5,544 research outputs found
Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data
The aim of this work is to estimate a quadratic functional of a unknown
Wigner function from noisy tomographic data. The Wigner function can be seen as
the representation of the quantum state of a light beam. The estimation of a
quadratic functional is done from result of quantum homodyne measurement
performed on identically prepared quantum systems. We start by constructing an
estimator of a quadratic functional of the Wigner function. We show that the
proposed estimator is optimal or nearly optimal in a minimax sense over a class
of infinitely differentiable functions. Parametric rates are also reached for
some values of the smoothness parameters and the asymptotic normality is given.
Then, we construct an adaptive estimator that does not depend on the smoothness
parameters and prove it is minimax over some set-ups
Nonquadratic estimators of a quadratic functional
Estimation of a quadratic functional over parameter spaces that are not
quadratically convex is considered. It is shown, in contrast to the theory for
quadratically convex parameter spaces, that optimal quadratic rules are often
rate suboptimal. In such cases minimax rate optimal procedures are constructed
based on local thresholding. These nonquadratic procedures are sometimes fully
efficient even when optimal quadratic rules have slow rates of convergence.
Moreover, it is shown that when estimating a quadratic functional nonquadratic
procedures may exhibit different elbow phenomena than quadratic procedures.Comment: Published at http://dx.doi.org/10.1214/009053605000000147 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Minimax estimation of linear and quadratic functionals on sparsity classes
For the Gaussian sequence model, we obtain non-asymptotic minimax rates of
estimation of the linear, quadratic and the L2-norm functionals on classes of
sparse vectors and construct optimal estimators that attain these rates. The
main object of interest is the class s-sparse vectors for which we also provide
completely adaptive estimators (independent of s and of the noise variance)
having only logarithmically slower rates than the minimax ones. Furthermore, we
obtain the minimax rates on the Lq-balls where 0 < q < 2. This analysis shows
that there are, in general, three zones in the rates of convergence that we
call the sparse zone, the dense zone and the degenerate zone, while a fourth
zone appears for estimation of the quadratic functional. We show that, as
opposed to estimation of the vector, the correct logarithmic terms in the
optimal rates for the sparse zone scale as log(d/s^2) and not as log(d/s). For
the sparse class, the rates of estimation of the linear functional and of the
L2-norm have a simple elbow at s = sqrt(d) (boundary between the sparse and the
dense zones) and exhibit similar performances, whereas the estimation of the
quadratic functional reveals more complex effects and is not possible only on
the basis of sparsity described by the sparsity condition on the vector.
Finally, we apply our results on estimation of the L2-norm to the problem of
testing against sparse alternatives. In particular, we obtain a non-asymptotic
analog of the Ingster-Donoho-Jin theory revealing some effects that were not
captured by the previous asymptotic analysis.Comment: 32 page
Optimal adaptive estimation of a quadratic functional
Adaptive estimation of a quadratic functional over both Besov and balls
is considered. A collection of nonquadratic estimators are developed which have
useful bias and variance properties over individual Besov and balls. An
adaptive procedure is then constructed based on penalized maximization over
this collection of nonquadratic estimators. This procedure is shown to be
optimally rate adaptive over the entire range of Besov and balls in the
sense that it attains certain constrained risk bounds.Comment: Published at http://dx.doi.org/10.1214/009053606000000849 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems
We consider the estimation of quadratic functionals in a Gaussian sequence
model where the eigenvalues are supposed to be unknown and accessible through
noisy observations only. Imposing smoothness assumptions both on the signal and
the sequence of eigenvalues, we develop a minimax theory for this problem. We
propose a truncated series estimator and show that it attains the optimal rate
of convergence if the truncation parameter is chosen appropriately.
Consequences for testing problems in inverse problems are equally discussed: in
particular, the minimax rates of testing for signal detection and
goodness-of-fit testing are derived.Comment: Corrected some typo
Quadratic functional estimation in inverse problems
We consider in this paper a Gaussian sequence model of observations ,
having mean (or signal) and variance which is
growing polynomially like , . This model describes a large
panel of inverse problems. We estimate the quadratic functional of the unknown
signal when the signal belongs to ellipsoids of both
finite smoothness functions (polynomial weights , ) and
infinite smoothness (exponential weights , , ). We propose a Pinsker type projection estimator in each case and study its
quadratic risk. When the signal is sufficiently smoother than the difficulty of
the inverse problem ( or in the case of exponential
weights), we obtain the parametric rate and the efficiency constant associated
to it. Moreover, we give upper bounds of the second order term in the risk and
conjecture that they are asymptotically sharp minimax. When the signal is
finitely smooth with , we compute non parametric upper
bounds of the risk of and we presume also that the constant is asymptotically
sharp
Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression
We consider the problem of testing a particular type of composite null
hypothesis under a nonparametric multivariate regression model. For a given
quadratic functional , the null hypothesis states that the regression
function satisfies the constraint , while the alternative
corresponds to the functions for which is bounded away from zero. On the
one hand, we provide minimax rates of testing and the exact separation
constants, along with a sharp-optimal testing procedure, for diagonal and
nonnegative quadratic functionals. We consider smoothness classes of
ellipsoidal form and check that our conditions are fulfilled in the particular
case of ellipsoids corresponding to anisotropic Sobolev classes. In this case,
we present a closed form of the minimax rate and the separation constant. On
the other hand, minimax rates for quadratic functionals which are neither
positive nor negative makes appear two different regimes: "regular" and
"irregular". In the "regular" case, the minimax rate is equal to
while in the "irregular" case, the rate depends on the smoothness class and is
slower than in the "regular" case. We apply this to the issue of testing the
equality of norms of two functions observed in noisy environments
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