Let P =
h ψ
2 1
, ..., ψ
2 p
i
T
and G =
h γ
1
, ..., γ
q
i
T
, an iterative system for the γ
j
and ψ
j
is given by
P
t+1
= p
n
1 n
n
X
i=1
u d
i
; c
t
d
2t i
−1
C
−1t
D
t
, G
t+1
= p
n
1 n
n
X
i=1
u d
i
; c
t
d
2t i
−1
M
−1t
N
t
,
with the matrices C, D, M and N given in respectively 26, 27, 29 and 30. Note that µ is calculated at each step with equation 6 and Σ from G
and P .
For the starting point of this iterative procedure, one can perform a confirmatory FA based on a robust “plug-in” covariance matrix that is fast to compute. Our choice goes to the orthogo-
nalized Gnanadesikan-Kettering OGK proposed by Maronna and Zamar 2002. Alternatively, for an even faster solution, one can perform an exploratory FA on e.g. the OGK and set the
resulting loadings γ
i j
that are constrained e.g. to zero to the constrained value e.g. zero. We sucessfully implemented this last alternative and used it for the simulations. The resulting
algorithm is as follows 1. Compute µ
start
= µ
OGK
and Σ
OGK
. 2. Compute Ψ
start
and Γ
start
by means of an exploratory factor analysis based on Σ
OGK
. Re- place the fixed γ
i j
in Γ
start
by their value e.g. zero. 3. Compute Σ
start
= Γ
start
Γ
T start
+ Ψ
start
. 4. Compute the Mahalanobis distance d
1 i
= q
x
i
− µ
start T
Σ
−1 start
x
i
− µ
start
. 5. Compute the different weights ud
1 i
; c. 6. Compute the mean vector
µ
1
= P
n i=1
u d
1 i
; cx
i
P
n i=1
u d
1 i
; c .
7. Compute the vector of loadings G and uniqueness P
G
1
= p
n
1 n
n
X
i=1
u d
i
; c
1
d
21 i
−1
M
−11
N
1
, P
1
= p
n
1 n
n
X
i=1
u d
i
; c
1
d
21 i
−1
C
−11
D
1
.
8. Using P
1
and G
1
compute Σ
1
. 9. Use a convergence criterion; if the conditions of convergence are met stop, otherwise start
again from 4 and put µ
start
= µ
1
and Σ
start
= Σ
1
. Repeat the procedure until convergence is reached.
3. Inference for the direct and two-stage estimators
3.1. Direct estimator
To compute the asymptotic covariance V
G
of the loadings estimator ˆ G
and V
P
of the variances estimator ˆ
P , we use the results given by Lopuha¨a 1989 for general S -estimators
4
together with the delta-method. The asymptotic covariance of √
n vec ˆ
Σ is
V
Σ
= e
3
I
p
2
+ K
p,p
Σ ⊗ Σ + e
4
vecΣvecΣ
T
, 8
where ⊗ denotes the Kronecker product and K
p,p
is a p
2
× p
2
-block matrix with i, j-block being a p
× p matrix with 1 at entry j, i and 0 elsewhere. The constants e
3
and e
4
are given by e
3
= p
p + 2E
Φ
[ψ
2
d; cd
2
] E
2 Φ
ψ
′
d; cd
2
+ p + 1ψd; cd ,
9 e
4
= −
2 p
e
3
+ 4E
Φ
[ρd; c − b
]
2
E
2 Φ
[ψd; cd] .
10 with p the dimension of Σ, ψd; c = ∂∂dρd; c. The expectations are taken at the standard-
ized p-variate normal distribution Φ. The values of e
3
and e
4
can be obtained by Monte-Carlo simulation.
The minimization of 7 provides the solutions ˆ G
and ˆ P
and ˆ Σ is actually a function of
the latter. To find the asymptotic variance of ˆ G
and ˆ P
one can use the delta-method. Let
D
Γ
=
∂ ∂G
T
vec ΣG , P
and D
Ψ
=
∂ ∂P
T
vec ΣG , P
, with ΣG , P
= ΓG Γ
T
G + ΨP
. We then have
V
G
= D
Γ T
D
Γ −1
D
Γ T
V
Σ
D
Γ
D
Γ T
D
Γ −1
, 11
and
V
P
= D
Ψ T
D
Ψ −1
D
Ψ T
V
Σ
D
Ψ
D
Ψ T
D
Ψ −1
. 12
For D
Γ
, we have
D
Γ
= ∂
∂γ
1
vecΣ . . .
∂ ∂γ
q
vecΣ =
∂ ∂γ
1
vecΓΓ
T
. . . ∂
∂γ
q
vecΓΓ
T
, =
vec ∂
∂γ
1
ΓΓ
T
. . . vec
∂ ∂γ
q
ΓΓ
T
, =
vec ∂
∂γ
1
ΓΓ
T
+ vec Γ ∂
∂γ
1
Γ
T
. . . vec
∂ ∂γ
q
ΓΓ
T
+ vec Γ ∂
∂γ
q
Γ
T
.
Because vecAB = I
p
⊗ AvecB = B
T
⊗ I
m
vecA where A is an m × n matrix and B is an
n × q matrix see e.g. Magnus and Neudecker, 1988, we obtain that
D
Γ
= I
p
⊗ ∂
γ
1
Γ vecΓ
T
+ ∂
γ
1
Γ
T T
⊗ I
p
vecΓ . . .
I
p
⊗ ∂
γ
q
Γ vecΓ
T
+ ∂
γ
q
Γ
T T
⊗ I
p
vecΓ .
13 3.2. Two-stage estimator
The two-stage estimator ˆθ is the result of the minimization of 3, i.e. the solution in θ of
∂σθ
∂θ
T
W ˆ
σ − σθ = 0.
14 5
The asymptotic variance of ˆθ is equal to see e.g. Yuan and Bentler, 1998
Ω = A
−1
ΠA
−1
, 15
with A =
∂σθ
∂θ
T
T
W
∂σθ
∂θ
T
, Π =
∂σθ
∂θ
T
T
WV
Σ
W
∂σθ
∂θ
T
and V
Σ
being the asymptotic variance of
√ n
ˆ σ
= √
n vec ˆ
Σ. 14 actually defines a class of GLS estimators within which the most
efficient and equivalent to the MLE is obtained when W = V
−1 Σ
, which is replaced in practice
by a sample estimate of V
Σ
. Hence for the MLE, 15 simplifies to Ω =
∂σθ
∂θ
T T
V
−1 Σ
∂σθ
∂θ
T −1
. 16
When θ contains only the loadings, 16 can be written as
V
G
= D
Γ T
V
−1 Σ
D
Γ −1
, 17
with V
Σ
given in 8. One can see that 11 and 17 are equal if D
Γ
is a square matrix, but not in the other cases. It is however not clear which of the asymptotic covariance matrices is smaller,
and hence which of the two-stage or direct estimators is the most efficient.
4. Simulation study
