We use Haff’s fundamental identity to express the expectation of Sp in lower-order terms, where S follows the central Wishart distribution.Peer Reviewe

Neudecker, Heinz

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Statistics & Operations Research TransactionsSORT 31 (2) July-December 2007, 201-206Statistics &Operations ResearchTransactionsA recursion formula for expected negative andpositive powers of the central Wishart distributionc© Institut d’Estadı´stica de Catalunyasort@idescat.esISSN: 1696-2281www.idescat.net/sortHeinz NeudeckerAmsterdam School of EconomicsAbstractWe use Haff’s fundamental identity to express the expectation of Sp in lower-order terms, where Sfollows the central Wishart distribution.MSC: primary 62H10, secondary 15A69Keywords: Haff’s Fundamental Identity, matrix differentiation, central Wishart distribution.1 The recursion formula (positive powers)Let S ∼ Wm(Ω, n). Hence S follows the Central Wishart distribution with scale matrixΩ > 0 and n degrees of freedom. We use Haﬀ’s Fundamental Identity (FI):ε F1Ω−1F2 = 2εF1 ∇ F2 + 2 (ε F′2 ∇ F′1)′+ (n − m − 1)ε F1 S −1 F2,with F1 = F1(S ) a diﬀerentiable matrix function of S and n > m + 1. Further ∇ is amatrix of diﬀerential operators with typical elementdi j =12(1 + δi j) ∂∂ si j(i, j = 1, . . . ,m),where δi j is a Kronecker delta: δii = 1 and δi j = 0 (i  j).Address for correspondence: Oosterstraat, 13. 1741 GH Schagen. The Netherlands.E-mail: H.Neudecker@uva.nlReceived: May 2004Accepted: December 2006202 A recursion formula for expected negative and positive powers of the central Wishart distributionAll matrices are square of dimension m. A very useful property is: when dF =P′(dS )Q then 2∇ F = PQ + (tr P)Q, where dF is the diﬀerential of F. For theseproperties see Neudecker (2001). Further ε is the expectation operator.We choose F1 = Im and F2 = S p, (p > 0). By Haﬀ’s FI we get:εΩ−1 S p = 2ε∇ S p + (n − m − 1)ε S p−1,because ∇ I = 0. For definition and computation of ∇(·) see also Neudecker (2000).Clearlyd S p =p−1∑k=0S k (dS )S p−k−1where S 0 = Im.Hence2∇ S p = (m + p)S p−1 +p−1∑k=1(tr S k)S p−k−1.ThenΩ−1ε S p = (n + p − 1)ε S p−1 +εp−1∑k=1(tr S k)S p−k−1or equivalentlyε S p = (n + p − 1)Ωε S p−1 + Ωεp−1∑k=1(tr S k)S p−k−1.2 Discussion and application (positive powers)The second recursion formula is not really suited for finding ε S p, because this requiresknowledge of ε (tr S k) S p−k−1 for 1 ≤ k ≤ p − 1. This expression has to be found byother methods. See Neudecker (2001). The formula is, however, suited for establishingLoewner orderings. Relevant is that Ωε S p = ε S pΩ > 0 (positive definite). Thisfollows from the first recursion formula, which yields the identity Ω−1ε S p = ε S pΩ−1and further the positive definiteness of the two expressions. Pre- and postmultiplicationof the identity by matrixΩ yields:Ωε S p = ε S pΩ. AsΩ andε S p are two commutingpositive definite matrices, they are simultaneously diagonizable by an orthogonalsimilarity transformation. Hence T ′ΩT = Λ and T ′ε S p T = M, say, and consequentlyHeinz Neudecker 203Ωε S p = T Λ T ′ T M T ′ = T ΛM T ′ > 0 (positive definite), as both Λ and M arediagonal positive definite. See e.g. Hadley (1972, ex. 7-14).We further conclude thatΩ−1ε S p > (n+p−1)ε S p−1 or equivalently (ε S )−1ε S p >n−1(n + p − 1)ε S p−1 as ε S = nΩ (p ≥ 2). The first inequality follows from the secondrecursion formula. Repeated substitution finally yields the inequalityε S p > (ε S )p3 The recursion formula (negative powers)With F1 = Im and F2 = S −p, we get by Haﬀ’s FIΩ−1ε S −p = 2ε∇S −p + (n − m − 1)ε S −(p+1) (1)For p = 1, 2, . . . , n − m − 2, we clearly havedS −p = −p∑k=1S −k(dS )S k−(p+1)Hence2∇ S −p =p∑k=1S −(p+1) −p∑k=1(tr S −k)S k−(p+1)= −p S −(p+1) −p∑k=1(tr S −k)S k−(p+1)Insertion in (1) leads to the recursion formulaΩ−1ε S −p = (n − m − p − 1)ε S −(p+1) −p∑k=1ε(tr S −k)S k−(p+1) (2)or equivalentlyε S −p = (n − m − p − 1)Ωε S −(p+1) −Ωp∑k=1ε(tr S −k)S k−(p+1) (3)204 A recursion formula for expected negative and positive powers of the central Wishart distribution4 Discussion and application (negative powers)Because of the symmetry of the RHS of (2) we haveΩ−1ε S −p = (ε S −p)Ω−1. (4)We shall prove that Ω−1ε S −p > 0 (positive definite).Proof. From the commutativity property (4) we conclude that ε S −p = T Λp T ′ sayand Ω−1 = T M T ′ where T is orthogonal and M and Λp are diagonal positive definite.HenceΩ−12 = T M 12 T ′andΩ−12 (ε S −p)Ω− 12 = T M 12 T ′ T Λp T ′ T M 12 T ′= T M 12 Λp M12 T ′ = T MΛp T ′ = T M T ′ T Λp T ′= Ω−1ε S −pHenceΩ−1ε S −p > 0 as Ω− 12 (ε S −p)Ω− 12 > 0 Because ε S −(p+1) > 0 andp∑k=1ε(tr S −k)S k−(p+1) > 0 we conclude from (2) thatΩ−1ε S −p < (n − m − p − 1)ε S −(p+1) (5)when p < n − m − 1. Or equivalently(ε S −1)ε S −p < (n − m − 1)−1(n − m − p − 1)ε S −(p+1) < ε S −(p+1) (6)asε S −1 = (n − m − 1)−1Ω−1and (n − m − 1)−1(n − m − p − 1) < 1.The inequality (6) ultimately leads by successive substitution to the inequalityε S −p >(ε S −1)p (p ≥ 2) (7)Heinz Neudecker 205As ε S −1 = n(n − m − 1)−1 (ε S )−1 > (ε S )−1, we can get the inequalityε S −p >(ε S −1)p> (ε S )−p (8)It is known thatε S p > (ε S )p p ≥ 2 (9)See the end of section 2.Combining (8) and (9) we finally have the inequalityε S −p >(ε S −1)p> (ε S )−p > (ε S p)−1 (10)5 ReferencesHadley, G. (1972). Linear Algebra. Addison-Wesley Publishing Company Inc.,Reading.Neudecker, H. (2000). «A note on the matrix Haﬃan». Qu¨estiio´, 24, 419-424.Neudecker, H. (2001). «Some applications of the matrix Haﬃan in connection withdiﬀerentiable matrix functions of a central Wishart variate». Qu¨estiio´, 25, 187-210.

A recursion formula for expected negative and positive powers of the central Wishart distribution

UPCommons

Revistes Catalanes amb Accés Obert

Statistics & Operations Research Transactions
SORT 31 (2) July-December 2007, 201-206
Statistics &
Operations Research
Transactions
A recursion formula for expected negative and
positive powers of the central Wishart distribution
c© Institut d’Estadı´stica de Catalunya
sort@idescat.esISSN: 1696-2281
www.idescat.net/sort
Heinz Neudecker
Amsterdam School of Economics
Abstract
We use Haff’s fundamental identity to express the expectation of Sp in lower-order terms, where S
follows the central Wishart distribution.
MSC: primary 62H10, secondary 15A69
Keywords: Haff’s Fundamental Identity, matrix differentiation, central Wishart distribution.
1 The recursion formula (positive powers)
Let S ∼ Wm(Ω, n). Hence S follows the Central Wishart distribution with scale matrix
Ω > 0 and n degrees of freedom. We use Haﬀ’s Fundamental Identity (FI):
ε F1Ω−1F2 = 2εF1 ∇ F2 + 2 (ε F′2 ∇ F′1
)′
+ (n − m − 1)ε F1 S −1 F2,
with F1 = F1(S ) a diﬀerentiable matrix function of S and n > m + 1. Further ∇ is a
matrix of diﬀerential operators with typical element
di j =
1
2
(
1 + δi j
) ∂
∂ si j
(i, j = 1, . . . ,m),
where δi j is a Kronecker delta: δii = 1 and δi j = 0 (i  j).
Address for correspondence: Oosterstraat, 13. 1741 GH Schagen. The Netherlands.
E-mail: H.Neudecker@uva.nl
Received: May 2004
Accepted: December 2006
202 A recursion formula for expected negative and positive powers of the central Wishart distribution
All matrices are square of dimension m. A very useful property is: when dF =
P′(dS )Q then 2∇ F = PQ + (tr P)Q, where dF is the diﬀerential of F. For these
properties see Neudecker (2001). Further ε is the expectation operator.
We choose F1 = Im and F2 = S p, (p > 0). By Haﬀ’s FI we get:
εΩ−1 S p = 2ε∇ S p + (n − m − 1)ε S p−1,
because ∇ I = 0. For definition and computation of ∇(·) see also Neudecker (2000).
Clearly
d S p =
p−1∑
k=0
S k (dS )S p−k−1
where S 0 = Im.
Hence
2∇ S p = (m + p)S p−1 +
p−1∑
k=1
(
tr S k
)
S p−k−1.
Then
Ω−1ε S p = (n + p − 1)ε S p−1 +ε
p−1∑
k=1
(
tr S k
)
S p−k−1
or equivalently
ε S p = (n + p − 1)Ωε S p−1 + Ωε
p−1∑
k=1
(
tr S k
)
S p−k−1.
2 Discussion and application (positive powers)
The second recursion formula is not really suited for finding ε S p, because this requires
knowledge of ε (tr S k) S p−k−1 for 1 ≤ k ≤ p − 1. This expression has to be found by
other methods. See Neudecker (2001). The formula is, however, suited for establishing
Loewner orderings. Relevant is that Ωε S p = ε S pΩ > 0 (positive definite). This
follows from the first recursion formula, which yields the identity Ω−1ε S p = ε S pΩ−1
and further the positive definiteness of the two expressions. Pre- and postmultiplication
of the identity by matrixΩ yields:Ωε S p = ε S pΩ. AsΩ andε S p are two commuting
positive definite matrices, they are simultaneously diagonizable by an orthogonal
similarity transformation. Hence T ′ΩT = Λ and T ′ε S p T = M, say, and consequently
Heinz Neudecker 203
Ωε S p = T Λ T ′ T M T ′ = T ΛM T ′ > 0 (positive definite), as both Λ and M are
diagonal positive definite. See e.g. Hadley (1972, ex. 7-14).
We further conclude thatΩ−1ε S p > (n+p−1)ε S p−1 or equivalently (ε S )−1ε S p >
n−1(n + p − 1)ε S p−1 as ε S = nΩ (p ≥ 2). The first inequality follows from the second
recursion formula. Repeated substitution finally yields the inequality
ε S p > (ε S )p
3 The recursion formula (negative powers)
With F1 = Im and F2 = S −p, we get by Haﬀ’s FI
Ω−1ε S −p = 2ε∇S −p + (n − m − 1)ε S −(p+1) (1)
For p = 1, 2, . . . , n − m − 2, we clearly have
dS −p = −
p∑
k=1
S −k(dS )S k−(p+1)
Hence
2∇ S −p =
p∑
k=1
S −(p+1) −
p∑
k=1
(
tr S −k
)
S k−(p+1)
= −p S −(p+1) −
p∑
k=1
(
tr S −k
)
S k−(p+1)
Insertion in (1) leads to the recursion formula
Ω−1ε S −p = (n − m − p − 1)ε S −(p+1) −
p∑
k=1
ε
(
tr S −k
)
S k−(p+1) (2)
or equivalently
ε S −p = (n − m − p − 1)Ωε S −(p+1) −Ω
p∑
k=1
ε
(
tr S −k
)
S k−(p+1) (3)
204 A recursion formula for expected negative and positive powers of the central Wishart distribution
4 Discussion and application (negative powers)
Because of the symmetry of the RHS of (2) we have
Ω−1ε S −p = (ε S −p)Ω−1. (4)
We shall prove that Ω−1ε S −p > 0 (positive definite).
Proof. From the commutativity property (4) we conclude that ε S −p = T Λp T ′ say
and Ω−1 = T M T ′ where T is orthogonal and M and Λp are diagonal positive definite.
Hence
Ω−
1
2 = T M 12 T ′
and
Ω−
1
2 (ε S −p)Ω− 12 = T M 12 T ′ T Λp T ′ T M 12 T ′
= T M 12 Λp M
1
2 T ′ = T MΛp T ′ = T M T ′ T Λp T ′
= Ω−1ε S −p
Hence
Ω−1ε S −p > 0 as Ω− 12 (ε S −p)Ω− 12 > 0 
Because ε S −(p+1) > 0 and
p∑
k=1
ε
(
tr S −k
)
S k−(p+1) > 0 we conclude from (2) that
Ω−1ε S −p < (n − m − p − 1)ε S −(p+1) (5)
when p < n − m − 1. Or equivalently
(
ε S −1
)
ε S −p < (n − m − 1)−1(n − m − p − 1)ε S −(p+1) < ε S −(p+1) (6)
as
ε S −1 = (n − m − 1)−1Ω−1
and (n − m − 1)−1(n − m − p − 1) < 1.
The inequality (6) ultimately leads by successive substitution to the inequality
ε S −p >
(
ε S −1
)p (p ≥ 2) (7)
Heinz Neudecker 205
As ε S −1 = n(n − m − 1)−1 (ε S )−1 > (ε S )−1, we can get the inequality
ε S −p >
(
ε S −1
)p
> (ε S )−p (8)
It is known that
ε S p > (ε S )p p ≥ 2 (9)
See the end of section 2.
Combining (8) and (9) we finally have the inequality
ε S −p >
(
ε S −1
)p
> (ε S )−p > (ε S p)−1 (10)
5 References
Hadley, G. (1972). Linear Algebra. Addison-Wesley Publishing Company Inc.,
Reading.
Neudecker, H. (2000). «A note on the matrix Haﬃan». Qu¨estiio´, 24, 419-424.
Neudecker, H. (2001). «Some applications of the matrix Haﬃan in connection with
diﬀerentiable matrix functions of a central Wishart variate». Qu¨estiio´, 25, 187-210.



A recursion formula for expected negative and positive powers of the central Wishart distribution

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