THE PYRAMID DISTRIBUTION

Transcription

1 Dept. of Math. Univ. of Oslo Pure Mathematics No. 39 ISSN December 5 THE PYRAMID DISTRIBUTION PAUL C. KETTLER Abstract. The paper introduces the pyramid probability distribution through its density in two dimensions, and investigates its properties and those of its copula. The research focuses on ways in which the pyramid distribution demonstrates dependence between its variables, primarily as revealed by its copula and related functions. The pyramid distribution bears an intimate relationship to the normal distribution, a relationship revealed and investigated. The pyramid density is built from the normal distribution function, making the pyramid the normal distribution once removed. Having normal margins, the pyramid returns to its foundation. The paper presents a general theory of this distribution, some formal, some discursive, including the presentation of a one-parameter family.. Introduction The paper introduces the pyramid probability distribution through its density in two dimensions, and investigates its properties and those of its copula. The distribution has normal margins, though it is not binormal. Properties the two distributions share are zero values in several statistics of concordance: Pearson's product-moment correlation, Kendall's tau, Spearman's rho, and Blomqvist's beta. As well, both distributions exhibit tail independence. The research, therefore, focuses on ways in which the pyramid distribution demonstrates dependence between its variables, primarily as revealed by its copula and related functions. The attractiveness of the pyramid distribution lies in its intimate relationship to the normal distribution. In brief, the pyramid density is built from the normal distribution function, making the pyramid, in a sense, the normal distribution once removed. The pyramid, having normal margins, in eect returns to its foundation. Many interesting questions arise about what other relationships between probability distributions, such as this, may appear upon examination, stimulating thought about a general theory. Some of these ideas come to the fore in Section 4 on a general transformation and again in Section 9 on conclusions. The task at hand, however, is to investigate this specic distribution in a quest to understand its nature. Following a section on motivation, which discusses a discrete version of the pyramid distribution, the development progresses through a description of the density, to the correlation of the variables, and to the distribution function with a determination of Kendall's tau. The Date: December 5. Mathematics Subject Classication. Primary 6E5, Secondary 6E. Key words and phrases. Constructed distributions; Marginally normal, but not binormal; Dependence; Zero concordance; Characteristic function; Copula; One-parameter family. The author wishes to thank Bernt Øksendal, Giulia DiNunno, Peter Tankov, and especially Fred Espen Benth for valuable insights and comments, and the Centre of Mathematics for Applications for the provision of an oce and related services.

2 PAUL C. KETTLER discussion then turns to the general transformation operating on an arbitrary density producing another, the latter being marginal to a bivariate distribution constructed in the manner of the pyramid. After that comes a calculation of the characteristic function, and brief treatments on innite divisibility and scalability. The functional form of the copula, including its quality of symmetric dependence, as dened, succeed. Calculation of Spearman's rho and Blomqvist's beta come next, along with copular density and tail independence. After that the paper presents a one-parameter family of distributions, and comments on Lévy copulas. Conclusions complete the discourse. Illuminating the text is a battery of 7 gures and a series of numerical calculations. For general discussions on dependence and related concepts see these references (Durbin and Stuart 95; Schweizer and Wol 98; Genest, Ghoudi, and Rivest 995; Joe 997; Embrechts, McNeil, and Straumann ; Breymann, Dias, and Embrechts 3; Lindskog, McNeil, and Schmock 3; Nyfeler 3; Tankov 3; Patton 4; Tankov 4). Papers emphasizing tail dependence are these (Schmidt and Stadtmüller 3; Schmidt 5). For material more specic to copulas see these (Fréchet 95; Sklar 959; Genest and MacKay 986; Genest and Rivest 993; Joe 993; Shih and Louis 995; Sklar 996; Nelsen 998; Embrechts, McNeil, and Lindskog 3; Carrière 4; Cherubini, Luciano, and Vecchiato 4; Cont and Tankov 4).. Motivation As an exercise leading to this study, the author investigated a discrete distribution, the binomial pyramid, in two dimensions constructed from binomial coecients of order n. This distribution has binomial margins, though is not bivariate binomial. In the limit as n the binomial pyramid frequency function p n (x, y), to be dened in Equation (.), induces a binomial distribution function which converges to the [continuous] pyramid distribution G(x, y) introduced by this paper in Equation (3.4). The pyramid density, provided in Equation (3.), has the shape of the normal distribution function scaled by / in each of its four axial branches. Though this continuous distribution stands on its own from its denition, the inspiration from the discrete model guided the further investigation and serves now as foundation. As a preliminary step consider a domain D n consisting of (n + ) (n + ) points on an integer lattice centered at the origin of R. As n is even these points have integer coordinates; as n is odd the points have half integer coordinates. This convention in dening D n allows for the consideration of both even and odd n at once. The plan is to construct a function P n (x, y) with binomial margins on this lattice, with level sets on the nested squares of D n. From P n (x, y) will follow the normalized frequency function p n (x, y), dened on a related lattice d n. Within D n are n+ nested squares, with the center square having either a degenerate point should n be even, or 4 points should n be odd. Thus the outermost square has 4n points. Within this square is another with 4(n ) points, etc. Now let { n I n :=, n ( )},..., min, be an index set. Then the domain of P n (x, y) conforms to D n := { (x, y) } (.) ( x, y ) I n I n

3 THE PYRAMID DISTRIBUTION 3 Next, dene P n (x, y) as the sum of binomial coecients on D n. To this end, let Then P n (x, y) := L := n max ( x, y ), (x, y) D n L ( ) n + k= k This last denition implies that P n (x, y) = ( ) n+ if either x = n/, or y = n/, i.e., on points of the outer square of D n. On the next included square of D n, P n (x, y) = ( ) n+ + ( n+ ) = + (n + ) = n +, etc. A simple inductive argument establishes that the common margins of P n (x, y), dened as P n ( ), are the scaled binomial frequencies as here, assuming equiprobable events. The mean of this distribution is zero, the variance σn = n/4, and therefore the standard deviation σ n = n/. so ( ) n P n (x) = (n + ) n x, ( ) n P n (y) = (n + ) n y The next task is to normalize P n (x, y) to p n (x, y). To start let Î n := I n σ n, where division by σ n is implied for each component of I n. Then in analogy to the denition for D n in Equation (.), let { d n := (x, y) } ( x, y ) În În = {(x, y) } (σ x, σ y ) I n I n As above, consider d n centered at the origin of R, and dene (.) p n (x, y) := P n(σ n x, σ n y)σn (n + ) n = P n n( x, n y) n+ ( n ) n + Then, p n (x, y) is a frequency function with common equiprobable binomial margins p n ( ). In particular, for x În, and y În, ( ) n p n (x) = σ n = ( ) n (n+) n, so n + σx n + n x ( ) n p n (y) = σ n n + σy = ( ) n (n+) n n + n y

4 4 PAUL C. KETTLER Remark. Note that the bivariate binomial frequency function p n (x, y) has a total mass of σn, and the common marginal binomial frequency function p n ( ) has a total mass of σ n. The necessity of this scaling devolves from the fact that the lattice points on which p n (x, y) is dened are separated by /σ n on each axis. Also, observe that the lower term of the binomial coecient in the expression for p n ( ) is always an integer if n is even, and a half ingeter if n is odd. In the latter case the coecient is calculated with reference to the gamma function. Lastly, let Then n n M := + x n n N := + y Φ n (x, y) := σ n Φ n (x) := σ n Φ n (y) := σ n Φ n (x, y) Φ n (x) Φ n (y) N M p n (x, y) M p n (x) N p n (y) D G(x, y) n D N,(x) n D N,(y) n N, ( ) is the normal distribution function, and the convergences are in distribution, after extending Φ n (x, y) and Φ n ( ) to R and R, respectively, by assigning probability zero to Borel sets which do not contain points of the lattices. For reference, see Figures and, which show p 6 (x, y) and its level curves. 3. The pyramid The pyramid distribution is an example of a two dimensional probability distribution that is normal on its margins, but is not bivariate normal. On the sample space (x, y) R it has a density not dierentiable on the diagonals y = ±x, and has tails thinner than the bivariate normal in all directions. As such, the distribution has an interesting copula, one that will gure in the sequel. See Figures 3, 4, and 5, which show, in order, the density of the pyramid distribution, the level curves of this density, and a scatter plot of points taken from the distribution. 3.. The density. The pyramid distribution is dened by its density. Given the normal density and distribution, respectively, as f(x) and F (x), the pyramid density is (3.) g(x, y) := F (x y x y) = F ( ( x y ) )

5 THE PYRAMID DISTRIBUTION 5 In words, the pyramid density has the shape of the normal distribution function of the lower half line on each of its four faces. These faces come to a point at the origin. That the tails are thin derives from the fact that F (x) = o[f(x)], as seen by a simple application of l'hôpital's Rule. From this denition, one calculates the distribution readily, the subject of Proposition 3.5 below. As well, this distribution is uncorrelated by Pearson's product-moment statistic, though the variables are dependent. First, the task is to prove that g(x, y) is a density, that is, that it integrates to, and that the marginal distributions are normal. Establishing the distribution follows. This lemma and corollary are useful. Lemma 3.. I (x) := I (x) := x x F (y) dy = xf (x) + f(x) yf (y) dy = [( x ) F (x) + xf(x) ] Proof. The two sides have the same derivative, and vanish at. Corollary 3.. I () = I () = F (y) dy = π yf (y) dy = 4 Remark. By Corollary 3., 4I () =, and therefore 4yF (y) is a density on the interval (, ]; by Lemma 3., 4I (x) is its distribution. This is the distribution of the pyramid { in the sense of Lebesgue integration, wherein the dierential of mass on the square (x, y) (y, x) x y }, taking y to be non-positive, is the perimeter 8y times the uniform density on the square F (y). This distribution therefore provides the total mass outside the centered square with edges of length x, x. Informally, this line of reasoning demonstrates that g(x, y) is a density. The discussion proceeds now by conventional means. Proposition 3.3. g(x, y) is a density and the marginal distributions are normal. Proof. First, choose x. The value of the marginal density g (x) is an integral through three sections of the joint density. Two of the sections, on the slopes in the positive and negative y axis directions, provide equal contributions to the marginal density. The third section, on the traverse of the slope in the direction of the negative x axis, provides a single

6 6 PAUL C. KETTLER contribution. Therefore, (3.) g (x) = x = = x = f(x) g(x, y) dy F (y) dy + x x F (x) dy F (y) dy + F (x)( x) by Lemma 3.. For x >, symmetry in the pyramid density implies g (x) = g ( x) = f( x) = f(x). The procedure for y is similar, as follows. (3.3) g (y) = y = = y = f(y) g(x, y) dx F (x) dx + y y F (y) dx F (x) dx + F (y)( y) by Lemma 3.. For y >, symmetry in the pyramid density implies g (y) = g ( y) = f( y) = f(y). As either marginal measure is a density, so is the bivariate measure, by Fubini's Theorem. 3.. Correlation. Proposition 3.4. The pyramid distribution is uncorrelated, by Pearson's product-moment statistic. Proof. Proceed to compute the covariance. Owing to symmetry of g(x, y), = xy g(x, y) dx dy = xy g(x, y) dx dy = The sum of these four terms is E[XY ] =. xy g(x, y) dx dy See Figure 6, which shows the pyramid distribution function. xy g(x, y) dx dy

7 THE PYRAMID DISTRIBUTION The distribution. Proposition 3.5. The pyramid distribution (3.4) G(a, b) = Pr {X a, Y b} = F (a b)(ab + ) + (a b)f(a b) if a + b G( a, b) + F (a) + F (b) if a + b > Proof. First consider the case a + b, a b. probability is the following. (3.5) G(a, b) = Pr {X a, Y b} = a b x Among the several approaches to this a a F (x) dy dx+ F (y) dx dy y = F (a)(ab + ) + bf(a) This is a straightforward exercise in integration by parts, using Lemma 3.. Similarly, in the case a + b, a > b, (3.6) G(a, b) = Pr {X a, Y b} = b a y b b F (y) dx dy+ x F (x) dy dx = F (b)(ab + ) + af(b) Using the same methods and combining results, the assertion for this case follows. The result for the case a + b > devolves from the fact that the marginal distributions are normal Kendall's tau. A commonly addressed non-parametric measure of a distribution's degree of association is Kendall's tau. This value is zero for the pyramid distribution, a consequence of the following lemma. Lemma 3.6. Kendall's tau τ F for a distribution F (x, y) is zero if the probability measure is invariant under rotation by π/. Proof. From the denition, τ F = Pr{(x x )(y y ) > } Pr{(x x )(y y ) < } A rotation of the measure by π/ takes (x, y ) ( y, x ) and (x, y ) ( y, x ), reversing the concordance/discordance relationship as follows. Pr{( y + y )(x x ) > } Pr{( y + y )(x x ) < } = [Pr{ (x x )(y y ) > } Pr{ (x x )(y y ) < }] = [Pr{(x x )(y y ) > } Pr{(x x )(y y ) < }] = τ F

8 8 PAUL C. KETTLER By the hypothesis, therefore, τ F = τ F, and τ F = Corollary 3.7. τ F =, when F (x, y) is the pyramid distribution. Proof. On rotation, f(x, y) = f( y, x), satisfying the hypothesis of Lemma (3.6). 4. A general transformation The results of the previous section suggest a process, or transformation, which takes the normal density back to itself. The process looks like this with several functions leading to others. f(x) Dist F (x) Cons g(x, y) Dist G(a, b) Marg g (x) = g (x) = f(x), where the superscripts `Dist', `Cons', and `Marg' stand respectively for the processes of computing the distribution from the density, constructing the pyramid density, and determining the marginal density. This process is a transformation T, with f(x) as a xed point, thus. T : f(x) f(x) To give meaning to T for a more general class of densities it is necessary to be more specic about the construction step. In the present instance it was sucient to dene g(x, y) as in Equation (3.). In the general case it is necessary to guarantee that the construction produces a bivariate density, i.e., that the resulting function integrates to over R. This is simply a matter of scaling, for if Equation (3.) had been expressed instead for a preliminary unnormalized ḡ(c; x, y) with coecient c instead of the coecient, as this, (4.) ḡ(c; x, y) := cf (x y x y) = cf ( ( x y ) ), then a coecient c = could have been determined readily as c = ḡ(, x, y) dx dy, so that g(x, y) = ḡ(c ; x, y) is a density. R With this scaling any density is a candidate for applying the transformation T. A reasonable avenue of research, then, is to investigate T and its properties. Of particular interest is the set of xed point densities, which would be those satisfying Equations (3.) and (3.3) for a general g(x, y). These are side issues for the present, but are revisited in Section 9 on conclusions below.

9 THE PYRAMID DISTRIBUTION 9 5. The characteristic function Calculating the characteristic function is a straightforward task. Natural symmetries in the density suggest a piecewise approach, starting with a West quadrant, follow by axial symmetry to an East quadrant, followed by rotational symmetry to South and North quadrants. The rst subsection denes this multiple domain and produces the function with corollary results. Following are brief comments on the distribution not having the innite divisibility property, and on scaling the domain along the axes. 5.. Calculation. The domain of the pyramid density naturally separates into four subdomains, those being the quadrants bounded by y = ±x in the plane on which boundaries the density is only of class C (elsewhere being of class C.) For convenience call these quadrants, respectively, West, East, South, and North, or abbreviated, W, E, S, and N. Specically these are W := { (x, y) x <, x < y < x } E := { (x, y) x >, x < y < x } S := { (x, y) y <, y < x < y } N := { (x, y) y >, y < x < y } Calculation of the characteristic function, stated formally as a proposition, rst needs the support of a lemma and corollary. Lemma 5.. If a, x [F (x) cosh ax ( ) ] a [F (x a) + F (x + a)] exp x F (y) sinh ay dy = a [F (x) sinh ax ( ) ] a [F (x a) F (x + a)] exp F (y) cosh ay dy = a Proof. In each case, the two sides have the same derivative, which vanishes at. In the second case establishing the result requires reliance on the fact that F (x) is o [exp( ax)] as x, a direct conclusion following an application of l'hôpital's Rule. Corollary 5.. If a, F (y) sinh ay dy = [ exp a ( )] a F (y) cosh ay dy = [F (a) ] exp a ( ) a Proposition 5.3. The pyramid distribution has characteristic function ϕ(ζ, η) = sinh(ζη) γ(ζ, η), ζη

10 PAUL C. KETTLER where γ(ζ, η) := exp ( ζ + η ) is the characteristic function of the independent binormal distribution. Proof. Insofar as the density is symmetric among these quadrants about the origin, one may develop the characteristic function by integrating over one of the quadrants, then exploit this symmetry to nish the calculation. The choice is arbitrary, so begin with the West subdomain. Let ϕ W (ζ, η) be the characteristic function restricted to the West subdomain, with parallel denitions, ϕ E (ζ, η), ϕ S (ζ, η), ϕ N (ζ, η), for the other subdomains. Then, (5.) (5.) (5.3) ϕ W (ζ, η) = x = = i η x e i(ζx+ηy) F (x) dy dx e iζx F (x) x x e iηy dy dx e iζx F (x) sinh iηx dx Similarly, ϕ E (ζ, η) has a denition (5.4) ϕ E (ζ, η) = x x e i(ζx+ηy) [ F (x)] dy dx Now, to get ϕ E (ζ, η) into a form analogous to ϕ W (ζ, η) do the following. Change the variable x to x, recognize that [ F (x)] = F ( x), and exchange the outer limits of integration. Arrive at the following expression, which diers from ϕ W (ζ, η) of Equation (5.) only on the sign of the rst exponent. (5.5) ϕ E (ζ, η) = e iζx F (x) x x e iηy dy dx Continuing, as above for ϕ W (ζ, η), calculate that (5.6) ϕ E (ζ, η) = i η e iζx F (x) sinh iηx dx This process prepares one to add the results for ϕ W (ζ, η) and ϕ E (ζ, η) of Equations (5.3) and (5.6) to get a combined result for the characteristic function including both the West and

11 THE PYRAMID DISTRIBUTION East subdomains. Call this sum ϕ W E (ζ, η). Then, as one readily computes, (5.7) ϕ WE (ζ, η) = ϕ W (ζ, η) + ϕ E (ζ, η) = i η F (x) cosh iζx sinh iηx dx The pyramid density g(x, y) is symmetric in its variables. Therefore, the analogous combined characteristic function including both the South and North subdomains, ϕ SN (ζ, η), is achieved by interchanging the roles, respectively, of x and y, and of ζ and η. (5.8) ϕ SN (ζ, η) = i ζ F (y) cosh iηy sinh iζy dy Next, convert the last expressions for ϕ W E (ζ, η) and ϕ SN (ζ, η), Equations (5.7) and (5.8), respectively, using the identity cosh ax sinh bx = [sinh(a + b)x sinh(a b)x] At the same time, recognize that the variables of integration are simply formal, and change y to x in the second expression. As well, negate the argument in the nal hyperbolic sine, so as to conform it to the argument above it, while reversing the sign on that term to maintain the expression. ϕ WE (ζ, η) = i η ϕ SN (ζ, η) = i ζ F (x) [sinh i(ζ + η)x sinh i(ζ η)] dx F (x) [sinh i(ζ + η)x + sinh i(ζ η)] dx One can resolve these integrations by recourse to the rst part of Corollary 5.. [ )] [ (ζ + η) ϕ WE (ζ, η) = exp ( exp ( η(ζ + η) η(ζ η) ϕ SN (ζ, η) = ζ(ζ + η) [ exp ( (ζ + η) )] + ζ(ζ η) [ exp ( (ζ η) (ζ η) A few steps now of collecting terms (vertically on the rst factors) provides the complete characteristic function ϕ(ζ, η) = ϕ W E (ζ, η) + ϕ SN (ζ, η), as follows. ϕ(ζ, η) = [ ) )] (ζ η) (ζ + η) exp ( exp ( ζη = sinh(ζη) exp ( ζ + η ) ζη = sinh(ζη) γ(ζ, η) ζη )] )]

12 PAUL C. KETTLER Corollary 5.4. ϕ(ζ, η) = + exp (ζη x) dx γ(ζ, η) Corollary 5.5. The marginal distributions are normal. ) ϕ(ζ, ) = ϕ(, ζ) = exp ( ζ =: γ(ζ) Remark. In the development thus far, the theory has focused attention only on the bivariate normal distribution in the independent case. In this regard one could note that γ(ζ, η) = γ(ζ)γ(η) and incorporate this fact into the analysis. See Figures 7, 8, 9, and, which show, in order, the characteristic function of the pyramid distribution, its level curves, the characteristic function of the normal distribution, and its level curves. The dierence of the pyramid and normal characteristic functions, ω(ζ, η) := ϕ(ζ, η) γ(ζ, η), has maxima symmetrically placed in the four quadrants. The maxima, determined numerically, occur at (ζ, η) = (±.659, ±.659) The value at those points is.75. See Figures and, which show the dierence of the pyramid and normal characteristic functions, and the level curves of this dierence. 5.. Innite divisibility. The question of innite divisibility arises in the context of Lévy processes. In particular, could the pyramid distribution be innitely divisible, and accordingly be the basis for dening directly a Lévy process with associated Lévy measure? The following proposition answers the question in the negative. However, see below to Section 8 for a discussion of the induction of Lévy copulas related to the pyramid distribution from its ordinary copula. These Lévy copulas create bivariate Lévy measures from marginal Lévy measures, thereby allowing the construction of bivariate Lévy processes. The laws of all Lévy processes are innitely divisible. For an excellent treatment of innite divisibility and related properties, see this (Itô 94). Proposition 5.6. The pyramid distribution is not innitely divisible. Proof. A result of Sato implies this conclusion (Sato 999, Proposition., p. 65). Specifically, a Lévy process with non-trivial Lévy measure must have non-trivial Lévy measure on at least one of the projected processes. Thus for the pyramid distribution to be innitely divisible it must be purely Gaussian, which it is not Scalability. The Pyramid distribution is scalable, with density ĝ(x, y) as follows. ĝ(x, y) = c g(ca x, c a y) where g(x, y) is the Pyramid density, a >, c >. All the results of this paper apply appropriately to the scaled distribution, except for symmetry if a.

13 THE PYRAMID DISTRIBUTION 3 6. The copula Let α = F (a), β = F (b). Then the copula for any distribution D(a, b) with margins D (a) and D (b) is a function H : [, ] [, ] [, ] H ( D (a), D (b) ) = H(α, β) = D(D For G(a, b) with margins F (a) and F (b) this copula becomes (α), D (β)) = D(a, b) H ( F (a), F (b) ) = H(α, β) = G(F (α), F (β)) = G(a, b) See Figure 3, which shows the copula of the pyramid distribution. Following the calculation of the specic functional form for the copula, this section proceeds to calculate Spearman's rho and Blomqvist's beta, produces the copular density, and discusses tail independence. 6.. Specic functional form. H(α, β) has a specic form implied by G(a, b) given by Proposition 3.5. First needed is this lemma. Lemma 6.. α + β a + b. Also, α β a b. Proof. Then, α + β α β F (α) F ( β) a b a + b α β F (α) F (β) a b (6.) H(α, β) = G(F (α), F (β)) = (α β) ( F (α)f (β) + ) + ( F (α) F (β) ) f ( F (α) F (β) ) if α + β G ( F (α), F (β) ) + α + β if α + β > (6.) One may compare this copula with that of the independent copula C : [, ] [, ] [, ] C (α, β) = αβ See Figure 4, which shows the product copula, which represents all independent distributions. Further, one may look to the dierence function (6.3) H : [, ] [, ] [, ] H (α, β) := H(α, β) C (α, β) See Figures 5 and 6, which show the dierence of the pyramid and independent copulas, and the level curves of this dierence. This copula dierence exhibits a property inspiring a denition, and then two propositions. Denition 6.. A bivariate distribution with copula K(α, β) is symmetrically dependent if its copula dierence to the independent copula K (α, β) := K(α, β) C (α, β) satises the following conditions:

14 4 PAUL C. KETTLER () K (α, β) = K ( α, β) = K ( α, β) = K (α, β), {α, β} [, ] [, ] () K (α, β) is not identically zero (the case of independence.) Proposition 6.3. For a symmetrically dependent distribution with copula dierence K (α, β) Proof. From Denition 6. K (α, β) dα dβ = = K (α, β) dα dβ = K (α, β) dα dβ = K (α, β) dα dβ K (α, β) dα dβ, whence the conclusion follows. Proposition 6.4. The pyramid distribution is symmetrically dependent. Proof. Part of 3 First establish that H (α, β) = H ( α, β). α + β >. By Lemma 6., a + b >. Without loss of generality assume H (α, β) = G [ F (α), F (β) ] C (α, β) = G(a, b) F (a)f (b) = [G( a, b) + F (a) + F (b) ] F (a)f (b) { } [( ) ] = ( a)( b) + F ( b) bf( a) + [ F ( a)] + [ F ( b)] [ F ( a)] [ F ( b)], by Proposition 3.5, = [( ) ] ( a)( b) + F ( b) bf( a) F ( a)f ( b) = G [ F ( α), F ( β) ] C ( α, β) = H ( α, β) Part of 3 Next, establish that H (α, β) = H (α, β). Examine two exhaustive cases, rst for α β. By Lemma 6., a b.

15 Case I: α + β. By Lemma 6., a + b. H [ F (α), F (β) ] = G(a, b) F (a)f (b) THE PYRAMID DISTRIBUTION 5 = [(ab + )F (a) + bf(a)] F (a)f (b), by Proposition 3.5, = [( ) ] a( b) + F (a) bf(a) + F (a) [ F (b)] = [G(a, b) F (a)f ( b)] [ = H F (α), F ( β) ] Case II: α + β >. By Lemma 6., a + b >. [ H F (α), F (β) ] = [G( a, b) + F (a) + F (b) ] F (a)f (b) { } [( ) ] = ( a)( b) + F ( b) + ( a)f( b) + F (a) + [ F ( b)] F (a) [ F ( b)], by Proposition 3.5, { = } [( ) ] a( b) F ( b) + af( b) + F (a) + [ F ( b)] F (a) + F (a)f ( b) = [( ) ] a( b) + F ( b) + af( b) + F (a)f ( b) = [G(a, b) F (a)f ( b)] [ = H F (α), F ( β) ] The two cases for α > β are analogous (interchanging α and β,) establishing this Part. Part 3 of 3 Finally, by Part, and by Part, H (α, β) = H ( α, β), H ( α, β) = H ( α, β) The copula dierence H (α, β) has four extrema, two maxima and two minima. Standard development to identify them leads to an analytically intractable equation; however, numeric solutions are available. The maxima occur at and the minima occur at (α, α) = (.9889,.9889) ( α, α) = (.89,.9889), (α, α) = (.9889,.89) ( α, α) = (.89,.9889) The respective maxima and minima are ±.384.

16 6 PAUL C. KETTLER 6.. Spearman's rho and Blomqvist's beta. Symmetric dependence, as exhibited in the pyramid distribution by Proposition (6.4) implies the Spearman's rho, a measure of rank correlation, is zero. That this is true is a consequence of a more general result, that Spearman's rho for any distribution is a simple scaling of the integral of the copula dierence over its domain. This result is known and was stated by Nelsen, but not fully developed by him, omitting a few steps included now (Nelsen 998, Theorem 5..6., page 35, and Equation (5..6), p. 38). Proposition 6.5. Spearman's rho ϱ K for a distribution with copula K(α, β) and copula difference to the independent copula K (α, β) := K(α, β) C (α, β) is Proof. ϱ K = ϱ K = From this it follows by Equation (6.) that K (α, β) dα dβ K(α, β) dα dβ 3 But then, ϱ C = = C (α, β) dα dβ 3 αβ dα dβ 3 = ϱ K = ϱ K ϱ C = = [K(α, β) C (α, β)] dα dβ K (α, β) dα dβ as asserted Remark. This theorem provides the insight to visualize Spearman's rho by looking at the copula dierence function and evaluating its integral. In addition, the theorem provides a basis for selecting a copula, and thereafter a distribution, with a desired value for Spearman's rho. Corollary 6.6. Spearman's rho for a symmetrically dependent distribution is zero. Corollary 6.7. Spearman's rho ϱ H for the pyramid distribution is zero. Remark. Of interest also are the Fréchet lower and upper limit copulas C (v, z) and C (v, z), respectively, representing complete negative and complete positive dependence. See (Cherubini, Luciano, and Vecchiato 4, pp. 556) for details. One easily calculates Spearman's

17 THE PYRAMID DISTRIBUTION 7 rho for these copulas by the method of Proposition 6.5. As expected, ϱ K =. C (v, z) := max(v + z, ) ϱ C = C (v, z) := min(v, z) ϱ C = + the Fréchet lower bound copula the Fréchet upper bound copula Blomqvist's beta is a valuation of a copula at it's center, scaled so that β ( C (v, z) ) = and β ( C (v, z) ) = +. As well, β ( C (v, z) ) =. Specically, for any copula C(v, z), β = 4 C (, ) In the present instance ( β = 4 G, ) = 4 4 =, which follows by reference to Equation (3.5) The copular density. One readily computes the density of the pyramid copula from the copula. If h(α, β) be this density, then h(α, β) = β α β Applying the chain rule to Equation (6.) gives α h(u, v) du dv = h(α, β) = g( F (α), F (β) ) f ( F (α) ) f ( F (β) ) H(α, β) α β Substituting the alternate pyramid density denition of Equation (3.), one has h(α, β) = [α β ( α) ( β)] f ( F (α) ) f ( F (β) ), or h(α, β) = [ ( α β )] f ( F (α) ) f ( F (β) )

18 8 PAUL C. KETTLER 6.4. Tail dependence. The tails of the pyramid distribution are independent by the ordinary denition. Here it is, tailored to the present circumstances, followed by a Proposition. Denition 6.8. A bivariate distribution is lower tail dependent with coecient λ L, λ L if lim Pr { Y a } G(a, a) H(α, α) X a = lim = lim = λ L a a F (a) α α A bivariate distribution is upper tail dependent with coecient λ U, λ U if the distribution of ( X, Y ) is lower tail dependent with coecient λ U. A distribution is either lower tail independent or upper tail independent, respectively, as λ L = or λ U =. A distribution is either lower tail completely dependent or upper tail completely dependent, respectively, as λ L = or λ U =. Proposition 6.9. The pyramid distribution is both lower and upper tail independent. Proof. By Proposition 3.5 λ L = G(a, a) lim = lim a F (a) a [ (a + ) + af(a) ] = F (a) Applying l'hôpital's Rule to this expression yields the result for the lower tail. The result for the upper tail follows by symmetry. 7. A one-parameter family On may dene a family of distributions based on linear combinations of the pyramid and binormal densities, thus. g θ (x, y) := θg(x, y) + ( θ)f(x, y), θ Then this scaling propagates through the distribution function, characteristic function, and copula, owing to the linearity of the integral operator. For completeness here are the formalities, giving rise to a few denitions along the way. The functions f(, ) and F (, ) are the bivariate normal density and distribution, respectively. Proposition 7.. The distribution function G θ (a, b) := a b g θ (x, y) dy dx = θg(a, b) + ( θ)f (a, b) The characteristic function ϕ θ (ζ, η) := e i(ζx+ηy) g θ (x, y) dy dx = θϕ(ζ, η) + ( θ)γ(ζ, η) The copula ( H θ (α, β) := G θ F (α), F (β) ) = θh(α, β) + ( θ)c (α, β) = C (α, β) + θ H (α, β)

19 THE PYRAMID DISTRIBUTION 9 Proof. The distribution function a b G θ (a, b) = g θ (x, y) dy dx a b a b = θ g(x, y) dy dx + ( θ) = θg(a, b) + ( θ)f (a, b) f(x, y) dy dx The characteristic function The copula ϕ θ (ζ, η) = = θ e i(ζx+ηy) g θ (x, y) dy dx e i(ζx+ηy) g(x, y) dy dx + ( θ) = θϕ(ζ, η) + ( θ)γ(ζ, η) H θ (α, β) = G θ ( F (α), F (β) ) e i(ζx+ηy) f(x, y) dy dx = θg ( F (α), F (β) ) + ( θ)f ( F (α), F (β) ) as above = θh(α, β) + ( θ)c (α, β) = C (α, β) + θ H (α, β) by Equation (6.3) Remark. Note that the ϕ(ζ, η) as calculated in Proposition 7. above is not the characteristic function of the sum of two random variables. Rather, it is the characteristic function of a single random variable dened by a density which is the convex combination of two other densities. As such, no questions of dependence arise. Also, observe that the family of densities has xed points placed symmetrically on the axes at (x, ) = (±.65, ) and (, y) = (, ±.65), where g θ (x, y) =.88. To see, take θ g θ(x, y) = and solve. As well, one could look toward extending the family to negative values of the parameter θ. This is not possible, for in such circumstances the density g θ (x, y) would be negative in a neighborhood of a point on each of the two tails of the major diagonal. To see this, locate the minima of g θ (x) := g θ (x, x). These are the points { x x f(x) = θ/ ( 4( θ )}. This equation has nite solutions for any θ <. Furthermore, g θ (x) < at these points as no stationary points more remote exist, and lim g θ(x) =. The conclusion readily follows. x ± A similar statement obtains for minima on the minor diagonal, but the analysis above suces. See Figure 7. This gure shows values of the density in the axial directions (x = ) (y = ) for choices of θ. The xed point appears.

20 PAUL C. KETTLER Lastly, the entire one-parameter family is tail independent. This is easily conjectured, for the pyramid distribution is tail independent by Proposition 6.9, and the multivariate normal distribution is known also to be. Formally, this is the statement, following a lemma. Lemma 7.. d da d da a a a f(x, y) dy dx = F (a)f(a) = f(x, y) dy dx = f(a) Proof. Standard calculus methods establish the results. d da F (a) Proposition 7.3. G θ (x, y) is both lower and upper tail independent. Proof. Let λ (θ) L be the lower tail dependency coecient. Then, λ (θ) L = lim θg(a, a) + ( θ)f (a, a) θg(a, a) + ( θ)f (a, a) = lim = a θf (a) + ( θ)f (a) a F (a) Applying l'hôpital's Rule and Lemma 7. to this expression yields the result for the lower tail. The result for the upper tail follows by symmetry. Remark. Note that Proposition 7.3 proves Proposition 6.9 again as a special case (for θ = ) and also proves tail independence for the uncorrelated bivariate normal distribution (for θ = ). 8. Lévy copulas Tankov in his Ph.D. thesis provided a result which enables one to construct a Lévy copula from an ordinary (probability) copula (Tankov 4, Theorem 5.). Applying his result here states that for the pyramid copula H(α, β), the induced function for a strictly increasing function L : [, ] d [, ] ( L(γ, δ) := ψ H ( ψ (γ), ψ (δ) )), ψ : [, ] [, ] having positive derivatives to order d on (, ), is a Lévy copula. Tankov oers as an example the function ψ(x) = x x. The Lévy copula is an important construct in dening and interpreting dependence relationships between and among Lévy processes. In this regard, the pyramid-derived Lévy copulas have interesting implications. The cited thesis, in addition to these works, is a good starting point to explore such ideas (Cont and Tankov 4, Chapter 5)(Tankov 3).

21 THE PYRAMID DISTRIBUTION The concept carries forward to a Lévy copula for any member of the parametric family. With assumptions as above the function ( L θ (γ, δ) := ψ (H θ ψ (γ), ψ (δ) )) is also a Lévy copula. The proof is omitted. 9. Conclusions The pyramid distribution stimulates interest in several threads of research. One is to investigate further the class of distributions with normal margins, made much more feasible by recent advances in copula theory. Another is to research distributions as building blocks for other distributions, generating margins which may have relations to the original blocks, as in the discussion of Section 4 on the general transformation T. A third is to seek applications for distributions with normal margins or other margins of interest, and for distributions constructed like architecture. Seen is a synergy between applications suggesting constructions, and constructions stimulating applications, including in nance, physics, and game theory, among others.

22 PAUL C. KETTLER Pyramid Density Discrete, n = 6.5. p(x,y) x values y values Figure. Pyramid Distribution Density, Discrete n = 6 Pyramid Density Level Discrete, n = y values p(x,y) x values -4 Figure. Pyramid Distribution Density Level Curves, Discrete n = 6

23 THE PYRAMID DISTRIBUTION 3 Pyramid Density.3.5. g(x,y) x-values y-values Figure 3. Pyramid Distribution Density Figure Pyramid Density Level y-values g(x,y) x-values Figure 4. Pyramid Distribution Density Level Curves

24 4 PAUL C. KETTLER Figure 9 Pyramid Density Scatter 3... Y Values X Values Figure 5. Scatter plot of points of the Pyramid Distribution between ±3 standard deviations in each variable Pyramid Distribution G(x,y) x-values y-values Figure 6. Pyramid Distribution Function

25 THE PYRAMID DISTRIBUTION 5 Pyramid Characteristic Function (,) values values Figure 7. Pyramid Distribution Characteristic Function Figure 3 Pyramid Characteristic Function Level values (,) values Figure 8. Pyramid Distribution Characteristic Function Level Curves

26 6 PAUL C. KETTLER Normal Characteristic Function (,) values values Figure 9. Normal Distribution Characteristic Function Figure 4 Normal Characteristic Function Level values (,) values Figure. Normal Distribution Characteristic Function Level Curves

27 THE PYRAMID DISTRIBUTION 7 Difference of Characteristic Functions...8 (,) values values Figure. Dierence of Pyramid and Normal Characteristic Functions Figure 5 Difference of Characteristic Functions Level values (,) values Figure. Dierence of Pyramid and Normal Characteristic Function Level Curves

28 8 PAUL C. KETTLER Pyramid Copula H(x,y) values values Figure 3. Pyramid Distribution Copula Independent Copula I(x,y) values values Figure 4. Independent Distribution (Product) Copula

29 THE PYRAMID DISTRIBUTION 9 Copula Difference Measure of Dependence... H(x,y) values values Figure 5. Dierence of Pyramid and Independent Copulas Figure 8 Copula Difference Level Measure of Dependence values H(x,y) values Figure 6. Dierence of Pyramid and Independent Copula Level Curves

30 3 PAUL C. KETTLER Axial Densities (both positive).3.5. All members of the family are equal here. x = -.65; f(x) = -F(x) =.88. Densities.5. theta = theta = Standard Deviations -.5 Figure 7. Axial Densities of the Pyramid Family, θ = +, θ =, θ =

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32 3 References Schmidt, R. and U. Stadtmüller (3). Nonparametric estimation of tail dependence. Research Report, London School of Economics. Schweizer, B. and E. F. Wol (98, Jul.). On nonparametric measures of dependence for random variables. Ann. Statist. 9 (4), Shih, J. H. and T. A. Louis (995, Dec.). Inferences on the association parameter in copula models for bivariate survival data. Biometrics 5 (4), Sklar, A. (959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 93. Sklar, A. (996). Random variables, distribution functions and copulas a personal look backward and forward. In L. Rüschendorf, B. Schweizer, and M. D. Taylor (Eds.), Distributions with Fixed Marginals and Related Topics, pp. 4. Hayward, California: Institute of Mathematical Statistics. Tankov, P. (3, Dec.). Dependence structure of spectrally positive multidimensional Lévy processes. Applied Probability Trust, Centre de Mathématiques Appliquées, École Polytechnique. Tankov, P. (4, Sep.). Lévy Processes in Finance: Inverse Problems and Dependence Modelling. Ph. D. thesis, Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 98 Palaiseau cedex, Paris. Rama Cont, advisor. Paul C. Kettler Centre of Mathematics for Applications Department of Mathematics University of Oslo P.O. Box 53, Blindern N36 Oslo Norway address: paulck@math.uio.no URL: paulck/