQD if the probability that they are simultaneously small or simultaneously large is at least as great as it would be were they independent.

Although in many studies of reliability, components are assumed to have independent lifetimes, it may be more realistic to assume some sort of dependence among components. For example, a system may have components that are subject to the same set of stresses or shocks, or in which the failure of one component results in an increased load on the surviving components.

In such a two-component system with lifetimes X and Y, we may wish to use a model in which regardless of the forms of the marginal distributions of X and Y small values of X tend to occur with small values of Y, i. Note that, like independence, quadrant dependence positive or negative is a property of the copula of continuous random variables, and consequently is invariant under strictly increasing transformations of the random variables. Also note that there are other interpretations of 5.

Many of the totally ordered one-parameter families of copulas that we encountered in Chapters 2 and 3 include P and hence have subfamilies of PQD copulas and NQD copulas. For example, If Cq is a member of the Mardia family 2. Some of the important consequences for measures of association for continuous positively quadrant dependent random variables are summarized in the following theorem. But recall from 5. If X and Y represent lifetimes of components in a reliability context, then this says that probability that Y has a short lifetime decreases to be precise, does not increase as the lifetime of X increases.

This behavior of the left tails of the distributions of X and Y and a similar behavior for the right tails based on 5. Each of the four tail monotonicity conditions implies positive quadrant dependence. Thus we have Theorem 5. If X and Y satisfy any one of the four properties in Definition 5. However, positive quadrant dependence does not imply any of the four tail monotonicity properties—see Exercise 5.

The next theorem shows that, when the random variables are continuous, tail monotonicity is a property of the copula. The proof follows immediately from the observation that univariate distribution functions are nondecreasing. Let X and Y be continuous random variables with copula C. Verifying that a given copula satisfies one or more of the conditions in Theorem 5.

In the preceding section, we saw that there was a geometric interpretation for the copula of positive quadrant dependent random variables—the graph of the copula must lie on or above the graph of P. There are similar geometric interpretations of the graph of the copula when the random variables satisfy one or more of the tail monotonicity properties—interpretations that involve the shape of regions determined by the horizontal and vertical sections of the copula.

To illustrate this, we first introduce some new notation. The shaded region in Fig. The next theorem expresses the criteria for tail monotonicity in terms of the shapes of the regions S1 u and S2 v determined by the vertical and horizontal sections of the copula. We prove part 1, leaving the proof of the remaining parts as an exercise. The proof proceeds along lines similar to those in 5. Because any one of the four tail monotonicity properties implies positive quadrant dependence, Theorem 5. By symmetry, V is not left tail decreasing in U, nor is V is not right tail increasing in U.

Furthermore, from Exercise 5. Suppose X and Y are random variables with the Marshall-Olkin bivariate exponential distribution with parameters l1 , l 2 , and l12 , as presented in Sect. In the next theorem, we show that when the random variables are continuous, stochastic monotonicity, like tail monotonicity and quadrant dependence, is a property of the copula.

The following geometric interpretation of stochastic monotonicity now follows [see Roberts and Varberg ]: Let Cq be a member of the Plackett family 3. The second part of the proof is analogous. As an immediate consequence, we have that the corner set monotonicity properties imply the corresponding tail monotonicity properties: Part 2 is similar. Let X and Y be continuous random variables with joint distribution function H: We prove part 1, the proof of part 2 is similar. Conversely, assume that 5. When the inequality in 5. In terms of total positivity, we have Corollary 5. Let X and Y be continuous random variables with joint distribution function H.

Let X and Y be random variables with the MarshallOlkin bivariate exponential distribution presented in Sect. Implications among the various dependence properties The final dependence property that we discuss in this section is likelihood ratio dependence Lehmann It differs from those considered above in that it is defined in terms of the joint density function rather than conditional distribution functions.

Let X and Y be continuous random variables with joint density function h x,y. This property derives its name from the fact that the inequality in 5. Negative likelihood ratio dependence is defined analogously, by reversing the sense of the inequality in 5. Of the dependence properties discussed so far, positive likelihood ratio dependence is the strongest, implying all of the properties in Fig.

To prove this statement, we need only prove Theorem 5. Let X and Y be random variables with an absolutely continuous distribution function. Proof Barlow and Proschan Recall that in Sect. Of the dependence properties discussed in this chapter, quadrant dependence is the weakest, whereas likelihood ratio dependence is the strongest. Thus the two most commonly used measures of association are related to two rather different dependence properties, a fact that may partially explain the differences between the values of rho and tau that we observed in several of the examples and exercises in earlier sections of this chapter.

The notion of positive likelihood ratio dependence can be extended to random variables whose joint distribution function fails to be absolutely continuous. To do so, we need some new notation: Let J and K denote intervals in R. We also write J 5. Show that Nelsen et al. Note that Cq is cubic in u and in v, so that Cq is given by 3. Suppose the copula of X and Y is Cq.

However, when C is an extreme value copula see Sect. Let X, Y, and h be as in Theorem 5. There are many other nonparametric measures of association that depend on the copula of the random variables. But one defect of such measures is that for the fourth property in Definition 5 Dependence 5. Examples abound in which a measure of concordance is zero but the random variables are dependent. A numeric measure d of association between two continuous random variables X and Y whose copula is C is a measure of dependence if it satisfies the following properties where we write d X ,Y or d C if convenient: As noted before, rC is proportional to the signed volume between the graphs of the copula C and the product copula P.

If in the integral 5. Then the quantity s C defined in 5. Proof Schweizer and Wolff It is easy to see from its definition that s satisfies the first two properties. If both a and b are almost surely strictly increasing, s satisfies the sixth property as a consequence of Theorem 2. If a is almost surely strictly increasing, and b almost surely strictly decreasing, s satisfies the sixth property as a consequence of Theorem 2. The remaining cases for the sixth property are similar.

Hence for many of the totally ordered families of copulas presented in earlier chapters e. Let X and Y be random variables with the circular uniform distribution presented in Sect. Because X and Y are jointly 5 Dependence symmetric, the measures of concordance t, r, b, and g are all 0 see Exercise 5. But clearly X and Y are not independent, and hence a measure of dependence such as s will be positive. The copula C of X and Y is given by 3.

Recall that the probability mass of Cq is distributed on two line segments, one joining 0,0 to q,1 and the other joining q,1 to 1,0 , as illustrated in Fig. However, X and Y are not independent—indeed, Y is a function of X. The same techniques used in the proof of Theorem 5. Elementary calculus then yields a maximum value q at the point q 2,1 2.

The principal diagonal of I2 is the support of M; while the secondary diagonal is support of W. Other l p distances yield other measures of association Conti Employing other L p distances between the diagonal sections of C and M and the secondary diagonal sections of C and W yields other meas- 5. Another such concept is tail dependence, which measures the dependence between the variables in the upper-right quadrant and in the lower-left quadrant of I2.

Let X and Y be continuous random variables with distribution functions F and G, respectively. The upper tail dependence parameter lU is the limit if it exists of the conditional probability that Y is greater than the t-th percentile of G given that X is greater than the t-th percentile of F as t approaches 1, i. If the limits in 5.

An Introduction to Copulas (Springer Series in Statistics)

The tail dependence parameters lU and l L are easily evaluated for some of the families of copulas encountered earlier: When a two-parameter family of Archimedean copulas is an interior or exterior power family associated with a generator j in W, the tail dependence parameters are determined by the parameters of the copula generated by j. The proof of the following theorem can be found in Nelsen Then the upper and lower tail dependence parameters of Ca ,1 are lU and l1La , respectively, and the upper and lower tail de1b pendence parameters of C1,b are 2 - 2 - lU and l1L b , respectively.

This pair of equations is invertible, so to find a member of the family 4. Now suppose that X and Y are continuous, with joint distribution function H, marginal distribution functions F and G, respectively, and copula C. As a consequence of 2. Let U and V be uniform random variables whose joint distribution function Cq is a member of the Plackett family 3. Note the special cases: Let C be an Archimedean copula with generator j in W. For example, for the Clayton family 4. The population versions can be expressed in terms of copulas—the sample versions will now be expressed in terms of empirical copulas and the corresponding empirical copula frequency function: We will show that the above expressions are equivalent to the expressions for r, t, and g that are usually encountered in the literature.

To show that 5. That is, the total contribution to the double sum in 5.

Note that the summand in 5. See Deheuvels , a,b for details. However, many of the dependence properties encountered in earlier sections of this chapter have natural extensions to the multivariate case. We shall examine only a few and provide references for others. X is positively orthant dependent POD if for all x in R n , both 5. As a consequence of Exercise 5. Let X be a three-dimensional random vector that assumes the four values 1,1,1 , 1,0,0 , 0,1,0 , and 0,0,1 each with 5.

Closely related to the notion of orthant dependence is multivariate concordance.

The multivariate version is similar: Let C1 and C2 be an n-copulas, and let C1 and C2 denote the corresponding n-dimensional joint survival functions. In the bivariate case, parts 1 and 2 of the above definition are equivalent see Exercise 2. Many of the measures of concordance in Sect. In general, however, each measure of bivariate concordance has several multidimensional versions.

See Joe ; Nelsen for details. There are also multivariate versions of some of the measures of dependence in Sect. Extensions of some of the other dependence properties to the multivariate case are similar. A note on notation: The following definitions are from Brindley and Thompson ; Harris ; Joe Two additional multivariate dependence properties are expressible in terms of the stochastic increasing property Joe In the bivariate case, the corner set monotonicity properties were expressible in terms of total positivity see Corollary 5.

The same is true in the multivariate case with the following generalization of total positivity: Lastly, X is positively likelihood ratio dependent if its joint ndimensional density h is MTP2. For implications among these and other dependence concepts, see Block and Ting ; Joe ; Block et al.

We close this section with the important observation that the symmetric relationship between positive and negative dependence properties that holds in two dimensions does not carry over to the n-dimensional case. In two dimensions, if the vector X,Y satisfies some positive dependence property, then the vector X,—Y satisfies the corresponding negative dependence property, and similarly for the vector —X,Y.

Furthermore, as a consequence of Theorem 2. The first, distributions with fixed margins, dates back to the early history of the subject. Another example, which also involves optimization when the margins are fixed, is the following. Secondly, we introduce quasi-copulas—functions closely related to copulas—which arise when finding bounds on sets of copulas. They also occur in the third section of this chapter, where we employ copulas and quasi-copulas to study some aspects of the relationship between operations on distribution functions and corresponding functions of random variables.

The final topic in this chapter is an application of copulas to Markov processes and leads to new interpretations of and approaches to these stochastic processes. For example, one of the central problems in statistics concerns testing the hypothesis that two random variables are independent, i. In such situations, it is often either assumed that the margins are normal, or no assumption at all is made concerning the margins.

In this section, we will be concerned with problems in which it is assumed that the marginal distributions of X and Y are known, that is, the margins of H are given distribution functions FX and FY , respec- 6 Additional Topics tively. Kolmogorov in Makarov , is typical: The problem can be solved without copulas Makarov , but the arguments are cumbersome and nonintuitive. The following theorem and proof are from Frank et al. Fix z in R, and let H denote the unknown joint distribution function of X and Y.

Illustrating the inequalities in the proof of Theorem 6. An Introduction to Copulas by Roger B. The study of copulas and their role in statistics is a new but vigorously growing field. With nearly a hundred examples and over exercises, this book is suitable as a text or for self-study. Nelsen , Hardcover, Revised. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions.

The only prerequisite is an upper level undergraduate course in probability and mathematical statistics, although some familiarity with nonparametric statistics would be useful. Knowledge of measure-theoretic probability is not required. He is also the author of "Proofs Without Words: This book is suitable as a text or for self-study. Additional Details Number of Volumes. Table Of Content Introduction. Reviews From the reviews of the second edition: With more than a hundred examples and over exercise, this book is suitable as a textbook or for self-study.

The only prerequisite is an upper level undergraduate course in probability and mathematical statistics. The second edition maintains the basic organizing of the material and the general level of presentation as the first one from The major additions are sections on: In addition to its primary use as an introductory book on copulas, this text could also serve as a complement to a graduate course or seminar in multivariate analysis focusing on dependence concepts. Dependence Modelling with Copulas , Hardcover.

The best introduction to the subject one can buy!

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