1
A PPENDIX
Sketch of the proof of Proposition III.1
Proof. Taking into account that Ω is Polish and that the convex
cone (in a certain Hilbert space) Fc (R) is isometric to fulfills
the completeness and separability properties, it is possible
to check that the sufficient conditions given by Brown and
Purves [1] for the Borel measurability of the mapping defining
the M-estimator of location are satisfied.
Proof of Theorem III-A.1
Proof. First, a lemma published by Kim and Scott [2] and
providing conditions to ensure the existence of the Gâteaux
differential will be recalled.
Take H a Hilbert space with associated norm k · kH , (Ω, A, P )
a probability space, and consider an associated Hilbert-valued
random element. Given a sample of independent observations
M [h ]
hn = (h1 , . . . , hn ) from such a random element, let g\
n
be the Hilbert value(s) that minimize(s) in H the function
n
1X
ρ(khi − gkH ).
J(g) =
n i=1
The Gâteaux differential of this function J at g ∈ H with
incremental h ∈ H is given by
one can express the M-estimate as
M [e
ge\
xn ] =
n
X
i=1
Pn
and trivially i=1 wi = 1. Furthermore, since φ(0) = 0 and
φ(x) = ρ0 (x)/x ≥ 0 if x > 0, and because ρ is assumed to
be non-decreasing, we have that
M [e
φ(Dθϕ (e
xi , ge\
xn ]))
≥ 0.
wi = P
n
ϕ
\
M [e
φ(D
(e
x
,
g
e
x
]))
j
n
j=1
θ
n
1 X
= lim
[ρ(khi − (g + ςh)kH ) − ρ(khi − gkH )] .
ς→0 n ς
i=1
Suppose the assumptions below are satisfied:
• ρ is non-decreasing, ρ(0) = 0 and limx→0 ρ(x)/x = 0,
• φ(0) , limx→0 φ(x) exists and it is finite.
Then, the Gâteaux differential of J at g ∈ H is δJ(g; h)
= − hV (g), hiH , where
V : H → H
n
1X
g 7→ V (g) =
φ(khi − gkH ) · (hi − g)
n i=1
Proof of Proposition III-B.1
Proof. Due to the translational invariance of the metric Dθϕ , it
holds that
M [e
M [e
e , ge\
e )).
φ(Dθϕ (e
xi , ge\
xn ])) = φ(Dθϕ (e
xi + U
xn ] + U
Hence, the weights wi are also invariant under such a translation. Consequently, because of the properties of the usual
fuzzy arithmetic,
M [e
e=
ge\
xn ] + U
n
X
e=
wi · x
ei + U
n
X
e ).
wi · (e
xi + U
i=1
i=1
δJ(g; h)
wi · x
ei ∈ Fc∗ (R),
M [e
e satisfies the necessary and sufficient condiThus, ge\
xn ] + U
tions to be a minimizer of the function J associated with the
e , so that it is the location M-estimate based on
en + U
sample x
this sample.
Proof of Proposition III-B.2
Proof. If c = 0, the result is trivial. Moreover, if c 6= 0 the
absolute scalability of the metric Dθϕ and the assumption on
φ ensure that
M [e
M [e
φ(Dθϕ (c · x
ei , c · ge\
xn ])) = φ(|c| · Dθϕ (e
xi , ge\
xn ]))
M [e
∝ φ(Dθϕ (e
xi , ge\
xn ])).
M [h ] to be a minimizer of J
and a necessary condition for g\
n
M [h ]) = 0.
Hence, the weights wi are invariant under such a scale
is V (g\
n
By using this lemma, the considered assumptions guarantee transformation. Therefore, using the properties of the usual
fuzzy arithmetic,
that
n
n
n
n
X
X
X
X
M [e
M [e
M [e
M [e
φ(Dθϕ (e
xi , ge\
xn ]))·e
xi =
φ(Dθϕ (e
xi , ge\
xn ]))·ge\
xn ].
c · ge\
xn ] = c ·
wi · x
ei =
wi · (c · x
ei ).
i=1
i=1
i=1
M [e
As ge\
xn ] does not depend on i, this yields
i=1
M [e
Thus, c · ge\
xn ] satisfies the necessary and sufficient conditions
to
be
a
minimizer
of the function J associated with the
M [e
M [e
M [e
φ(Dθϕ (e
xi , ge\
xn ]))·e
xi = ge\
xn ]·
φ(Dθϕ (e
xi , ge\
xn ])). sample c · x
en , so that it is the location M-estimate based on
i=1
i=1
this sample.
Pn
ϕ
\
M
As i=1 φ(Dθ (e
xi , ge [e
xn ])) > 0, then we obtain
Proof of Proposition III-B.3
n
ϕ
Proof. Under the assumptions required in Theorem III-A.1, M\
M
X
φ(Dθ (e
xi , ge [e
xn ]))
M [e
ge\
xn ] =
·x
ei .
estimates of LR fuzzy numbers sharing L and R functions can
Pn
ϕ
M [e
xj , ge\
xn ]))
i=1
j=1 φ(Dθ (e
be expressed as weighted means of the observations. Since the
sum and the product by scalars are closed operators in the class
By denoting
of LR fuzzy numbers sharing the same L and R functions,
M [e
φ(Dθϕ (e
xi , ge\
xn ]))
the corresponding M-estimate will also fall in the LR family.
wi = P
,
n
ϕ
\
M
xj , ge [e
xn ]))
j=1 φ(Dθ (e
n
X
n
X
2
Proof of Proposition III-B.4
Proof. From the translational and rotational equivariance of
the M-estimates, we obtain that
•
•
en ],
geM\
[e
xn − c] = geM\
[c − x
whence
M [e
M [e
ge\
xn ] − c = c − ge\
xn ].
M [e
By adding ge\
xn ]+c to the two members in the last equality,
M [e
M [e
M [e
2 ge\
xn ] = 2c + ge\
xn ] − ge\
xn ].
1
M [e
,
ge\
xn ] = c + · Oge\
M [e
xn ]
2
eα − sup U
eα , sup U
eα − inf U
eα ], which
= [inf U
M [e
leads to a symmetric fuzzy number about 0. Therefore, ge\
xn ]
α
is a symmetric fuzzy number about c.
it is possible to prove that the M-estimator will always
belong to a compact set unless the perturbed sample
contains more than n1 b n+1
2 c observations.
R EFERENCES
Hence,
where OUe
When ρ has linear upper and lower bounds, it is possible
to preserve the inequalities between distances and follow
a reasoning like in Sinova and Van Aelst [8].
When ρ is the Hampel loss function ρa,b,c satisfying
n − 2b n−1
2 c
·ρa,b,c (c)
ρa,b,c
max Dθϕ (e
xi , x
ej ) <
n−1
1≤i,j≤n
n−b 2 c−1
Sketch of the proof of Theorem III-B.1
Proof. The proof consists of the verification of Huber’s sufficient conditions for consistency (see Huber [3]).
First, the second countability and locally compactness of
(Fc ([a, b]), Dθϕ ) are checked. Secondly, the proof follows the
same scheme as in Sinova and Van Aelst [5] (where ρ(x)
= |x|), noticing that
• When ρ is subadditive apart from non-decreasing, it
is possible to preserve the relations and the way of
reasoning with distances. Furthermore, as ρ is assumed
to be unbounded, the proof of Huber’s assumption A-5
(iii) in Sinova and Van Aelst [5] can be adapted.
• When ρ is the Huber loss function ρa , for any a > 0, it
is possible to bound it by means of two lines with the
same slope for all x > 0
a2
ax −
≤ ρa (x) ≤ ax
2
and this allows us to finally work with distances.
• When ρ is the Hampel loss function ρa,b,c , for any c >
b > a > 0, it is possible to bound it as follows
1
0 ≤ ρa,b,c (x) ≤ a(b + c − a).
2
In this case, it is also necessary to notice that there
is a compact set C in the parameter space such that
the sequence of M-estimators of location almost surely
ultimately stays in C, thanks to the parameter space being
locally compact and Hausdorff and the strong laws of
large numbers.
Sketch of the proof of Theorem III-B.2
Proof. M-estimators defined through the representer theorem
are translational equivariant by Proposition III-B.1. Therefore,
the proof of the upper bound n1 b n+1
2 c can be carefully
extended from the real-valued settings taking care of the
semilinearity of Fc (R).
Sketch of the proof of Theorem III-B.3
Proof. Some ideas of the proof are based on the results by
Lopuhaä and Rousseeuw [7].
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