Page 16 - Contributed Paper Session (CPS) - Volume 4

CPS2101 Bertail Patrice et al.
Following [8], it is easy to estimate ) by a Nadaraya-Watson
estimator, denoted by
Then we have an estimated efficient score function given by

.

where ) is the estimated value of .
Let the following conditions:
H
H7 κ admits a 3rd order continuous derivative, square integrable and
ultimately monotone.
Theorem (Consistency of the estimator) Under the assumptions H1,...H7,
the estimator which is the solution of the estimating equation

̂

say , ∑ ̂ ( ) = 0, is efficient and asymptotically that is to say :

̂
,

4. Model selection
Here, we consider the sequence of nested zero-noise NMF models,
indexed by K ∈ {1, ..., F}, parametrized by the set . Let

where () = −2 ( ;) = (( −1 )) Π =1 ( −1 ) + () where

an(K) is a penalty satisfying the following assumption :
H8 Let the penalization an(K), be an increasing sequence (in K) such that, for
anyK1 > K2 > 0, an(K1)− an(K2) → ∞ and such that, for any K,
.
This is for instance the case for the BIC penalization criterion obtained with
an(K) = log(n) · dK, where the dimension dK is increasing in K.
We will show the consistency of for general penalization under the
̂

−1
following additional assumptions. H9 The function g(W v) satisfies an uniform
Lipschitz condition of the form
,
for some M∗ > 0.
−1
This is true in particular, if the derivative of g(W v) with respect to W is
bounded over the sets W , K =1,...,F.
K
Theorem 1 (Consistent estimation of the cone dimension)
Suppose that the assumptions H1,...,H5 are satisfied, as well as the additional
assumptions H8,H9 and H10. Then, as n → ∞, we almost surely have Kn → K .
∗
A toy numerical experiment. We chose F = K = 2 and 200 observations
have been generated based on model (2), where the hkn’s are i.i.d. r.v.’s drawn

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