Mathematical and Statistical Methods for Actuarial Sciences and Finance

144 C. Franceschini and N. Loperfido

2 Third moment

The third moment of ap-dimensional random vectorzis defined asμ 3 (z) =
E

(

z⊗zT⊗zT

)

,where⊗denotes the Kronecker (tensor) product and third mo-
ment is finite [9, page 177]. The third central moment is defined in a similar way:
μ 3 (z)=μ 3 (z−μ),whereμdenotes the expectation ofz.Forap-dimensional ran-
dom vector the third moment is ap×p^2 matrix containingp(p+ 1 )(p+ 2 )/6 possibly
distinct elements. As an example, letz=(Z 1 ,Z 2 ,Z 3 )Tandμijk=E

(

ZiZjZk

)

,

fori,j,k= 1 ,...,3. Then the third moment ofzis

μ 3 (z)=

⎛

⎝

μ 111 μ 112 μ 113 μ 211 μ 212 μ 213 μ 311 μ 312 μ 313

μ 121 μ 122 μ 123 μ 221 μ 222 μ 223 μ 321 μ 322 μ 323

μ 131 μ 132 μ 133 μ 231 μ 232 μ 233 μ 331 μ 332 μ 333

⎞

⎠.

In particular, if all components ofzare standardised, its third moment is scale-free,
exactly like many univariate measures of skewness.
Moments of linear transformationsy=Azadmit simple representations in terms
of matrix operations. For example, the expectationE(y)=AE(z)is evaluated via
matrix multiplication only. The varianceV(y)=AV(z)ATis evaluated using both
the matrix multiplication and transposition. The third momentμ 3 (y)is evaluated
using the matrix multiplication, transposition and the tensor product:

Proposition 1.Let z be a p-dimensional randomvector with finite third moment
μ 3 (z)and let A be a k×p real matrix. Then the third moment of Az isμ 3 (Az)=
Aμ 3 (z)

(

AT⊗AT

)

.

The third central moment of a random variable is zero, when it is finite and the
corresponding distribution is symmetric. There are several definitions of multivariate
symmetry. For example, a random vectorzis said to be centrally symmetric atμif
z−μandμ−zare identically distributed [15]. The following proposition generalises
this result to the multivariate case.

Proposition 2.If the randomvector z is centrally symmetric and the third central
moment is finite, it is a null matrix.

Sometimes it is more convenient to deal with cumulants, rather than with moments.
The following proposition generalises to the multivariate case a well known identity
holding for random variables.

Proposition 3.The third central moment of a randomvector equals its third cumulant,
when they both are finite.

The following proposition simplifies the task of finding entries ofμ 3 (z)correspond-
ing toE

(

ZiZjZk

)

.

Proposition 4.Let z=

(

Z 1 ,...,Zp

)T

be a randomvector whose third moment

μ 3 (z)is finite. Thenμ 3 (z)=(M 1 ,...,Mp

)

,whereMi=E

(

ZizzT

)

.