Martín Luther King, Jr
6. Límites llenos de gracia
parameter estimates which is based on the distribution of the considered point estimates, e.g. the maximum likelihood estimators. The second idea is to use several statistical estimator to create an uncertainty set.
4.2
Elliptical distributions
In this section we introduce some fundamental characteristics of elliptical distri- butions that can be used to model a financial market. We only collect the basic definitions and properties, and refer to Fang, Kotz and Ng [29] and Fang and Zhang [30] for further results and more details.
Definition 4.1. A random vector R ∈ Rnis said to have a spherical distribution if
OR = Rd
for every orthogonal matrix O ∈ Rn×n, and with “=” denoting equality of distri-d butions.
The following theorem summarizes some useful equivalence properties for the basic class of spherical distributions which will later be extended to the class of elliptical distributions.
Theorem 4.2. Let R ∈ Rn be a random vector. Then, the following statements are equivalent:
(i) OR= R for every orthogonal matrix O ∈ Rd n×n.
(ii) There exists a function φ : R → R, called the characteristic generator, such that the characteristic function ψ of R has the form
ψ(t) = EheitTRi= φ(tTt).
(iii) The vector R has a stochastic representation of the form R= Zud (n)
with the generating random variable Z ∈ R, Z ≥ 0 being independent of u(n), a uniformly distributed random vector on the unit sphere in Rn. Proof. See Fang, Kotz and Ng [29], Theorem 2.5.
A spherically distributed random vector R does not necessarily have a prob- ability density function (pdf). But in case a density function ϕR : Rn → R
exists, it must be of the form ϕR(x) = ξR(xTx) (analogous to φ(tTt)) for some
ξR : R+ → R+ which is called the density generator3. Furthermore, we obtain
the following results.
Proposition 4.3. Let R = Zud (n) be spherically distributed. Then, R has a density generator ξR : R → R if and only if the generating variate Z has a
probability density function ϕZ : R → R. Furthermore, the relationship between
these two functions is analytically given by ϕZ(z) =
2πn2
Γ(n2) · z
n−1ξ
R(z2).
Additionally, if R possesses a probability density function, then all the marginal densities exist as well.
Proof. See Fang, Kotz and Ng [29], Theorems 2.9 and 2.10.
Remark 4.4. Inverting the formula in the above proposition, we can equivalently express the density generator of R in terms of the pdf of Z by
ξR(t) = Γ(n 2) 2πn2 · t−n−12 ϕZ( √ t).
Notation 4.5. To denote that the vector R ∈ Rn is spherically distributed with the characteristic generator φ, we will write R ∼ Sn(φ). When dealing with a
density generator ξ, this will analogously be denoted by R ∼ Sn(ξ).
After having briefly introduced spherical distributions, we now extend the concept to elliptically symmetric distributions. In the literature elliptically sym- metric distributions are often called “elliptically contoured distributions”, as the level curves of the density (e.g. in a contour plot) are ellipses. In the following we will simply use the term elliptical distributions instead of elliptically symmetric distributions or elliptically contoured distributions.
Definition 4.6. A random vector R ∈ Rn is said to be elliptically distributed
with the parameters µ ∈ Rn and Σ ∈ Rn×n if
R = µ + Ad TY, Y ∼ Sk(φ)
with A ∈ Rk×n such that ATA = Σ and rank(Σ) = k. To abbreviate R be-
ing elliptically distributed with the characteristic generator φ, we will write R ∼ En(µ, Σ, φ).
3Note that both in the book of Fang and Zhang [30] and in Fang, Kotz and Ng [29] the letter
to denote the probability density function and the density generator is the same. The generator has the (scalar) argument xTx, and the pdf has the argument x, the function description
4.2. ELLIPTICAL DISTRIBUTIONS 97 Remark 4.7. Note that the spherical distribution equals the elliptical distribution with µ = 0 and A = Σ = I.
Similar to the above Theorem 4.2 we get the following statements with respect to elliptical distributions.
Theorem 4.8. Let R ∼ En(µ, Σ, φ) and let rank(Σ) = k. It holds:
(i) There exists a function φ : R → R such that the characteristic function ψ of R has the form
ψ(t) = EheitTRi = eitTµφ(tTΣt).
(ii) The vector R has a stochastic representation of the form R= µ + ZAd Tu(k)
with Z ≥ 0 being independent of u(k) and ATA = Σ. Proof. See Fang, Kotz and Ng [29], page 32.
Remark 4.9. Any scalar function φ fulfilling a certain integrability condition (for the exact condition, see [29] or [30]) can determine an elliptical distribution ([30], Theorem 2.6.1). As φ is therefore not unique, we can without loss of generality assume that φ is chosen such that
−2φ0
(0) = 1 (4.1)
holds, see Fang and Zhang [30], page 67.
The next proposition summarizes several useful results about the moments, marginals and combinations of elliptical distributions.
Proposition 4.10. Let R ∼ En(µ, Σ, φ) and E[Z2] < ∞ with Z as given in the
representation formula in Theorem 4.8, part (ii). Then, the following holds: (i) The expected value and the covariance matrix of R are given by
E[R] = µ, Cov[R] = E[Z 2] rank(Σ) · Σ = −2φ 0 (0)Σ = Σ.
where the last equality holds due to the normalization assumption in Equa- tion 4.1 in the above remark.
(ii) Any linear transformation of an elliptically distributed variable is again elliptically distributed, more precisely:
Let R ∼ En(µ, Σ, φ), rank(Σ) = k, B ∈ Rm×n and b ∈ Rm. Then
BR + b ∼ Em(Bµ + b, BΣBT, φ).
(iii) Any marginal distributions of an elliptically distributed variable are again elliptical, more precisely: Let R ∼ En(µ, Σ, φ) and partition R, µ and Σ
into R = R1 R2 , µ = µ1 µ2 , Σ =Σ11 Σ12 Σ21 Σ22
with appropriate dimensions k and n − k such that R1 ∈ Rk and R2 ∈ Rn−k.
Then it holds that
R1 ∼ Ek(µ1, Σ11, φ),
R2 ∼ En−k(µ2, Σ22, φ).
(iv) The conditional distribution of an elliptically distributed variable is again elliptical. Formally, this is stated as follows:
Let R = µ + ZAd Tu(n) ∼ En(µ, Σ, φ) with Σ = ATA being positive definite.
Consider again the partitioning as given in part (iii). Then it holds that (R1|R2 = x2) ∼ Ek(˜µ1, ˜Σ1, ˜φ) with ˜ µ1 = µ1+ Σ12Σ−122(x2− µ2), (4.2) ˜ Σ1 = Σ11− Σ12Σ−122Σ21 (4.3)
and ˜φ appropriate (for details see [29], page 45).
(v) Let Rs ∼ En(µ, Σ, φ), s = 1, . . . , S independent and identically distributed.
Then it holds that
Y = S X s=1 Rs∼ En(Sµ, Σ, φS) with φS =QS s=1φ.
Proof. See Fang, Kotz and Ng [29], Section 2.5 for the parts (i) to (iv), part (v) follows from Theorem 4.1 in Hult and Lindskog [39].
4.2. ELLIPTICAL DISTRIBUTIONS 99 The moments of an elliptical distribution are needed in our application of portfolio optimization when determining parameter estimates for the vector of expected returns and the covariance matrix, the input parameters of the opti- mization problem. Furthermore, the marginals being again elliptical guarantees the proper modelling of the individual assets. Finally, explicitly having the dis- tribution of a sum of independent and identically elliptically distributed variables allows us to describe the distribution of selected parameter estimates. For ex- ample, with the formula of part (v) it is known that the maximum likelihood estimator for the mean follows again an elliptical distribution if the realizations in the sample of historical data are elliptically distributed, and furthermore, the moments are given as well. Hence, we can use this information to create a con- fidence ellipsoid and use this as an uncertainty set for the vector of expected returns. This will be done in more detail in Section 5.2.
Remark 4.11. As in the case of spherical distributions, an elliptically distributed variable does not necessarily have a probability density function. If a density exists, then it must hold that rank(Σ) = n. Furthermore, as the probability density function of Y ∼ Sn(φ) is of the form ϕY(y) = ξY(yTy), the pdf of R = µ+ATY ∼
En(µ, Σ, φ) is of the form
ϕR(x) = |Σ|− 1
2ξY (x − µ)TΣ−1(x − µ) ,
see Fang, Kotz and Ng [29], page 46.
A sometimes useful result gives the following proposition which links the den- sity function of the elliptically distributed random variable and the density of its generating variate, similar to Proposition 4.3.
Proposition 4.12. Let R ∼ En(µ, Σ, φ) with Σ = ATA positive definite, and let
R possess a density function. Then R can be represented as R = µ + ZAd Tu(n)
(Theorem 4.8). Assume furthermore that the cumulative density function (cdf ) of Z is absolutely continuous (hence, Z possesses a probability density function). Then, the probability density function ϕR of R is given by
ϕR(x) = p det(Σ−1) · ξ Z (x − µ)TΣ−1(x − µ) , x 6= µ with ξZ(t) := Γ n2 2πn2 · t−n−12 · ϕ Z( √ t). Proof. See Frahm [31], Corollary 4.
Using this just stated result about the explicit expression of the density func- tion, it is rather straightforward to show symmetry with respect to the mean µ. This fact is of importance in the subsequent sections, as we will be investigating different estimators for µ which are only meaningful substitutes for the mean in case of symmetric distributions.
Proposition 4.13. Let R ∼ En(µ, Σ, φ) with Σ = ATA positive definite, let
R possess a density function and let Z (the generating variate of R) possess a density function. Then the probability density function of R is symmetric with respect to the mean vector µ = E[R].
Proof. From Proposition 4.12 we have that the probability density function can be expressed as ϕR(x) = p det(Σ−1) · Γ n 2 2πn2 · (x − µ)TΣ−1(x − µ)− n−1 2 · ϕZ( p (x − µ)TΣ−1(x − µ))
with ϕZ being the density of the generating variate Z. As it holds for all x ∈
Rn, x 6= 0 that ϕR(µ − x) = ϕR(µ + x) =pdet(Σ−1) ·Γ n 2 2πn2 · xTΣ−1 x−n−12 · ϕ Z( √ xTΣ−1x),
symmetry with respect to µ is proved.
To close the section about elliptical distribution, we use the multivariate stan- dard normal distribution to explicitly state the various generators and all the different introduced notations and calculations linking them.
Example 4.14. We start with a multivariate standard normally distributed ran- dom variable. Let Y ∼ N (0, I), i.e. Y ∼ Sn(φ) for some characteristic generator
φ. From Theorem 4.2 we have that Y can be expressed as Y = Zu(n). Fur- thermore, as the normal distribution is a continuous distribution, it holds that P(Y = 0) = 0. With these two prerequisites Corollary 1 on page 57 in Fang and Zhang [30] states that
Z = kY k.d As W := kY k2 = YTY ∈ R is known to follow a χ2
n-distribution with the pdf (see
e.g. [46], page 416) ϕW(w) = 1 Γ n2 2n2 e−w2w n 2−1,
the probability density function of Z is obtained by a transformation of the density and calculates to
ϕZ(z) =
1 Γ n2 2n2
4.3. PARAMETER ESTIMATION 101