1,661 research outputs found
Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression
We study the problem of nonparametric regression when the regressor is
endogenous, which is an important nonparametric instrumental variables (NPIV)
regression in econometrics and a difficult ill-posed inverse problem with
unknown operator in statistics. We first establish a general upper bound on the
sup-norm (uniform) convergence rate of a sieve estimator, allowing for
endogenous regressors and weakly dependent data. This result leads to the
optimal sup-norm convergence rates for spline and wavelet least squares
regression estimators under weakly dependent data and heavy-tailed error terms.
This upper bound also yields the sup-norm convergence rates for sieve NPIV
estimators under i.i.d. data: the rates coincide with the known optimal
-norm rates for severely ill-posed problems, and are power of
slower than the optimal -norm rates for mildly ill-posed problems. We then
establish the minimax risk lower bound in sup-norm loss, which coincides with
our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV
estimators. This sup-norm rate optimality provides another justification for
the wide application of sieve NPIV estimators. Useful results on
weakly-dependent random matrices are also provided
On rate optimality for ill-posed inverse problems in econometrics
In this paper, we clarify the relations between the existing sets of
regularity conditions for convergence rates of nonparametric indirect
regression (NPIR) and nonparametric instrumental variables (NPIV) regression
models. We establish minimax risk lower bounds in mean integrated squared error
loss for the NPIR and the NPIV models under two basic regularity conditions
that allow for both mildly ill-posed and severely ill-posed cases. We show that
both a simple projection estimator for the NPIR model, and a sieve minimum
distance estimator for the NPIV model, can achieve the minimax risk lower
bounds, and are rate-optimal uniformly over a large class of structure
functions, allowing for mildly ill-posed and severely ill-posed cases.Comment: 27 page
Spectral calibration of exponential Lévy Models [1]
We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.European option, jump diffusion, minimax rates, severely ill-posed, nonlinear inverse problem, spectral cut-off
On rate optimality for ill-posed inverse problems in econometrics
In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases.We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model,can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.
Nonlinear estimation for linear inverse problems with error in the operator
We study two nonlinear methods for statistical linear inverse problems when
the operator is not known. The two constructions combine Galerkin
regularization and wavelet thresholding. Their performances depend on the
underlying structure of the operator, quantified by an index of sparsity. We
prove their rate-optimality and adaptivity properties over Besov classes.Comment: Published in at http://dx.doi.org/10.1214/009053607000000721 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Early stopping for statistical inverse problems via truncated SVD estimation
We consider truncated SVD (or spectral cut-off, projection) estimators for a
prototypical statistical inverse problem in dimension . Since calculating
the singular value decomposition (SVD) only for the largest singular values is
much less costly than the full SVD, our aim is to select a data-driven
truncation level only based on the knowledge of
the first singular values and vectors. We analyse in detail
whether sequential {\it early stopping} rules of this type can preserve
statistical optimality. Information-constrained lower bounds and matching upper
bounds for a residual based stopping rule are provided, which give a clear
picture in which situation optimal sequential adaptation is feasible. Finally,
a hybrid two-step approach is proposed which allows for classical oracle
inequalities while considerably reducing numerical complexity.Comment: slightly modified version. arXiv admin note: text overlap with
arXiv:1606.0770
Stable soft extrapolation of entire functions
Soft extrapolation refers to the problem of recovering a function from its
samples, multiplied by a fast-decaying window and perturbed by an additive
noise, over an interval which is potentially larger than the essential support
of the window. A core theoretical question is to provide bounds on the possible
amount of extrapolation, depending on the sample perturbation level and the
function prior. In this paper we consider soft extrapolation of entire
functions of finite order and type (containing the class of bandlimited
functions as a special case), multiplied by a super-exponentially decaying
window (such as a Gaussian). We consider a weighted least-squares polynomial
approximation with judiciously chosen number of terms and a number of samples
which scales linearly with the degree of approximation. It is shown that this
simple procedure provides stable recovery with an extrapolation factor which
scales logarithmically with the perturbation level and is inversely
proportional to the characteristic lengthscale of the function. The pointwise
extrapolation error exhibits a H\"{o}lder-type continuity with an exponent
derived from weighted potential theory, which changes from 1 near the available
samples, to 0 when the extrapolation distance reaches the characteristic
smoothness length scale of the function. The algorithm is asymptotically
minimax, in the sense that there is essentially no better algorithm yielding
meaningfully lower error over the same smoothness class. When viewed in the
dual domain, the above problem corresponds to (stable) simultaneous
de-convolution and super-resolution for objects of small space/time extent. Our
results then show that the amount of achievable super-resolution is inversely
proportional to the object size, and therefore can be significant for small
objects
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