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CPS2101 Bertail Patrice et al.
p
−1
where vol(W ) = det(W (W ) )
−1
−1 t

3. Scores and Tangent spaces of the semiparametric model
The main idea of semiparametric model is to consider square root of
density as element of the Hilbert space L2(λ) . In the following we will consider
densities which are differentiable in quadratic mean (DQM) that is such that
for any parametric model pt t ∈ [0,1] in PW +,G, there exists a score function s
such that

.
In particular when it is assumed that g ∈ G and G is regular and
differentiable in (that is any density in G admit a score function sg) then pW,g(v)
−1
−1
= vol(W )g(W v) is automatically differentiable in quadratic mean. The
efficiency bounds and the efficient score may be obtained by computing
respectively the scores with respect to the parameter of interest (for us W) and
with respect to the nuissance parameter (for us g) and then by projecting the
score function with respect to W into the orthogonal space of the tangent
space engendered by the scores with respect to nuisance parameters. It
follows that

with

and

Estimation of the parameters. Let v1, ..., vn be i.i. d. PW,g . In theory an
(oracle) estimator would be given by solving the M-equation

However since these quantities depends on g as well as some other
unknown non-parametric quantities depending on g mainly
, we will replace the efficient score by an
estimated score, using the same ideas as in [8]. We will focus here on the
simple Nadaraya estimator, with smoothing parameter bn and kernel density κ
given by

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