Download The Bivariate Normal Copula

arXiv:0912.2816v1 [math.PR] 15 Dec 2009
The Bivariate Normal Copula
Christian Meyer∗†
December 15, 2009
Abstract
We collect well known and less known facts about the bivariate normal
distribution and translate them into copula language. In addition, we
prove a very general formula for the bivariate normal copula, we compute
Gini’s gamma, and we provide improved bounds and approximations on
the diagonal.
1
Introduction
When it comes to modelling dependent random variables, in practice one often
resorts to the multivariate normal distribution, mainly because it is easy to parameterize and to deal with. In recent years, in particular in the quantitative
finance community, we have also witnessed a trend to separate marginal distributions and dependence structure, using the copula concept. The normal (or
Gauss, or Gaussian) copula has even come to the attention of the general public
due to its use in the valuation of structured products and the decline of these
products during the financial crisis of 2007 and 2008.
The multivariate normal distribution has been studied since the 19th century.
Many important results have been published in the 1950s and 1960s. Nowadays,
quite some effort is spent on rediscovering some of them.
In this note we will proceed as follows. We will concentrate on the bivariate normal distribution because it is the most important case in practice
(applications in quantitative finance include pricing of options and estimation
of asset correlations; the impressive list by Balakrishnan & Lai (2009) mentions applications in agriculture, biology, engineering, economics and finance,
the environment, genetics, medicine, psychology, quality control, reliability and
survival analysis, sociology, physical sciences and technology). We will give an
extensive view on its properties. Everything will be reformulated in terms of
the associated copula.
We will also provide new results (at least if the author has not been rediscovering himself...), including a very general formula implying other well known
∗ DZ BANK AG, Platz der Republik, D-60265 Frankfurt. The opinions or recommendations
expressed in this article are those of the author and are not representative of DZ BANK AG.
† E-Mail: [email protected]
1
formulas, a derivation of Gini’s gamma, and improved and very simple bounds
for the diagonal.
For collections of facts on the bivariate (or multivariate) distribution we refer
to the books of Balakrishnan & Lai (2009), of Kotz, Balakrishnan & Johnson
(2000), and of Patel & Read (1996), and to the survey article of Gupta (1963a)
with its extensive bibliography (Gupta 1963b). For theory on copulas we refer
to the book of Nelsen (2006).
We will use the symbols P, E and V for probabilities, expectation values and
variances.
2
Definition and basic properties
Denote by
2
x
1
,
ϕ(x) := √ exp −
2
2π
Φ(h) :=
Z
h
ϕ(x) dx
−∞
the density and distribution function of the standard normal distribution, and
by
2
1
x − 2̺xy + y 2
p
ϕ2 (x, y; ̺) :=
,
exp −
2(1 − ̺2 )
2π 1 − ̺2
Z h Z k
ϕ2 (x, y; ̺) dy dx,
Φ2 (h, k; ̺) :=
−∞
−∞
the density and distribution function of the bivariate standard normal distribution with correlation parameter ̺ ∈ (−1, 1). The bivariate normal (or Gauss,
or Gaussian) copula with parameter ̺ is then defined by application of Sklar’s
theorem, cf. Section 2.3 of (Nelsen 2006):
(2.1)
C(u, v; ̺) := Φ2 Φ−1 (u), Φ−1 (v); ̺
For ̺ ∈ {−1, 1}, the correlation matrix of the bivariate standard normal distribution becomes singular. Nevertheless, the distribution, and hence the normal
copula, can be extended continuously. We may define
C(u, v, −1) := lim C(u, v; ̺) = max(u + v − 1, 0),
(2.2)
C(u, v, +1) := lim C(u, v; ̺) = min(u, v).
(2.3)
̺−→−1
̺−→+1
Hence C(·, ·; ̺), for ̺ −→ −1, approaches the lower Fréchet bound,
W (u, v) := max(u + v − 1, 0),
and, for ̺ −→ 1, approaches the upper Fréchet bound,
M (u, v) := min(u, v).
2
Furthermore, we have
C(u, v; 0) = u · v =: Π(u, v),
(2.4)
the independence copula. What happens in between can be described using the
following differential equation derived by Plackett (1954):
d
d2
Φ2 (x, y; ̺) = ϕ2 (x, y; ̺) =
Φ2 (x, y; ̺)
d̺
dx dy
(2.5)
We find
C(u, v; ̺) − C(u, v; σ) = I(u, v; σ, ̺) :=
Z
̺
σ
and in particular,
ϕ2 Φ−1 (u), Φ−1 (v); r dr,
C(u, v; ̺) = W (u, v) + I(u, v; −1, ̺),
= Π(u, v) + I(u, v; 0, ̺),
= M (u, v) − I(u, v; ̺, 1).
(2.6)
(2.7)
(2.8)
(2.9)
In other words, the bivariate normal copula allows comprehensive total concordance ordering with respect to ̺, cf. Section 2.8 of (Nelsen 2006):
W (·, ·) = C(·, ·; −1) ≺ C(·, ·; ̺) ≺ C(·, ·; σ) ≺ C(·, ·; 1) = M (·, ·)
for −1 ≤ ̺ ≤ σ ≤ 1, i.e., for all u, v,
W (u, v) = C(u, v; −1) ≤ C(u, v; ̺) ≤ C(u, v; σ) ≤ C(u, v, 1) = M (u, v).
1
2
into (2.8) we obtain
Z ̺
1
1
1
1
√
C 12 , 12 ; ̺ = +
dr = +
arcsin(̺),
2
4
4
2π
2π
1
−
r
0
By plugging u = v =
(2.10)
a result already known to Stieltjes (1889).
Mehler (1866) and Pearson (1901a), among other authors, obtained the socalled tetrachoric expansion in ̺:
C(u, v; ̺) = uv +̺ Φ
−1
∞
X
̺k+1
−1
(u) ̺ Φ (v)
Hek Φ−1 (u) Hek Φ−1 (v)
(k + 1)!
k=0
(2.11)
where
[k/2]
Hek (x) =
X
i=0
are the Hermite polynomials.
k!
i!(k − 2i)!
3
i
1
−
xk−2i
2
The bivariate normal copula inherits the symmetries of the bivariate normal
distribution:
C(u, v; ̺) = C(v, u; ̺)
= u − C(u, 1 − v; −̺)
= v − C(1 − u, v; −̺)
= u + v − 1 + C(1 − u, 1 − v; ̺)
(2.12)
(2.13)
(2.14)
(2.15)
Here, (2.12) is a consequence of exchangeability, and (2.15) is a consequence of
radial symmetry, cf. Section 2.7 of (Nelsen 2006).
In the following sections we will discuss formulas for the bivariate normal
copula, its numerical evaluation, bounds and approximations, measures of concordance, and univariate distributions related to the bivariate normal copula.
It will be convenient to assume ̺ 6∈ {−1, 0, 1} unless explicitly stated otherwise
(cf. (2.2), (2.4), (2.3) for the simple formulation in the missing cases).
3
Formulas
If (X, Y ) are bivariate standard normally distributed with correlation ̺ then
Y conditional on X = x is normally distributed with expectation ̺x and variance 1 − ̺2 . This translates into the following formulas for the conditional
distributions of the bivariate normal copula:
!
∂
Φ−1 (v) − ̺ · Φ−1 (u)
p
,
(3.1)
C(u, v; ̺) = Φ
∂u
1 − ̺2
!
∂
Φ−1 (u) − ̺ · Φ−1 (v)
p
(3.2)
C(u, v; ̺) = Φ
∂v
1 − ̺2
The copula density is given by:
ϕ2 Φ−1 (u), Φ−1 (v); ̺
∂2
c(u, v; ̺) :=
C(u, v; ̺) =
∂u ∂v
ϕ (Φ−1 (u)) ϕ (Φ−1 (v))
1
= p
exp
1 − ̺2
2̺Φ−1 (u)Φ−1 (v) − ̺2 Φ−1 (u)2 + Φ−1 (v)2
2(1 − ̺2 )
(3.3)
!
In terms of its copula density, the bivariate normal copula can be written as
Z uZ v
c(s, t; ̺) dt ds.
(3.4)
C(u, v; ̺) =
0
0
4
Now let α, β, γ ∈ (−1, 1) with αβγ = ̺. Then the following holds:
C(u, v; ̺)
Z ∞Z
=
−∞
∞
Φ
−∞
Φ−1 (u) − α · x
√
1 − α2
Φ
Φ−1 (v) − β · y
p
1 − β2
!
ϕ2 (x, y; γ) dy dx (3.5)
!
Φ−1 (v) − β · Φ−1 (t)
Φ−1 (u) − α · Φ−1 (s)
√
p
c(s, t; γ) dt ds
Φ
=
Φ
1 − α2
1 − β2
0
0
Z 1Z 1
∂
∂
=
C(u, s; α)
C(t, v; β) c(s, t; γ) dt ds
∂s
∂t
0
0
Z
1
Z
1
The right hand side of (3.5) has occurred in credit risk modelling but without
the link to the bivariate normal copula, cf. (Bluhm & Overbeck 2003). A proof
in that context will be provided in Section A.1.
Other formulas for C(u, v; ̺) are obtained by carefully studying the limit as
some of the variables α, β, γ are approaching the value one. The interesting
cases, not regarding symmetry, are listed in Table 1.
α
=α
=1
=α
=1
=1
β
=β
=β
=β
=̺
=1
γ
=γ
=γ
=1
=1
=̺
reference
(3.5)
(3.6)
(3.7)
(3.8)
(3.4)
Table 1: Limiting cases of (3.5)
By approaching α = 1 in (3.5) we obtain:
!
Φ−1 (v) − β · y
p
C(u, v; ̺) =
Φ
ϕ2 (x, y; γ) dy dx
1 − β2
−∞
−∞
!
Z uZ 1
Φ−1 (v) − β · Φ−1 (t)
p
c(s, t; γ) dt ds
=
Φ
1 − β2
0
0
Z uZ 1
∂
=
C(u, t; β) c(s, t; γ) dt ds
∂t
0
0
Z
Φ−1 (u)
Z
∞
(3.6)
However, Equation (3.6) may be considered rather unattractive. By approaching
5
γ = 1 in (3.5) instead we obtain a more interesting formula:
!
Z ∞ −1
Φ−1 (v) − β · z
Φ (u) − α · z
√
p
ϕ(z) dz
(3.7)
C(u, v; ̺) =
Φ
Φ
1 − α2
1 − β2
−∞
!
Z 1 −1
Φ (u) − α · Φ−1 (t)
Φ−1 (v) − β · Φ−1 (t)
√
p
=
Φ
dt
Φ
1 − α2
1 − β2
0
Z 1
∂
∂
=
C(t, v; α)
C(u, t; β) dt
∂t
∂t
0
Equation (3.7) seems to have been discovered repeatedly, sometimes in more
general context (e.g., multivariate, cf. (Steck & Owen 1962)), sometimes for
special cases. Gupta (1963a) gives credit to Dunnett & Sobel (1955).
By approaching α = γ = 1 in (3.5) we obtain:
!
Z u
Z u
∂
Φ−1 (v) − ̺ · Φ−1 (t)
p
dt =
C(u, v; ̺) =
Φ
C(t, v; ̺) dt
(3.8)
2
∂t
1−̺
0
0
!
Z v
Z v
∂
Φ−1 (u) − ̺ · Φ−1 (t)
p
dt =
C(u, t; ̺) dt
(3.9)
=
Φ
2
1−̺
0 ∂t
0
These formulas can also be derived from (3.1) and (3.2). Finally, by approaching
α = β = 1 in (3.5) we rediscover (3.4).
In order to evaluate the bivariate normal distribution function numerically,
Owen (1956) defined the following very useful function, to which we will refer
as Owen’s T -function:
Z a
Z a
exp − 12 h2 (1 + x2 )
1
ϕ(hx)
T (h, a) =
dx
=
ϕ(h)
dx
(3.10)
2
2π 0
1 + x2
0 1+x
He proved that
u+v
− T Φ−1 (u), αu − T Φ−1 (v), αv − δ(u, v)
2
(
1
, if u < 12 , v ≥ 21 or u ≥ 12 , v < 21
δ(u, v) := 2
0, else
C(u, v; ̺) =
where
(3.11)
(3.12)
and
1
αu = p
1 − ̺2
Φ−1 (v)
−̺ ,
Φ−1 (u)
1
αv = p
1 − ̺2
In particular, on the lines defined by v =
hold:
1
2; ̺
u
= −T
2
1
2
Φ−1 (u)
−̺ .
Φ−1 (v)
and by u = v, the following formulas
−1
̺
!
,
(u), − p
1 − ̺2
r
1−̺
C(u, u; ̺) = u − 2 · T Φ−1 (u),
1+̺
C u,
Φ
6
(3.13)
(3.14)
(3.15)
From (3.11) and (3.14) we can derive the useful formula
C(u, v; ̺) = C u, 12 ; ̺u + C v, 21 ; ̺v − δ(u, v),
(3.16)
where
αu
̺u = − p
= sin(arctan(−αu )),
1 + α2u
αv
= sin(arctan(−αv )).
̺v = − p
1 + α2v
On the diagonal u = v, (3.16) reads:
C(u, u; ̺) = 2 · C
u,
1
2; −
r
1−̺
2
Inversion of (3.17) using (2.13) gives:
(
1
C(u, u; 1 − 2̺2 ),
1
C u, 2 ; ̺ = 2 1
u − 2 C(u, u; 1 − 2̺2 ),
!
(3.17)
̺ < 0,
̺ > 0.
(3.18)
Applying (3.8) to (3.17) we obtain, cf. also (Steck & Owen 1962),
Z u
C(u, u; ̺) = 2 ·
g(t; ̺) dt
(3.19)
r
(3.20)
0
with
g(u; ̺) := Φ
We find
1−̺
−1
· Φ (u) .
1+̺
d
C(u, u; ̺) = 2 · g(u; ̺).
(3.21)
du
The function g will become important in Section 5. Note that if U , V are
uniformly distributed on [0, 1] with copula C(·, ·; ̺) then
g(u; ̺) = P(V ≤ u | U = u).
Below we list some properties of g:
lim g(u; ̺) = 0,
(3.22)
lim g(u; ̺) = 1,
(3.23)
g( 12 ; ̺) = 12 ,
(3.24)
u−→0+
u−→1−
g(1 − u; ̺) = 1 − g(u; ̺),
g(g(u; ̺); −̺) = u
(3.25)
(3.26)
In particular, (3.22) and (3.23) show that the bivariate normal copula does not
exhibit (lower or upper) tail dependence (cf. Section 5.2.3 of Embrechts, Frey
& McNeil (2005)).
Substitution of t = g(s; ̺) in (3.19) and application of (3.26) lead to the
identity, cf. also (Steck & Owen 1962):
C(u, u; ̺) = 2u · g(u; ̺) − C (g(u; ̺), g(u; ̺); −̺)
7
(3.27)
4
Numerical evaluation
The bivariate normal copula has to be evaluated numerically. To my knowledge,
in the literature there is no direct approach. Hence for the time being we have
to rely on the definition of the bivariate normal copula (2.1) and numerical evaluation of Φ2 and Φ−1 . There are excellent algorithms available for evaluation of
Φ−1 , cf. (Acklam 2004); the main problem is evaluation of the bivariate normal
distribution function Φ2 .
In the literature on evaluation of Φ2 there are basically two approaches:
application of a multivariate method to the bivariate case, and explicit consideration of the bivariate case. For background on multivariate methods we refer
to the recent book by Bretz & Genz (2009). In most cases, bivariate methods
will be able to obtain the desired accuracy in less time. In the following we
will provide an overview on the literature. We will concentrate on methods
and omit references dealing with implementation only. Comparisons of different approaches in terms of accuracy and running time have been provided by
numerous authors, e.g., Aǧca & Chance (2003), Terza & Welland (1991), and
Wang & Kennedy (1990).
Before the advent of widely available computer power, extensive tables of the
bivariate normal distribution function had to be created. Using (3.16) or similar
approaches, the three-dimensional problem (two variables and the correlation
parameter) was reduced to a two-dimensional one.
Pearson (1901a) used the tetrachoric expansion (2.11) for small |̺|, and
quadrature for large |̺|. Nicholson (1943), building on ideas of Sheppard (1900),
worked with a two-parameter function, denoted V -function. Owen (1956) introduced the T -function (3.10) which is closely related to Nicholson’s V -function.
For many years, quadrature of the T -function was the method of choice for
evaluation of the bivariate normal distribution. Numerous authors, e.g. Borth
(1973), Daley (1974), Young & Minder (1974), and Patefield & Tandy (2000),
have been working on improvements, e.g. by dividing the plane into many
regions and choosing specific quadrature methods in each region.
Sowden & Ashford (1969) applied Gauss-Hermite quadrature to (3.7) and
Simpson’s rule to (3.8). Drezner (1978) used (3.16) and Gauss quadrature. Divgi
(1979) relied on polar coordinates and an approximation to the univariate Mills’
ratio. Vasicek (1998) proposed an expansion which is more suitable for large |̺|
than the tetrachoric expansion (2.11).
Drezner & Wesolowsky (1990) applied Gauss-Legendre quadrature to (2.8)
for |̺| ≤ 0.8, and to (2.9) for |̺| > 0.8. Improvements of their method in terms
of accuracy and robustness have been provided by Genz (2004) and West (2005).
Most implementations today will rely on variants of the approaches of Divgi
(1979) or of Drezner & Wesolowsky (1990). The method of Drezner (1978),
although less reliable, is also very common, mainly because it is proposed in
(Hull 2008) and other prevalent books.
8
5
Bounds and approximations
Nowadays, high-precision numerical evaluation of the bivariate normal copula
is usually available and there is not much need for low-precision approximations
anymore, in particular if the mathematics is hidden behind strange constants
derived from some optimization procedure.
On the other hand, if the mathematics is general and transparent then the
resulting approximations are often rather weak. As an example, take approximations to the multivariate Mills’ ratios applied to the bivariate case, cf. (Lu
& Li 2009) and the references therein.
Bounds, if they are not too weak, are more interesting than approximations
because they can be used, e.g., for checking numerical algorithms. Moreover,
derivation of bounds often provides valuable insight into the mathematics behind
the function to be bounded.
In the following we concentrate on bounds and approximations explicitly
derived for the bivariate case. Throughout this section we will only consider the
case ̺ > 0, 0 < u = v < 1/2. Note that by successively applying, if required,
(3.16), (3.18), (3.27) and (2.15), we can always reduce C(u, v; ̺) to a sum of two
terms of that form. Any approximation or bound given for the special case can
be translated to an approximation or bound for the general case, with at most
twice the absolute error. Note also that for many existing approximations and
bounds the diagonal u = v may be considered a worst case, cf. (Willink 2004).
Mee & Owen (1983) elaborated on the so-called conditional approach proposed by Pearson (1901b). If (X, Y ) are bivariate standard normally distributed
with correlation ̺ then we can write
Φ2 (h, k; ̺) = Φ(h) · P(Y ≤ k | X ≤ h).
(5.1)
The distribution of Y conditional on X = h is normal but the distribution of Y
conditional on X ≤ h is not. Nevertheless, it can be approximated by a normal
distribution with the same mean and variance. In terms of the bivariate normal
copula, the resulting approximation is
!
u · Φ−1 (u) + ̺ · ϕ Φ−1 (u)
C(u, u; ̺) ≈ u · Φ p
. (5.2)
u2 − ̺2 · ϕ (Φ−1 (u)) · (u · Φ−1 (u) + ϕ (Φ−1 (u)))
The approximation works well for |̺| not too large. For |̺| large there are
alternative approximations, e.g. (Albers & Kallenberg 1994).
The simpler of the two approximations proposed by Cox & Wermuth (1991)
replaces the second factor in (5.1) by the mean of the conditional distribution
(the more complicated approximation adds a term of second order). In terms
of the bivariate normal copula, the resulting approximation is
!
r
1 − ̺ u · Φ−1 (u) + ̺ · ϕ Φ−1 (u)
.
·
C(u, u; ̺) ≈ u · Φ
1+̺
(1 + ̺) · u
9
Mallows (1959) gave two approximations to Owen’s T -function (3.10). In
terms of the bivariate normal copula, the simpler one reads
r
1−̺
3
u 1
C(u, u; ̺) ≈ 2u · Φ
− Φ−1
.
· Φ−1
+
1+̺
2 4
4
Further approximations were derived by Cadwell (1951) and Pólya (1949).
There are not too many bounds available in the literature. For ̺ > 0 there
are, of course, the trivial bounds (2.2) and (2.3):
u2 ≤ C(u, u; ̺) ≤ u
The upper bound given by Pólya (1949) is just the one above. His lower bound is
too weak (even negative) on the diagonal. A recent overview on known bounds,
and derivation of some new ones, is provided by Willink (2004). We will present
some of his bounds below in more general context.
Theorem 5.1 Let ̺ ≥ 0 and 0 ≤ u ≤ 1/2. Then C(u, u; ̺) is bounded as
follows, where g(u; ̺) is defined as in (3.20):
C(u, u; ̺) ≥ u · g(u; ̺)
(5.3)
C(u, u; ̺) ≤ u · g(u; ̺) · 2
(5.4)
The lower bound (5.3) is tight for ̺ = 0 or u = 0. The maximum error of (5.3)
equals 1/4 and is obtained for u = 1/2, ̺ = 1. The upper bound (5.4) is tight
for ̺ = 1 or u = 0. The maximum error of (5.4) equals 1/4 and is obtained for
u = 1/2, ̺ = 0.
A proof of Theorem 5.1 is provided in Section A.2, together with a proof of
the following refinement:
Theorem 5.2 Let ̺ > 0 and 0 ≤ u ≤ 1/2. Then C(u, u; ̺) is bounded as
follows, where g(u; ̺) is defined as in (3.20):
2
(5.5)
C(u, u; ̺) ≥ u · g(u; ̺) · 1 + arcsin(̺)
π
C(u, u; ̺) ≤ u · g(u; ̺) · (1 + ̺)
(5.6)
These bounds are the optimal ones of the form u · g(u) · a(̺). They are tight for
̺ =q
0, ̺ = 1, or u = 0. The maximum error of (5.6) is obtained for u = 1/2,
̺ = 1 − π42 ≈ 0.7712, the value being
1
4
r
4
2
1 − 2 − arcsin
π
π
r
4
1− 2
π
The lower bound (5.5) is tight for u = 1/2.
10
!!
≈ 0.05263.
The bounds (5.3) and (5.6) have been discussed, without explicit computation of the maximum error, by Willink (2004). The maximum error of (5.5) is
difficult to grasp analytically. Numerically, the error always stays below 0.006.
An alternative upper bound is given by the following theorem, the proof of
which is provided in Section A.3:
Theorem 5.3 Let ̺ > 0 and 0 ≤ u ≤ 1/2. Then
u C(u, u; ̺) ≤ 2u · g
;̺ .
2
The bound is tight for ̺ = 0, ̺ = 1, or u = 0. The maximum error is obtained
for u = 1/2,
r
1−̺
1+̺
−1 1
−
=ϕ
· Φ (4) ,
1+̺
2π · Φ−1 ( 14 )
i.e., ̺ ≈ 0.5961, the value being approx. 0.015.
It is also possible to derive good approximations to C(u, u; ̺) by considering
the family C(u, u; ̺) ≈ u · g(u; ̺) · (a(̺) + b(̺)u). In particular, the choice
2
4
a(̺) := 1 + ̺,
b(̺) := 2 · 1 + arcsin(̺) − (1 + ̺) = arcsin(̺) − 2̺
π
π
is attractive because the resulting approximation
4
arcsin(̺) − 2̺ u
C(u, u; ̺) ≈ u · g(u; ̺) · 1 + ̺ +
π
(5.7)
is tight for ̺ = 0, ̺ = 1, u = 0, or u = 1/2, and for u −→ 0+ it has the same
asymptotic behaviour as (5.6). By visual inspection we may conjecture that it
is even an upper bound, with an error almost cancelling the error of the lower
bound (5.5) most of the time. Consequently, an even better approximation (but
not a lower bound, for ̺ large) is given by
̺
2
C(u, u; ̺) ≈ u · g(u; ̺) · 1 + +
arcsin(̺) − ̺ u .
(5.8)
2
π
Again, (5.8) is tight for ̺ = 0, ̺ = 1, u = 0, or u = 1/2. Numerically,
the absolute error always stays below 0.0006. Hence (5.8) is comparable in
performance with (5.2), and much better for ̺ large.
6
Measures of concordance
In the study of dependence between (two) random variables, properties and
measures that are scale-invariant, i.e., invariant under strictly increasing transformations of the random variables, can be expressed in terms of the (bivariate)
copula of the random variables. Among these are the so-called measures of
11
concordance, in particular Kendall’s tau, Spearman’s rho, Blomqvist’s beta and
Gini’s gamma. For background and general definitions and properties we refer to
Section 5 of (Nelsen 2006). In this section we will provide formulas for measures
of concordance for the bivariate normal copula, depending on the correlation
parameter ̺.
Blomqvist’s beta follows immediately from (2.10):
β(̺) := 4 · C 12 , 21 ; ̺ − 1
2
= arcsin(̺)
(6.1)
π
For the bivariate normal copula, Kendall’s tau equals Blomqvist’s beta:
Z 1Z 1
τ (̺) := 4
C(u, v; ̺) dC(u, v; ̺) − 1
0
0
Z 1Z 1
∂
∂
C(u, v; ̺)
C(u, v; ̺) du dv
=1−4
∂v
∂u
0
0
2
= arcsin(̺)
π
(6.2)
For a proof of (6.1) and (6.2) cf. Section 5.3.2 of (Embrechts et al. 2005). Both
Blomqvist’s beta and Kendall’s tau can be generalized to (copulas of) elliptical
distributions, cf. (Lindskog, McNeil & Schmock 2003). This is not the case for
Spearman’s rho, cf. (Hult & Lindskog 2002), which is given by:
Z 1Z 1
C(u, v; ̺) − uv du dv
̺S (̺) := 12
0
= 12
Z
0
=
0
1
Z
0
1
C(u, v; ̺) du dv − 3
̺
6
arcsin
π
2
(6.3)
For proofs of (6.3) cf. (Kruskal 1958) or Section 5.3.2 of (Embrechts et al. 2005).
Gini’s gamma for the bivariate normal copula is given as follows:
Z 1
Z 1
1
γ(̺) := 4
C(u, 1 − u; ̺) du −
C(u, u; ̺) du +
2
0
0
Z 1
Z 1
1
=4
u − C(u, u; −̺) du −
C(u, u; ̺) du +
2
0
0
2
1+̺
1−̺
=
arcsin
− arcsin
(6.4)
π
2
2
√
√
1+̺
1−̺
4
=
arcsin
− arcsin
(6.5)
π
2
2
p
1 p
4
= arcsin
(1 + ̺)(3 + ̺) − (1 − ̺)(3 − ̺)
(6.6)
π
4
12
I have not been able to find proofs in the literature, hence they will be provided
in Section A.4.
Equation (6.6) can be inverted which may be useful for estimation of ̺ from
an estimate for γ(̺):
r
π
π
̺ = sin γ(̺) ·
3 − tan γ(̺) ·
(6.7)
4
4
Finally, we propose a new measure of concordance, similar to Gini’s gamma
but based on the lines u = 1/2, v = 1/2 instead of the diagonals. For a bivariate
copula C it is defined by:
Z 1
Z 1
v
u
(6.8)
C 21 , v − dv
γ̃(C(·, ·)) := 4
C u, 21 − du +
2
2
0
0
Z 1
Z 1
1
C 21 , v dv −
=4
C u, 21 du +
2
0
0
For the bivariate normal copula we obtain
γ̃(̺) := γ̃(C(·, ·; ̺)) =
4
arcsin
π
̺
√
2
.
(6.9)
A proof is given implicitly in Section A.4.
7
Univariate distributions
In this section we will discuss two univariate distributions being closely related
to the bivariate normal copula (or distribution).
7.1
The skew-normal distribution
A random variable X on R is skew-normally distributed with skewness parameter
λ ∈ R if it has a density function of the form
fλ (x) = 2ϕ(x)Φ(λx).
(7.1)
The skew-normal distribution was introduced by O’Hagan & Leonard (1976) and
studied and made popular by Azzalini (1985, 1986). Its cumulative distribution
function is given by
P(X ≤ x) =
Z
x
2ϕ(x)Φ(λx) = 2
Z
0
−∞
13
Φ(x)
Φ λΦ−1 (t) dt.
(7.2)
In the light of (3.19) and (3.17), cf. also (Azzalini & Capitanio 2003), we find
λ
(7.3)
P(X ≤ x) = 2Φ2 x, 0; − √
1 + λ2
 1 − λ2


,
λ ≥ 0,
Φ2 x, x;
2
1+λ
=
(7.4)
1 − λ2


,
λ
≤
0.
1 − Φ2 −x, −x;
1 + λ2
In particular, the bounds given in Theorem 5.2 can be applied.
7.2
The Vasicek distribution
A random variable P on the interval [0, 1] is Vasicek distributed with parameters
p ∈ (0, 1) and ̺ ∈ (0, 1) if Φ−1 (P ) is normally distributed with mean
Φ−1 (p)
E(Φ−1 (P )) = √
1−̺
(7.5)
̺
.
1−̺
(7.6)
and variance
V(Φ−1 (P )) =
In Section A.1 implicitly it is proved that
E(P ) = p,
E(P 2 ) = C(p, p; ̺),
so that
V(P ) = E(P 2 ) − E(P )2 = C(p, p; ̺) − p2 .
Furthermore, we have
P(P ≤ q) = P(Φ
−1
(P ) ≤ Φ
−1
√
1 − ̺ · Φ−1 (q) − Φ−1 (p)
.
(q)) = Φ
√
̺
The (one-sided) α-Quantile qα of P , with α ∈ (0, 1), is therefore given by
√
̺ · Φ−1 (α) + Φ−1 (p)
√
qα = Φ
.
(7.7)
1−̺
In particular, the median of P is simply
−1 Φ (p)
= Φ E(Φ−1 (P )) .
q0.5 = Φ √
1−̺
The density of P is
d
P(P ≤ q) =
dq
r
1−̺
·ϕ
̺
√
1 − ̺ · Φ−1 (q) − Φ−1 (p)
1
·
.
√
̺
ϕ (Φ−1 (q))
14
(7.8)
The distribution is unimodal with the mode at
√
1−̺
Φ
· Φ−1 (p)
1 − 2̺
for ̺ < 0.5, monotone for ̺ = 0.5, and U-shaped for ̺ > 0.5.
Let P̃ be Vasicek distributed with parameters p̃, ̺˜, and let
corr Φ−1 (P ), Φ−1 (P̃ ) = γ.
Then
and
s
̺˜
̺
·
,
1 − ̺ 1 − ̺˜
p
E P · P̃ = C p, p̃; γ · ̺ · ̺˜ ,
cov Φ−1 (P ), Φ−1 (P̃ ) = γ ·
p
cov P, P̃ = C p, p̃; γ · ̺ · ̺˜ − p · p̃.
The Vasicek distribution does not offer immediate advantages over other
two-parametric continuous distributions on (0, 1), such as the beta distribution.
Its importance stems from its occurrence as mixing distribution in linear factor
models set up as in Section A.1. It is a special case of a probit-normal distribution; it is named after Vasicek who introduced it into credit risk modeling.
For (different) details on the material in this section we refer to Vasicek
(1987, 2002) and Tasche (2008). Estimation of the parameters p and ̺ is also
discussed by Meyer (2009).
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A
A.1
Proofs
Proof of (3.5)
Let
p
1 − α2 · ǫ,
p
X̃ = β · Ỹ + 1 − β 2 · ǫ̃,
X =α·Y +
where α, β ∈ (−1, 1) \ {0} are parameters and where Y , Ỹ , ǫ, ǫ̃ are all standard
normal and pairwise independent, except
γ := corr Y, Ỹ = cov Y, Ỹ .
By construction, X and X̃ are standard normal again with
corr X, X̃ = cov X, X̃ = αβγ.
We define indicator variables Z = Z(X) ∈ {0, 1}, Z̃ = Z̃(X̃) ∈ {0, 1} calibrated
to expectation values u, v:
Z=1
:⇐⇒
Z̃ = 1
:⇐⇒
19
X ≤ Φ−1 (u),
X̃ ≤ Φ−1 (v)
Conditional on (Y = y, Ỹ = ỹ), Z and Z̃ are independent. We find
P(Z = 1 | Y = y) = P X ≤ Φ−1 (u) | Y = y
p
= P α · y + 1 − α2 · ǫ ≤ Φ−1 (u)
−1
Φ (u) − α · y
Φ−1 (u) − α · y
√
√
=Φ
.
=P ǫ≤
1 − α2
1 − α2
Now we define the random variables
P := P (Y ) := Φ
Φ−1 (u) − α · Y
√
1 − α2
Φ−1 (v) − β · Y
p
1 − β2
P̃ := P̃ (Ỹ ) := Φ
We find
!
,
.
u = E(Z) = E(E(Z | Y )) = E(P(Z = 1 | Y )) = E(P ),
v = E(P̃ )
and
P Z = 1, Z̃ = 1 = P X ≤ Φ−1 (u), X̃ ≤ Φ−1 (v)
= Φ2 Φ−1 (u), Φ−1 (v), cov(X, X̃)
= Φ2 Φ−1 (u), Φ−1 (v), αβγ .
On the other hand,
P Z = 1, Z̃ = 1 = P Z Z̃ = 1 = E Z Z̃
= E E(Z Z̃ | Y, Ỹ ) = E E(Z | Y, Ỹ ) · E(Z̃ | Y, Ỹ )
= E E(Z | Y ) · E(Z̃ | Ỹ ) = E P · P̃
Z ∞Z ∞
=
P (x) · P̃ (y) · ϕ2 (x, y, γ) dx dy
−∞
=
Z
∞
−∞
A.2
−∞
Z
∞
−∞
Φ
Φ−1 (u) − α · x
√
1 − α2
Φ
Φ−1 (v) − β · y
p
1 − β2
!
Proof of Theorems 5.1 and 5.2
We will assume ̺ as fixed and write
C(u) := C(u, u; ̺),
20
g(u) := g(u; ̺).
ϕ2 (x, y, γ) dx dy.
The upper bound (5.4) follows from (3.27). Regarding the lower bound (5.3)
we note that
r
1−̺
̺
′
g (u) =
· exp
· Φ−1 (u)2 > 0,
1+̺
1+̺
2̺
Φ−1 (u)
g ′′ (u) = g ′ (u) ·
·
< 0.
1 + ̺ ϕ(Φ−1 (u))
Hence g is increasing and concave on (0, 1/2) and we conclude
Z u
1
C(u) = 2
g(t) dt ≥ 2 · · u · g(u) = u · g(u).
2
0
Now we define
Da (u) := u · g(u) · a
with a ∈ [1, 2], u ∈ [0, 1/2]. We start by noting that
C ′ (u) = 2g(u) > 0,
C ′′ (u) = 2g ′ (u) > 0,
hence C is increasing and convex on (0, 1/2). Furthermore,
Da′ (u) = a · (g(u) + u · g ′ (u)) > 0,
Da′′ (u) = a · (2 · g ′ (u) + u · g ′′ (u))
1
u · Φ−1 (u)
= 2a · g ′ (u) · 1 +
·
1 + ̺ ϕ (Φ−1 (u))
1
· H −Φ−1 (u)
= 2a · g ′ (u) · 1 −
1+̺
with H(x) = x · R(x), where
1 − Φ(x)
ϕ(x)
R(x) =
is Mills’ ratio. Pinelis (2002) has shown that H ′ (x) > 0 for x > 0, H(0) = 0,
and limx−→∞ H(x) = 1. Hence Da (u) is increasing and convex on (0, 1/2) as
well.
For u ∈ (0, 1/2), C ′ (u) = Da′ (u) is equivalent with
f (u) :=
2−a
u · g ′ (u)
=
,
g(u)
a
or
a=
2
.
1 + f (u)
We will show that f is strictly increasing on (0, 1/2). We have
Φ Φ−1 (u) · ϕ λ · Φ−1 (u)
f (u) = λ ·
= λ · Fλ −Φ−1 (u)
−1
−1
ϕ (Φ (u)) · Φ (λ · Φ (u))
q
with λ = 1−̺
1+̺ ∈ [0, 1] and
Fλ (x) :=
R(x)
,
R(λ · x)
21
x ≥ 0.
We find
R′ (x) · R(λ · x) − R(x) · R′ (λ · x)
R(λ · x)2
′
R (x)
R′ (λ · x)
= Fλ (x) ·
<0
−λ·
R(x)
R(λ · x)
Fλ′ (x) =
for λ < 1. Here we have used that Fλ (x) > 0 and that the function
y 7→ y ·
R′ (y)
R(y)
is strictly decreasing on (0, ∞), cf. (Baricz 2008). We conclude
Fλ′ −Φ−1 (u)
′
f (u) = −λ ·
> 0.
ϕ (−Φ−1 (u))
Furthermore, we find
f
and
1
2
R(0)
=
=λ·
R(0)
r
1−̺
1+̺
g ′ (u) + u · g ′′ (u)
g ′ (u)
u−→0+
u · Φ−1 (u)
2̺
·
= lim + 1 +
1 + ̺ ϕ(Φ−1 (u))
u−→0
2̺
· H −Φ−1 (u)
= lim 1 −
1+̺
u−→0+
2̺
1−̺
=1−
=
.
1+̺
1+̺
lim f (u) = lim
u−→0+
We have C(0) = Da (0) = 0, C ′ (0) = Da′ (0) = 0, and
lim +
u−→0
D′ (u)
a
a
Da (u)
= lim + a′
= lim + (1 + f (u)) =
.
C(u)
1+̺
u−→0 C (u)
u−→0 2
By standard calculus we conclude that
• For a ≥ 1 + ̺ we have Da′ (u) ≥ C ′ (u), and hence Da (u) ≥ C(u), for all
u ∈ [0, 1/2];
q
−1
• For a ≤ 2 · 1 + 1−̺
we have Da′ (u) ≤ C ′ (u), and hence Da (u) ≤
1+̺
C(u), for all u ∈ [0, 1/2];
• For
a∈
!
r
−1
1−̺
,1 + ̺
2· 1+
1+̺
22
there exists u0 ∈ (0, 1/2) with Da′ (u) < C ′ (u) for u ∈ (0, u0 ), and Da′ (u) >
C ′ (u) for u ∈ (u0 , 1/2). Consequently, the best lower bound for C of
the form Da is obtained if C(1/2) = Da (1/2), i.e., a = 1 + π2 arcsin(̺).
Moreover, the upper bound Da with a = 1 + ̺ can not be improved.
The maximum error of Da with a = 1 + ̺ is attained if
d
1
1
= 0,
[Da (1/2) − C(1/2)] = − p
d̺
4 2π 1 − ̺2
q
which is equivalent with ̺ = 1 − π42 .
A.3
Proof of Theorem 5.3
We will assume ̺ as fixed and write
C(u) := C(u, u; ̺),
We have
C(u) = 2 ·
with
Z
0
g(u) := g(u; ̺).
u
g(t) dt = 2u · g(v(u))
v(u) := v(u; ̺) ≤ u.
Since g is concave and increasing, we even know that
u
v(u) ≤
2
and hence
u
.
C(u) = 2u · g(v(u)) ≤ 2u · g
2
Moreover, for the same reason we have
u
u
u u d · g′
2u · g
− C(u) = 2
− g(u) − g
≥ 0,
du
2
2
2
2
and hence, for ̺ fixed, the maximum error is obtained for u = 1/2, the value
being
r
1−̺
1
1
1
− −
· Φ−1
arcsin(̺).
Φ
1+̺
4
4 2π
Derivation of the above expression with respect to ̺ gives the result.
Note that by (3.26) we can write
C(u)
; −̺
v(u; ̺) = g
2u
Unfortunately, for large ̺, the function v is not convex, and the approximation
u C( 1 )
2u · g
· 12
2
g( 4 )
is not an upper bound for C(u).
23
A.4
Proof of (6.4), (6.5), (6.6)
In a first step, using (2.8) we find:
Z 1
Z 1
Z ̺
Φ−1 (u)2
1
1
2
√
dr du
exp −
C(u, u; ̺) du =
u +
2π 0
1+r
1 − r2
0
0
Z ̺
Z ∞
1
1
v2
1
√
= +
dv dr
ϕ(v) exp −
3 2π 0
1+r
1 − r2 −∞
r
Z ̺
Z ∞
1
1
1
1+r
√
ds dr
= +
ϕ(s)
3 2π 0
3+r
1 − r2 −∞
Z ̺
1
1
1
p
= +
dr
3 2π 0
(1 − r)(3 + r)
Z 1+̺
2
1
1
1
√
dr
= +
3 2π 12
1 − r2
1+̺
1
1
1
arcsin
− arcsin
= +
3 2π
2
2
1
1
1+̺
= +
.
arcsin
4 2π
2
We conclude that
Z 1
Z 1
1
γ(̺) = 4
u − C(u, u; −̺) du −
C(u, u; ̺) du +
2
0
0
1 1
1
1
1
1+̺
1−̺
1
+ − −
−
+
arcsin
arcsin
=4
4 2π
2
2 4 2π
2
2
2
1+̺
1−̺
=
arcsin
− arcsin
.
π
2
2
In a similar way, using again (2.8), we can compute
Z 1
Z ̺
Z 1
Φ−1 (u)2
1
1
u
√
exp −
C u, 21 ; ̺ du =
dr du
+
2π 0
2(1 − r2 )
1 − r2
0
0 2
1
̺
1
,
arcsin √
= +
4 2π
2
which, using (3.17), leads to
√
Z 1
Z 1 q
1
1
1−̺
C(u, u; ̺) du = 2 ·
C u, 21 ; − 1−̺
.
−
arcsin
du
=
2
2 π
2
0
0
We obtain alternative formulas for Gini’s gamma, the second one using the
addition theorem for the arcsin function:
√
√
4
1+̺
1−̺
γ(̺) =
arcsin
− arcsin
π
2
2
p
p
1
4
= arcsin
(1 + ̺)(3 + ̺) − (1 − ̺)(3 − ̺)
π
4
24