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1.
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
2.
Speed of convergence in nonparametric kernel estimation of a regression function and its derivatives
Alexander A. Georgiev 《Annals of the Institute of Statistical Mathematics》1984,36(1):455-462
Summary The objective in nonparametric regression is to infer a functiong(x) and itspth order derivativesg
(g)(x),p≧1 fixed, on the basis of a finite collection of pairs {x
i, g(xi)+Z
i}
i=1
n
, where the noise componentsZ
i satisfy certain modest assumptions and the domain pointsx
i are selected non-randomly. This paper exhibits a new class of kernel estimatesg
n
(p)
,p≧0 fixed. The main theoretical results of this study are the rates of convergence obtained for mean square and strong consistency
ofg
n
(p)
each of them being uniform on the (0,1). 相似文献
3.
Accuracy of several multidimensional refinable distributions 总被引:3,自引:0,他引:3
Carlos Cabrelli Chritopher Heil Ursula Molter 《Journal of Fourier Analysis and Applications》2000,6(5):483-502
Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced
from linear combinations of translates of f1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=Σ
i=1
r
Σk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ. 相似文献
4.
We say that n independent trajectories ξ1(t),…,ξ
n
(t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ
i
(t
i
) and ξ
j
(t
j
) is at least ɛ, for some indices i, j and for all large enough t
1,…,t
n
, with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition
function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by
the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct
−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ(α) are asymptotically separated, where ξ(α) is the α-process generated by −(−Δ)α/2.
Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000
RID="*"
ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782
RID="**"
ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573 相似文献
5.
U. Stadtmüller 《Periodica Mathematica Hungarica》1986,17(2):83-108
We consider a class of nonparametric estimators for the regression functionm(t) in the model:y
i
=m(t
i
) +
i
, 1 i n, t
i
[0, 1], which are linear in the observationsy
i
. Several limit theorems concerning local and global stochastic and a.s. convergence and limit distributions are given. 相似文献
6.
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
7.
In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of
subsets {A
t
| t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f
t
of G under which #{t∈ℕ | f
t
x∈A
t
} is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical
Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov.
Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999 相似文献
8.
ZHENGZUKANG 《高校应用数学学报(英文版)》1997,12(1):33-38
Let X,i.i.d. and Y1i. i.d. be two sequences of random variables with unknown distribution functions F(x) and G(y) respectively. X, are censored by Y1. In this paper we study the uniform consistency of the Kaplan-Meier estimator under the case ey=sup(t:F(t)<1)>to=sup(t2G(t)<1) The sufficient condition is discussed. 相似文献
9.
We study non-parametric tests for checking parametric hypotheses about a multivariate density f of independent identically distributed random vectors Z1,Z2,… which are observed under additional noise with density ψ. The tests we propose are an extension of the test due to Bickel and Rosenblatt [On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973) 1071-1095] and are based on a comparison of a nonparametric deconvolution estimator and the smoothed version of a parametric fit of the density f of the variables of interest Zi. In an example the loss of efficiency is highlighted when the test is based on the convolved (but observable) density g=f*ψ instead on the initial density of interest f. 相似文献
10.
Jussi Klemelä 《Probability Theory and Related Fields》2006,134(4):539-564
Estimation of a quadratic functional of a function observed in the Gaussian white noise model is considered. A data-dependent
method for choosing the amount of smoothing is given. The method is based on comparing certain quadratic estimators with each
other. It is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the “regular” and “irregular”
region. We consider lp bodies and construct bounds for the risk of the estimator which show that for p=4 the estimator is exactly optimal and for example when p ∈[3,100], then the upper bound is at most 1.055 times larger than the lower bound. We show the connection of the estimator
to the theory of optimal recovery. The estimator is a calibration of an estimator which is nearly minimax optimal among quadratic
estimators.
Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/6-2, CIES, France, and Jenny
and AnttiWihuri Foundation. 相似文献
11.
Incompleteness and minimality of complex exponential system 总被引:3,自引:0,他引:3
Guan-tie DENG School of Mathematical Sciences Beijing Normal University Beijing China 《中国科学A辑(英文版)》2007,50(10):1467-1476
A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series. 相似文献
12.
Siddhartha Bhattacharya 《Israel Journal of Mathematics》2003,137(1):211-221
Let Γ be a discrete group and fori=1,2; letα
i be an action of Γ on a compact abelian groupX
i by continuous automorphisms ofX
i. We study measurable equivariant mapsf: (X
1,α
1)→(X
2,α
2), and prove a rigidity result under certain assumption on the order of mixing of the underlying actions. 相似文献
13.
Ibrahim A. Ahmad 《Annals of the Institute of Statistical Mathematics》1979,31(1):279-288
Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several
authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of
Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables.
The results obtained are then applied to two estimates of the functional Δ(f)=∫f
2
(x)dx were strong consistency is established. One of the suggested two estimates of Δ(f) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where
they established consistency in the second mean of the estimate.
Research supported in part by the National Research Council of Canada and in part by McMaster University Research Board. Now
at Memphis State University, Memphis, Tennessee 38152, U.S.A. 相似文献
14.
Samir Lababidi 《Statistical Inference for Stochastic Processes》2003,6(1):15-24
This paper deals with a nonparametric estimation problem of an integral-type functional from indirect observations where the observation Y
(t) is a sum of a known function of an unobservable process X
(t) and a Gaussian white noise, and X
(t) is a sum of an unknown function a(t) and a Gaussian process. The minimax lower bound on the quality of nonparametric estimation is derived and an asymptotically efficient estimator is proposed. The paper concludes with some examples including one about reduction to parameter estimation. 相似文献
15.
For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H
0 : f≡ 0, (the signal f is zero) against the nonparametric alternative H
1 : f∈Λɛ where Λɛ is a set of functions on R
1 of the form Λɛ = {f : f∈?, ϕ(f) ≥ Cψɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C
* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C
* and is not possible for C < C
*. We propose asymptotically minimax test statistics.
Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000 相似文献
16.
Let A = d/dθ denote the generator of the rotation group in the space C(Γ), where Γ denotes the unit circle. We show that the stochastic Cauchy problem
, where b is a standard Brownian motion and f ∈ C(Γ) is fixed, has a weak solution if and only if the stochastic convolution process t ↦ (f * b)t has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination
with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all f ∈ C(Γ) outside a set of the first category. 相似文献
((1)) |
17.
Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic
diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained
for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression
experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation.
For one direction of the equivalence an explicit Markov kernel is constructed. 相似文献
18.
Jinhong You 《Journal of multivariate analysis》2006,97(4):844-873
We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators. 相似文献
19.
Der-Shin Chang Yuang-Chin Chiang 《Annals of the Institute of Statistical Mathematics》1980,32(1):275-281
Consider a realization of the process
on the intervalT=[0,1] for functionsf
1(t),f
2(t),...,f
n
(t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f
k
(t)}
k=1
n
to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD
s-optimal design problems are discussed. 相似文献
20.
A nonparametric statistical model of small diffusion type is compared with its discretization by a stochastic Euler difference
scheme. It is shown that the discrete and continuous models are asymptotically equivalent in the sense of Le Cam's deficiency
distance for statistical experiments, when the discretization step decreases with the noise intensity ε.
Received: 12 April 1996 / Revised version: 29 October 1997 相似文献