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1.
f be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals
of f, specifically, L
r
norms ||f||
r
of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable
in L
2
) functional or estimating a singular functional like the value of f at certain point or the maximum of f. In the first case, the convergence rate typically is n
−1/2, n being the number of observations. In the second case, the rate of convergence coincides with the one of estimating the function
f itself in the corresponding norm.
We show that the case of estimating ||f||
r
is in some sense intermediate between the above extremes. The optimal rate of convergence is worse than n
−1/2
but is better than the rate of convergence of nonparametric estimates of f. The results depend on the value of r. For r even integer, the rate occurs to be n
−β/(2β+1−1/r)
where β is the degree of smoothness. If r is not an even integer, then the nonparametric rate n
−β/(2β+1)
can be improved, but only by a logarithmic in n factor.
Received: 6 February 1996hinspaceairsp/Revised version: 10 June 1998 相似文献
2.
Risk bounds for model selection via penalization 总被引:11,自引:0,他引:11
Andrew Barron Lucien Birgé Pascal Massart 《Probability Theory and Related Fields》1999,113(3):301-413
Performance bounds for criteria for model selection are developed using recent theory for sieves. The model selection criteria
are based on an empirical loss or contrast function with an added penalty term motivated by empirical process theory and roughly
proportional to the number of parameters needed to describe the model divided by the number of observations. Most of our examples
involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression
function. We show that the quadratic risk of the minimum penalized empirical contrast estimator is bounded by an index of the accuracy of the sieve. This accuracy index quantifies the trade-off among the candidate models
between the approximation error and parameter dimension relative to sample size.
If we choose a list of models which exhibit good approximation properties with respect to different classes of smoothness,
the estimator can be simultaneously minimax rate optimal in each of those classes. This is what is usually called adaptation. The type of classes of smoothness in which one gets adaptation depends heavily on the list of models. If too many models
are involved in order to get accurate approximation of many wide classes of functions simultaneously, it may happen that the
estimator is only approximately adaptive (typically up to a slowly varying function of the sample size).
We shall provide various illustrations of our method such as penalized maximum likelihood, projection or least squares estimation.
The models will involve commonly used finite dimensional expansions such as piecewise polynomials with fixed or variable knots,
trigonometric polynomials, wavelets, neural nets and related nonlinear expansions defined by superposition of ridge functions.
Received: 7 July 1995 / Revised version: 1 November 1997 相似文献
3.
Abstract. Let be open,X a Banach space and . We show that every is holomorphic if and only if every set inX is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that is holomorphic for all , where W is a separating subspace of to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in a weak
sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive
Lebesgue measure, then the same is true for the interior points of , uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality.
Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small
multipliers).
Received January 29, 1998; in final form March 8, 1999 / Published online May 8, 2000 相似文献
4.
5.
We consider the problem of nonparametric identification for a multi-dimensional functional autoregression y
t
= f(y
t
−1, …,y
t−d
) + e
t
on the basis of N observations of y
t
. In the case when the unknown nonlinear function f belongs to the Barron class, we propose an estimation algorithm which provides approximations of f with expected L
2 accuracy O(N
1/4ln1/4
N). We also show that this approximation rate cannot be significantly improved.
The proposed algorithms are “computationally efficient”– the total number of elementary computations necessary to complete
the estimate grows polynomially with N.
Received: 23 September 1997 / Revised version: 28 January 1999 相似文献
6.
We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on ℓ2 (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a
real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w
* closure of the set of its points of Gateaux differentiability is norm separable.
Received May 31, 1999 / final version received February 16, 2000?Published online April 19, 2000 相似文献
7.
Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian
noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent
for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically
distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a
value f(t
i
) of a regression function f at a grid point t
i
(nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result
is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional
version of the Hungarian construction for the partial sum process.
Received: 4 February 1997 相似文献
8.
9.
Christopher Boyd 《Monatshefte für Mathematik》2000,130(3):177-188
We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′
b
locally Asplund we show that the space of n-homogeneous polynomials on (E′
b
)′
b
is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.
(Received 24 March 1999; in final form 14 February 2000) 相似文献
10.
11.
Summary. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms.
Received March 28, 2000 / Revised version received June 23, 2000 / Published online March 8, 2002
RID="*"
ID="*" Supported by the National Science Foundation under grant DMS-9870187
RID="**"
ID="**" Supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160 相似文献
12.
Summary.
We consider convex interpolation with
cubic splines on
grids built by adding two knots in each subinterval of neighbouring data
sites. The additional knots have to be variable in order to get a chance
to always retain convexity. By means of the staircase algorithm we
provide computable intervals for the added knots such that all knots
from these intervals allow convexity preserving spline interpolation of
continuity.
Received
May 31, 1994 / Revised version received December 22, 1994 相似文献
13.
Jon F. Carlson 《manuscripta mathematica》1998,97(3):303-307
Let P be a finite p-group, and let k a field of characteristic p>0. We prove that a finietly generated kG-module M is endotrivial if and only if
where PHom (-,-) is the set of endomorphisms which factor through projective modules.
Received: 23 January 1998 相似文献
14.
Yasuyuki Nagatomo 《Mathematische Zeitschrift》2003,243(2):243-261
We prove the completeness of an instanton moduli space on quaternion-K?hler manifold . A point in the boundary of the moduli represents an ASD bundle with a particular singular set. It is shown that the singular
set is a quaternion submanifold of and the Poincaré dual of the homology class represented by is the second Chern class o f the instanton bundle.
Received: 4 July 2000 / Published online: 2 December 2002
Current Address: Faculty of Mathematics, Kyushu University, Ropponmatsu, Fukuoka, 810-8560, Japan (e-mail: nagatomo@math.kyushu-u.ac.jp) 相似文献
15.
16.
The L
p
minimax risks (1≤p<∞) are studied for statistical estimation in the Gaussian white noise model. The asymptotic rate and constants are given,
and the optimal estimator is proposed. This, together with the work of Golubev, Levit and Tsybakov (1996) establishes the
classification of the L
p
minimax constants on the classes of analytical functions.
Received: 10 December 1996 / Revised version: 14 December 1997 相似文献
17.
Let
So is the collection of all n + 1 term exponential sums with constant first term. We prove the following two theorems.
Theorem 1 (Remez-type inequality for
$E_n$
at 0).
Let
$s \in \left( 0, \frac 12 \right]\,.$
There are absolute constants
$c_1 > 0$
and
$c_2 > 0$
such that
where the supremum is taken for all
$f \in E_n$
satisfying
Theorem 2 (Nikolskii-type inequality for
$E_n$
).
There are absolute constants
$c_1 > 0$
and
$c_2 > 0$
such that
for every
$a < y < b$
and
$q > 0\,.$
It is quite remarkable that, in the above Remez- and Nikolskii-type inequalities, behaves like , where denotes the collection of all algebraic polynomials of degree at most n with real coefficients.
Received: 4 November 1998 / in final form: 2 March 1999 相似文献
18.
Martin R. Bridson 《Mathematische Annalen》2000,317(4):629-633
Abstract. We construct finitely presented subgroups of GL that have infinitely many conjugacy classes of finite subgroups. This answers a question of Grunewald and Platonov. We suggest
a variation on their question.
Received: 26 August 1999 / Revised: 28 September 1999 / Published online: 8 May 2000 相似文献
19.
Juan José Torrens 《Numerische Mathematik》1997,76(1):69-85
Résumé. On établit des majorations de l'erreur d'approximation par éléments finis à partir de données de Lagrange pour des fonctions
appartenant à un espace de Sobolev d'ordre convenable, lorsque les degrés de liberté sont approchés à l'aide de la méthode
des plaquettes splines introduite par A. Le Méhauté (cf. [13], [14], [15]). Les résultats obtenus s'appliquent notamment
à la construction de surfaces de classe .
Received May 29, 1995 / Revised version received August 20, 1995 相似文献